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Interactive Notebook Foldables, Organizers and Templates

Interactive Notebook Foldables, Organizers and Templates

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  • 1. Contributing AuthorDinah ZikeGeometryConcepts and ApplicationsConsultantDouglas Fisher, PhDDirector of Professional DevelopmentSan Diego State UniversitySan Diego, CA
  • 2. Copyright © by The McGraw-Hill Companies, Inc. All rights reserved. Printed in theUnited States of America. Except as permitted under the United States CopyrightAct, no part of this book may be reproduced in any form, electronic or mechanical,including photocopy, recording, or any information storage or retrieval system,without prior written permission of the publisher.Send all inquiries to:The McGraw-Hill Companies8787 Orion PlaceColumbus, OH 43240-4027ISBN: 0-07-869020-X Geometry: Concepts and Applications (Virginia Student Edition)Noteables™: Interactive Study Notebook with Foldables™1 2 3 4 5 6 7 8 9 10 024 09 08 07 06 05 04
  • 3. Geometry: Concepts and Applications iiiFoldables. . . . . . . . . . . . . . . . . . . . . . . . . . 1Vocabulary Builder . . . . . . . . . . . . . . . . . . 21-1 Patterns and Inductive Reasoning. . 41-2 Points, Lines, and Planes . . . . . . . . . 71-3 Postulates . . . . . . . . . . . . . . . . . . . . 91-4 Conditional Statements . . . . . . . . . 111-5 Tools of the Trade . . . . . . . . . . . . . 131-6 A Plan for Problem Solving . . . . . . 16Study Guide . . . . . . . . . . . . . . . . . . . . . . 19Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 23Vocabulary Builder . . . . . . . . . . . . . . . . . 242-1 Real Numbers and Number Lines . 262-2 Segments and Properties ofReal Numbers . . . . . . . . . . . . . . . . 282-3 Congruent Segments . . . . . . . . . . . 312-4 The Coordinate Plane . . . . . . . . . . 342-5 Midpoints . . . . . . . . . . . . . . . . . . . 37Study Guide . . . . . . . . . . . . . . . . . . . . . . 40Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 43Vocabulary Builder . . . . . . . . . . . . . . . . . 443-1 Angles . . . . . . . . . . . . . . . . . . . . . . 463-2 Angle Measure . . . . . . . . . . . . . . . 483-3 The Angle Addition Postulate . . . . 513-4 Adjacent Angles & Linear Pairs . . . 543-5 Comp. and Supp. Angles. . . . . . . . . 563-6 Congruent Angles. . . . . . . . . . . . . . 583-7 Perpendicular Lines. . . . . . . . . . . . . 60Study Guide . . . . . . . . . . . . . . . . . . . . . . 62Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 65Vocabulary Builder . . . . . . . . . . . . . . . . . 664-1 Parallel Lines and Planes . . . . . . . . . 684-2 Parallel Lines and Transversals . . . . . 704-3 Transversals and Corres. Angles. . . . 734-4 Proving Lines Parallel . . . . . . . . . . . . 764-5 Slope. . . . . . . . . . . . . . . . . . . . . . . . . 784-6 Equations of Lines . . . . . . . . . . . . . 80Study Guide . . . . . . . . . . . . . . . . . . . . . . 83Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 87Vocabulary Builder . . . . . . . . . . . . . . . . . 885-1 Classifying Triangles. . . . . . . . . . . . 905-2 Angles of a Triangle . . . . . . . . . . . 935-3 Geometry in Motion . . . . . . . . . . . 955-4 Congruent Triangles . . . . . . . . . . . 985-5 SSS and SAS . . . . . . . . . . . . . . . . . 1005-6 ASA and AAS . . . . . . . . . . . . . . . . 102Study Guide . . . . . . . . . . . . . . . . . . . . . 104Foldables. . . . . . . . . . . . . . . . . . . . . . . . 107Vocabulary Builder . . . . . . . . . . . . . . . . 1086-1 Medians . . . . . . . . . . . . . . . . . . . . 1106-2 Altitudes and Perp. Bisectors . . . . 1126-3 Angle Bisectors of Triangles. . . . . 1156-4 Isosceles Triangles . . . . . . . . . . . . 1176-5 Right Triangles. . . . . . . . . . . . . . . 1196-6 The Pythagorean Theorem . . . . . 1216-7 Distance on a Coordinate Plane . . 123Study Guide . . . . . . . . . . . . . . . . . . . . . 125Foldables. . . . . . . . . . . . . . . . . . . . . . . . 129Vocabulary Builder . . . . . . . . . . . . . . . . 1307-1 Segments, Angles, Inequalities . . . 1317-2 Exterior Angle Theorem . . . . . . . 1347-3 Inequalities Within a Triangle . . . 1377-4 Triangle Inequality Theorem . . . . 139Study Guide . . . . . . . . . . . . . . . . . . . . . 142Foldables. . . . . . . . . . . . . . . . . . . . . . . . 145Vocabulary Builder . . . . . . . . . . . . . . . . 1468-1 Quadrilaterals . . . . . . . . . . . . . . . 1488-2 Parallelograms . . . . . . . . . . . . . . . 1508-3 Tests for Parallelograms. . . . . . . . 1528-4 Rectangles, Rhombi, and Squares . 1548-5 Trapezoids . . . . . . . . . . . . . . . . . . 156Study Guide . . . . . . . . . . . . . . . . . . . . . 159Contents
  • 4. iv Geometry: Concepts and ApplicationsFoldables. . . . . . . . . . . . . . . . . . . . . . . . 163Vocabulary Builder . . . . . . . . . . . . . . . . 1649-1 Using Ratios and Proportions . . . 1669-2 Similar Polygons. . . . . . . . . . . . . . 1699-3 Similar Triangles. . . . . . . . . . . . . . 1729-4 Proportional Parts and Triangles . 1759-5 Triangles and Parallel Lines . . . . . 1779-6 Proportional Parts andParallel Lines . . . . . . . . . . . . . . . . 1799-7 Perimeters and Similarity. . . . . . . 181Study Guide . . . . . . . . . . . . . . . . . . . . . 183Foldables. . . . . . . . . . . . . . . . . . . . . . . . 187Vocabulary Builder . . . . . . . . . . . . . . . . 18810-1 Naming Polgons. . . . . . . . . . . . . . 19010-2 Diagonals and Angle Measure . . 19210-3 Areas of Polygons. . . . . . . . . . . . . 19410-4 Areas of Triangles and Trapezoids. 19610-5 Areas of Regular Polygons. . . . . . 19810-6 Symmetry. . . . . . . . . . . . . . . . . . . 20010-7 Tessellations. . . . . . . . . . . . . . . . . 202Study Guide . . . . . . . . . . . . . . . . . . . . . 203Foldables. . . . . . . . . . . . . . . . . . . . . . . . 207Vocabulary Builder . . . . . . . . . . . . . . . . 20811-1 Parts of a Circle . . . . . . . . . . . . . . 21011-2 Arcs and Central Angles . . . . . . . 21211-3 Arcs and Chords . . . . . . . . . . . . . . 21411-4 Inscribed Polygons . . . . . . . . . . . . 21611-5 Circumference of a Circle . . . . . . 21811-6 Area of a Circle . . . . . . . . . . . . . . 220Study Guide . . . . . . . . . . . . . . . . . . . . . 223Foldables. . . . . . . . . . . . . . . . . . . . . . . . 227Vocabulary Builder . . . . . . . . . . . . . . . . 22812-1 Solid Figures . . . . . . . . . . . . . . . . 23012-2 SA of Prisms and Cylinders . . . . . 23312-3 Volumes of Prisms and Cylinders. . 23612-4 SA of Pyramids and Cones . . . . . . 23812-5 Volumes of Pyramids and Cones . 24112-6 Spheres . . . . . . . . . . . . . . . . . . . . 24312-7 Similarity of Solid Figures . . . . . . 245Study Guide . . . . . . . . . . . . . . . . . . . . . 247Foldables. . . . . . . . . . . . . . . . . . . . . . . . 251Vocabulary Builder . . . . . . . . . . . . . . . . 25213-1 Simplifying Square Roots. . . . . . . 25413-2 45°-45°-90° Triangles . . . . . . . . . . 25713-3 30°-60°-90° Triangles . . . . . . . . . . 25913-4 Tangent Ratio. . . . . . . . . . . . . . . . 26113-5 Sine and Cosine Ratios. . . . . . . . . 264Study Guide . . . . . . . . . . . . . . . . . . . . . 266Foldables. . . . . . . . . . . . . . . . . . . . . . . . 269Vocabulary Builder . . . . . . . . . . . . . . . . 27014-1 Inscribed Angles. . . . . . . . . . . . . . 27214-2 Tangents to a Circle . . . . . . . . . . . 27514-3 Secant Angles . . . . . . . . . . . . . . . 27714-4 Secant-Tangent Angles . . . . . . . . 27914-5 Segment Measures. . . . . . . . . . . . 28114-6 Equations of Circles . . . . . . . . . . . 283Study Guide . . . . . . . . . . . . . . . . . . . . . 285Foldables. . . . . . . . . . . . . . . . . . . . . . . . 289Vocabulary Builder . . . . . . . . . . . . . . . . 29015-1 Logic and Truth Tables. . . . . . . . . 29215-2 Deductive Reasoning . . . . . . . . . . 29515-3 Paragraph Proofs . . . . . . . . . . . . . 29715-4 Preparing for Two-Column Proofs. 29915-5 Two-Column Proofs . . . . . . . . . . . 30215-6 Coordinate Proofs . . . . . . . . . . . . 305Study Guide . . . . . . . . . . . . . . . . . . . . . 308Foldables. . . . . . . . . . . . . . . . . . . . . . . . 313Vocabulary Builder . . . . . . . . . . . . . . . . 31416-1 Graph Systems of Equations . . . . 31516-2 Solving Systems of Equationsby Using Algebra . . . . . . . . . . . . . 31816-3 Translations . . . . . . . . . . . . . . . . . 32116-4 Reflections . . . . . . . . . . . . . . . . . . . 32216-5 Rotations . . . . . . . . . . . . . . . . . . . 32416-6 Dilations. . . . . . . . . . . . . . . . . . . . 326Study Guide . . . . . . . . . . . . . . . . . . . . . 328
  • 5. ©Glencoe/McGraw-HillGeometry: Concepts and Applications vOrganizing Your FoldablesMake this Foldable to help you organize andstore your chapter Foldables. Begin with onesheet of 11" ϫ 17" paper.FoldFold the paper in half lengthwise. Then unfold.Fold and GlueFold the paper in half widthwise and glue all of the edges.Glue and LabelGlue the left, right, and bottom edges of the Foldableto the inside back cover of your Noteables notebook.Reading and Taking Notes As you read and study each chapter, recordnotes in your chapter Foldable. Then store your chapter Foldables insidethis Foldable organizer.Foldables Organizer
  • 6. ©Glencoe/McGraw-Hillvi Geometry: Concepts and ApplicationsThis note-taking guide is designed to help you succeed in Geometry: Conceptsand Applications. Each chapter includes:Surface Area and Volume©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R12Geometry: Concepts and Applications 227NOTE-TAKING TIP: When taking notes, explaineach new idea or concept in words and give oneor more examples.Chapter12FoldFold the paper inthirds lengthwise.OpenOpen and fold a 2"tab along the shortside. Then fold therest in fifths.DrawDraw lines alongfolds and label asshown.Begin with a plain piece of 11" ϫ 17" paper.SurfaceAreaVolumeCh.12PrismsCylindersPyramidsConesSpheresVocabulary TermFoundDefinitionDescription oron Page Example©Glencoe/McGraw-Hill228 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 12. Asyou complete the study notes for the chapter, you will see Build Your Vocabularyreminders to complete each term’s definition or description on these pages.Remember to add the textbook page number in the second column for referencewhen you study.C H A P T E R12BUILD YOUR VOCABULARYaxiscomposite solidconecubecylinder[SIL-in-dur]edgefacelateral area[LAT-er-ul]lateral edgelateral facenetoblique cone[oh-BLEEK]oblique cylinderoblique prismThe Chapter Openercontains instructions andillustrations on how to makea Foldable that will help youto organize your notes.A Note-Taking Tipprovides a helpfulhint you can usewhen taking notes.The Build Your Vocabularytable allows you to writedefinitions and examplesof important vocabularyterms together in oneconvenient place.Within each chapter,Build Your Vocabularyboxes will remind youto fill in this table.
  • 7. Geometry: Concepts and Applications viiFind the volume of the rectangularpyramid.B ϭ ഞwϭ (10)(4) orV ϭ ᎏ13ᎏBh Theorem 12-11ϭ ᎏ13ᎏ(40)(12) Substitutionϭ cm3Find the volume of the cone to the nearest hundredth.Find the height hh2ϩ 212ϭ 352h2ϩ 441 ϭ 1225h2ϭ 784͙h2ෆ ϭ ͙784ෆh ϭV ϭ ᎏ13ᎏ␲r2h Theorem 12-11ϭ ᎏ13ᎏ␲(21)2(28) SubstitutionϷ in342 in.35 in.©Glencoe/McGraw-HillVolumes of Pyramids and Cones12–5Geometry: Concepts and Applications 241Use the boxes forVolume of Pyramids andCones. Sketch and labela pyramid and a cone.Then write the formulafor finding the volumesof a pyramid and acone.SurfaceAreaVolumeCh.12PrismsCylindersPyramidsConesSpheresORGANIZE ITTheorem 12-11 Volume of a PyramidIf a pyramid has a volume of V cubic units and a height ofh units and the area of the base is B square units, thenV ϭ ᎏ13ᎏBh.12 cm10 cm4 cmTheorem 12-12 Volume of a ConeIf a cone has a volume of V cubic units, a radius of r units,and a height of h units, then V ϭ ᎏ13ᎏ␲r2h.• Find the volumes ofpyramids and cones.WHAT YOU’LL LEARNSOL G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve practical problems. Calculators will be used to finddecimal approximations for results.BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillBUILD YOURVOCABULARYUse your Chapter 12 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 12, go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder(pages 228–229) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R12Geometry: Concepts and Applications 247Complete each sentence.1. Two faces of a polyhedron intersect at a(n) .2. A triangular pyramid is called a .3. A is a figure that encloses a part of space.4. Three faces of a polyhedron intersect at a point called a(n).Find the lateral area and surface area ofeach solid to the nearest hundredth.5. a regular pentagonal prism with apothema ϭ 4, side length s ϭ 6, and height h ϭ 12a. L ϭb. S ϭ6. a cylinder with radius r ϭ 42 and height h ϭ 10a. L Ϸ b. S Ϸ12-1Solid Figures12-2Surface Areas of Prisms and Cylinders4612STUDY GUIDE©Glencoe/McGraw-Hill11–3Geometry: Concepts and Applications 215Page(s):Exercises:HOMEWORKASSIGNMENTIn circle W, find XV if UW៮៮ Ќ XV៮៮, VW ϭ 35, and WY ϭ 21.ЄVYW is a angle. Definition of perpendicular᭝VYW is a right triangle. Definition of right triangle(WY)2ϩ (YV)2ϭ ΂ ΃2Pythagorean Theorem212ϩ (YV)2ϭ 352Replace WY and VW.ϩ (YV)2ϭ 1225(YV)2ϭ Subtract.͙(YV)2ෆ ϭ Take the square root ofeach side.YV ϭ ϭ XY Theorem 11-5XV ϭ YV ϩ XY Segment additionXV ϭ ϩ 28 SubstitutionXV ϭIn circle G, if CෆGෆ Ќ AෆEෆ, EG ϭ 20, CG ϭ 12, find AE.Your TurnWUX Y2135VA CGE1220Lessons cover the content ofthe lessons in your textbook.As your teacher discusses eachexample, follow along andcomplete the fill-in boxes.Take notes as appropriate.Foldables featurereminds you to takenotes in your Foldable.Your Turn Exercises allowyou to solve similarexercises on your own.Bringing It All TogetherStudy Guide reviews themain ideas and keyconcepts from each lesson.©Glencoe/McGraw-HillEach lesson iscorrelated to theVirginia SOLs.Examples parallel theexamples in your textbook.
  • 8. viii Geometry: Concepts and Applications©Glencoe/McGraw-HillYour notes are a reminder of what you learned in class. Taking good notescan help you succeed in mathematics. The following tips will help you takebetter classroom notes.• Before class, ask what your teacher will be discussing in class. Reviewmentally what you already know about the concept.• Be an active listener. Focus on what your teacher is saying. Listen forimportant concepts. Pay attention to words, examples, and/or diagramsyour teacher emphasizes.• Write your notes as clear and concise as possible. The following symbolsand abbreviations may be helpful in your note-taking.• Use a symbol such as a star (5) or an asterisk (*) to emphasis importantconcepts. Place a question mark (?) next to anything that you do notunderstand.• Ask questions and participate in class discussion.• Draw and label pictures or diagrams to help clarify a concept.• When working out an example, write what you are doing to solve theproblem next to each step. Be sure to use your own words.• Review your notes as soon as possible after class. During this time, organizeand summarize new concepts and clarify misunderstandings.Note-Taking Don’ts• Don’t write every word. Concentrate on the main ideas and concepts.• Don’t use someone else’s notes as they may not make sense.• Don’t doodle. It distracts you from listening actively.• Don’t lose focus or you will become lost in your note-taking.NOTE-TAKING TIPSWord or PhraseSymbol orAbbreviationWord or PhraseSymbol orAbbreviationfor example e.g. not equalsuch as i.e. approximately ഠwith w/ therefore Іwithout w/o versus vsand ϩ angle Є
  • 9. Reasoning in Geometry©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R1Geometry: Concepts and Applications 1NOTE-TAKING TIP: When you are taking notes, besure to be an active listener by focusing on whatyour teacher is saying.FoldFold lengthwise in fourths.DrawDraw lines along thefolds and label eachcolumn sequences,patterns, conjectures,and conclusions.Begin with a sheet of 8ᎏ12ᎏ" ϫ 11" paper.sequences patterns conjectures conclusionsChapter1
  • 10. ©Glencoe/McGraw-Hill2 Geometry: Concepts and ApplicationsC H A P T E R1BUILD YOUR VOCABULARYThis is an alphabetical list of new vocabulary terms you will learn in Chapter 1.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.Vocabulary TermFoundDefinitionDescription oron Page Examplecollinear[co-LIN-ee-ur]compassconclusionconditional statementconjecture[con-JEK-shoor]constructioncontrapositive[con-tra-PAS-i-tiv]conversecoplanar[co-PLAY-nur]counterexampleendpointformula
  • 11. Vocabulary TermFoundDefinitionDescription oron Page Example©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY1Geometry: Concepts and Applications 3hypothesis[hi-PA-the-sis]if-then statementinductive reasoning[in-DUK-tiv]inverse[in-VURS]lineline segmentmidpointnoncollinearnoncoplanarplanepointpostulate[PAS-chew-let]ray
  • 12. Patterns and Inductive Reasoning©Glencoe/McGraw-Hill4 Geometry: Concepts and Applications1–1Find the next three terms of the sequence 11.2, 9.2,7.2, . . . .Study the pattern in the sequence.11.2 9.2 7.2Each term is less than the term before it. Assumethis pattern continues.11.2 9.2 7.2The next three items are .Find the next three terms of eachsequence.a. 3.7, 5.7, 7.7, . . . b. 1, 3, 9, . . .Your Turn• Identify patterns anduse inductive reasoning.WHAT YOU’LL LEARNWhen you make conclusions based on a ofexamples or past events, you are using inductive reasoning.BUILD YOUR VOCABULARY (page 3)Write a sequence and ageometric pattern inyour Foldable. Explainhow to find the next3 terms of each.sequences patterns conjectures conclusionsORGANIZE IT
  • 13. ©Glencoe/McGraw-Hill1–1Geometry: Concepts and Applications 5Find the next three terms of the sequence 101, 102, 105,110, 117, . . . .101 102 105 110 117Notice the pattern. To find the next three terms in thesequence, add , , and .101 102 105 110 117 126 137 150The next three terms are .Find the next four terms in the sequence 51,53, 57, 63, 71, 81, 93, . . .Draw the next figure in the pattern.There are two patterns to study.• The first pattern is size of thesquares. The next squareshould be the area of theprevious square.• The second pattern is shadedor unshaded. The next squareshould be .Your Turn
  • 14. ©Glencoe/McGraw-Hill1–16 Geometry: Concepts and ApplicationsDraw the next figure in the pattern.Minowa studied the data below and made the followingconjecture. Find a counterexample for her conjecture.Multiplying a number byϪ1 produces a productthat is less than Ϫ1.The product of Ϫ2 and Ϫ1 is 2 but 2 Ϫ1. So, theconjecture is .Find a counterexample for this statement:Division of a positive number by another positive numberproduces a quotient less than the dividend.Your TurnYour TurnA conjecture is a based on inductivereasoning.An example that shows that a conjecture is notis a counterexample.BUILD YOUR VOCABULARYNumber ϫ(Ϫ1) Product5(Ϫ1) Ϫ515(Ϫ1) Ϫ15100(Ϫ1) Ϫ100300(Ϫ1) Ϫ300(page 2)Page(s):Exercises:HOMEWORKASSIGNMENT. . .
  • 15. ©Glencoe/McGraw-Hill1–2Geometry: Concepts and Applications 7Points, Lines, and PlanesName two points on the line.Two points are point andpoint .Give three names for the line.Any two points on the line or the script letter can be used toname it. Three names are .Refer to the figure shown.a. Name two points on the line.b. Give three names for the line.Your TurnKL m©Glencoe/McGraw-Hill• Identify and drawmodels of points,lines, and planes,and determine theircharacteristics.WHAT YOU’LL LEARN A point is the basic unit of geometry.A series of points that extends without end indirections is a line.Points that lie on the same are said to becollinear.Points that do not lie on the same line are said to benoncollinear.A ray is part of a line that has a definite starting pointand extends without end in direction.A line segment has a definite beginning and .BUILD YOUR VOCABULARY (pages 2–3)AdB
  • 16. ©Glencoe/McGraw-Hill1–28 Geometry: Concepts and ApplicationsName three points that are collinearand three points that are noncollinear.Points M, P, and Q, are .Points N, P, and Q are.Name three segments and one ray.Three of the segments are.One ray is ray .Refer to the figure.a. Name three collinear points andthree noncollinear points.b. Name three segments and one ray.Your TurnD E FGBCFEADGN MPQPage(s):Exercises:HOMEWORKASSIGNMENTA plane is a surface that extends withoutend in all directions.Points that lie on the same are coplanar.Points that do not lie on the same arenoncoplanar.BUILD YOUR VOCABULARY (pages 2–3)REMEMBER ITThe order of theletters that identify aline can be switched butthe order of the lettersthat identify a raycannot.
  • 17. ©Glencoe/McGraw-Hill1–3Geometry: Concepts and Applications 9PostulatesIn the figure, points K, L, and M arenoncollinear.Name all of the different lines that canbe drawn through these points.There is only one line through each pair of points.Therefore, the lines that contain points K, L, and M,taken two at a time, are .Name the intersection of KL៭៮៬ and KM៭៮៬.The intersection of KL៭៮៬ and KM៭៮៬ is .Refer to the figure.a. Name three different lines.b. Name the intersection of AC៭៮៬ and BH៭៮៬.Your Turn• Identify and use basicpostulates about points,lines, and planes.WHAT YOU’LL LEARNKLMFIEHAGDLCBJ KPostulate 1–1 Two points determine a unique line.Postulate 1–2 If two distinct lines intersect, then theirintersection is a point.Postulate 1–3 Three noncollinear points determine aunique plane.Postulates are in geometry that areaccepted as .BUILD YOUR VOCABULARY (page 3)
  • 18. ©Glencoe/McGraw-Hill1–310 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTName all of the planes that are represented in the prism.There are eight points, A, B, C, D, E, F, G, and H.There is only plane that contains three noncollinearpoints. The different planes are planes.Name four different planes in the figure.Name the intersection of planeABC and plane DEF.The intersection is .Name theintersection of plane ABD andplane DJK.Your TurnFIEHAGDLCBJ KYour TurnFHA BGECDEPostulate 1–4 If two distinct planes intersect, then theirintersection is a line.REMEMBER ITThree noncollinearpoints determine aunique plane.ABCGFEDHFIEHAGDLCBJ K
  • 19. ©Glencoe/McGraw-HillConditional Statements and Their Converses1–4Geometry: Concepts and Applications 11• Write statements inif-then form and writethe converses of thestatements.WHAT YOU’LL LEARN If-then statements join two statements based on acondition.If-then statements are also known as conditionalstatements.In a conditional statement the part following if is thehypothesis. The part following then is the conclusion.BUILD YOUR VOCABULARY (pages 2–3)Identify the hypothesis and conclusion in thisstatement.If it is raining, then we will read a book.Hypothesis:Conclusion:Write two other forms of this statement.If two lines are parallel, then they never intersect.All never intersect.Lines never if they are .a. Identify the hypothesis and conclusion in this statement.If you ski, then you like snow.b. Write two other forms of this statement. If a figure is arectangle, then it has four angles.Your TurnHow can you show thata conjecture is false?(Lesson 1-1)REVIEW ITSOL G.1.a The student will construct and judge the validity of a logical argumentconsisting of a set of premises and a conclusion. This will include identifying theconverse, inverse, and contrapositive of a conditional statement.
  • 20. ©Glencoe/McGraw-Hill1–412 Geometry: Concepts and ApplicationsThe converse of a conditional statement is formed byexchanging the .BUILD YOUR VOCABULARY (page 2)Write the converse of this statement.If today is Saturday, then there is no school.If there is , then .Write the converse of this statement.If it is Ϫ30° F, then it is cold.Write the statement in if-then form. Then write theconverse of the statement.Every member of the jazz band must attend therehearsal on Saturday.If-then form: If a is a member of the jazz band,then he or she must attend.Converse: If a studenton Saturday, then he or she is amember.Write the statement in if-then form. Then writethe converse of the statement. People who live in glass housesshould not throw stones.Your TurnYour TurnPage(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITThe converse of atrue statement is notnecessarily true.
  • 21. ©Glencoe/McGraw-HillGeometry: Concepts and Applications 13Tools of the Trade• Use geometry tools.WHAT YOU’LL LEARNA straightedge is an object used to draw a line.A compass is commonly used for drawing arcs and.In geometry, figures drawn using only aand a are constructions.The midpoint is the in the of aline segment.BUILD YOUR VOCABULARYFind two lines or segments in a classroom that appearto be parallel. Use a ruler to determine whether theyare parallel.The opposite sides of a textbook represent two segments thatappear to be parallel.• Choose two points on one side of the textbook.• Place the 0 mark of the ruler on each point. Make sure theruler is perpendicular to the side at each chosen point.• Measure the distance to the second side. If the distances are, then the sides are .Find another pair of lines or segments in aclassroom that appear to be parallel. Use a ruler or a yardstickto determine if they are parallel.Your Turn(pages 2–3)1–5SOL G.2.a The student will use pictorial representations, including computer software, con-structions, and coordinate methods, to solve problems involving symmetry and transformation.This will include investigating and using formulas for finding distance, midpoint, and slope.
  • 22. ©Glencoe/McGraw-Hill14 Geometry: Concepts and ApplicationsOn the figure shown, mark a point C on line ᐉ that youjudge will create BෆCෆ that is the same length as AෆBෆ. Thenmeasure to determine how accurate your guess was.To draw an exact recreation of the length, place the point of acompass on point B. Place the point of the pencil on point. Then draw a small arc on line ᐉ without changing thesetting of the compass. This duplicates the measure of .Use a compass and a straightedge to construct asix-pointed star.Use the compass to draw a circle. Then using the samecompass setting, put the compass point on the circle and drawa small arc on the circle.Move the compass point to the arc and, without changing thecompass setting, draw another arc along the circle. Continueuntil there are six arcs.Draw two triangles by connecting alternating marks, resultingin a six-pointed star.1–5AB ᐉREMEMBER ITAn arc is part of acircle.
  • 23. ©Glencoe/McGraw-HillGeometry: Concepts and Applications 15a. On the figure given, mark point Z on line m that you judgewill create WෆZෆ that is the same length as XෆYෆ. Thenmeasure to determine the accuracy of your guess.b. Use a compass and a straightedge to construct a trianglewith sides of equal length.XmW YZYour Turn1–5Page(s):Exercises:HOMEWORKASSIGNMENT
  • 24. ©Glencoe/McGraw-Hill16 Geometry: Concepts and ApplicationsA Plan for Problem Solving• Use a four-step plan tosolve problems thatinvolve the perimetersand areas of rectanglesand parallelograms.WHAT YOU’LL LEARNA formula is an that shows how certainquantities are related.BUILD YOUR VOCABULARY (page 2)Perimeter of a RectangleThe perimeter P of arectangle is the sum ofthe measures of its sides.It can also be expressedas two times the lengthᐉ plus two times thewidth w.Area of a RectangleThe area A of arectangle is the productof the length ᐉ and thewidth w.KEY CONCEPTSa. Find the perimeter of a rectangle with length12 centimeters and width 3 centimeters.P ϭ 2ᐉ ϩ 2wP ϭ 2 ϩ 2P ϭ ϩ or centimetersb. Find the perimeter of a square with side 10 feet long.P ϭ 2ᐉ ϩ 2wP ϭ 2(10) ϩ 2(10)P ϭ ϩ or feeta. Find the area of a rectangle with length12 kilometers and width 3 kilometers.A ϭ ᐉwA ϭ΂ ΃΂ ΃A ϭ square kilometersb. Find the area of a square with sides 10 yards long.A ϭ ᐉwA ϭ΂ ΃΂ ΃A ϭ square yards1–6
  • 25. What is the differencebetween perimeter andarea?WRITE IT©Glencoe/McGraw-Hill1–6Geometry: Concepts and Applications 17a. Find the perimeter of a rectangle with length 11 meters andwidth 4 meters.b. Find the perimeter of a square with sides 7 centimeters long.c. Find the area of a rectangle with length 14 inches andwidth 4 inches.d. Find the area of a square with sides 11 feet long.Find the area of a parallelogram with a height of4 meters and a base of 5.5 meters.A ϭ bhA ϭ ΂ ΃΂ ΃A ϭ square metersFind the area of a parallelogram with a heightof 6.4 inches and a base length of 10 inches.Your TurnYour TurnArea of a ParallelogramThe area of aparallelogram is theproduct of the base band the height h.KEY CONCEPT
  • 26. 18 Geometry: Concepts and ApplicationsA door is 3-feet wide and 6.5-feet tall. Chad wants topaint the front and back of the door. A one-pint can ofpaint will cover about 15 ft2. Will two one-pint cans ofpaint be enough?EXPLORE You know the dimensions of the door and thatone-pint can of paint covers about .PLAN Use the formula for the area of ato find the total area of the two sides of the doorto be covered with paint.SOLVE Area of both sides of the doorA ϭ 2ᐉwA ϭ 2΂ ΃΂ ΃ϭOne pint covers 15 ft2. Two one-pint cans cover2(15) or ft2. So, two one-pint cans willbe enough.EXAMINE Since the area of one side of the door is (3)(6.5) or19.5 ft2the answer is reasonable.Chad will need one-pint cans of paint.A building contractor needs to build arectangular deck with an area of 484 ft2. The side lengthsmust be whole numbers. The perimeter must be less than260 ft. What are the possible dimensions for the deck?Your Turn©Glencoe/McGraw-Hill1–6Page(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITAbbreviations forunits of area haveexponent 2.Square foot ϭ ft2Square meter ϭ m2Problem-Solving Plan1. Explore the problem.2. Plan the solution.3. Solve the problem.4. Examine the solution.KEY CONCEPT
  • 27. Geometry: Concepts and Applications 19BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillFind the next three terms in the sequence.1. 1, 1, 2, 3, 5, . . . 2. Ϫ1, 2, Ϫ4, 8, Ϫ16, . . .3. Draw the next figure in the pattern.Use the figure to match the example to the correct term.4. collinear points5. segment6. plane7. rayG FAPBCDEBUILD YOURVOCABULARYUse your Chapter 1 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 1, go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder (pages 2–3)to help you solve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R1STUDY GUIDE1-1Patterns and Inductive Reasoning1-2Points, Lines, and Planesa. GFCb. PෆBෆc. AD៮៮៬d. PEᠭ៮៮៬e. GBE
  • 28. ©Glencoe/McGraw-Hill20 Geometry: Concepts and ApplicationsChapter BRINGING IT ALL TOGETHERComplete the sentence.8. A(n) is a statement in geometry that isaccepted as true without proof.Identify three planes in the figure shown.9. 10.11.12. Refer to the above figure. Where do planes ACF andDEF intersect?a. point F b. DF៭---៬ c. plane DEF d. point DUnderline the correct term that completes each sentence.13. The “if” part of the if-then statement is thehypothesis/conclusion.14. The “then” part of the if-then statement is thehypothesis/conclusion.15. Rewrite the statement in if-then form.Students who complete all assignments score higher on tests.16. Write the converse of the statement.If it is Saturday, then there is no school.FABCDE11-3Postulates1-4Conditional Statements and Their Converses
  • 29. ©Glencoe/McGraw-HillGeometry: Concepts and Applications 21Match the geometry tool to its function.17. compass18. straightedge19. protractor20. patty paper21. Indicate whether the statement is true or false.A conjecture is a special drawing that is created using only astraightedge and compass.Complete each sentence.22. The is the distance around the edges of a figure.23. The formula for the area of a rectangle is .24. is the formula to find the area of aparallelogram.25. Find the area of a rectangle with length 8 feet and width9 feet.26. A framer must frame a piece of art. The frame is 1ᎏ12ᎏ incheswide, and its outer edge measures 24 inches by 36 inches.What is the area of the piece of art displayed in the center ofthe frame?Chapter BRINGING IT ALL TOGETHER11-5Tools of the Trade1-6A Plan for Problem Solvinga. to plot pointsb. to draw arcs and circlesc. to measure anglesd. to draw lines in constructionse. to find the midpoint in constructions
  • 30. Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 1 Practice Test on page 45of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 1 Study Guide and Reviewon pages 42–44 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 1 Practice Test onpage 45 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 1 Foldable.• Then complete the Chapter 1 Study Guide and Review onpages 42–44 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 1 Practice Test onpage 45 of your textbook.©Glencoe/McGraw-Hill22 Geometry: Concepts and ApplicationsARE YOU READY FORTHE CHAPTER TEST?Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 1.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R1Checklist
  • 31. Segment Measure and Coordinate Graphing©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R2Geometry: Concepts and Applications 23NOTE-TAKING TIP: When taking notes, it ishelpful to record the main ideas as you listen toyour teacher, or read through a lesson.FoldFold lengthwise to theholes.CutCut along the top line andthen cut 10 tabsLabelLabel each tab with ahighlighted term from thechapter. Store the Foldablein a 3-ring binder.Begin with a sheet of notebook paper.Chapter2
  • 32. ©Glencoe/McGraw-Hill24 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 2.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R2BUILD YOUR VOCABULARYVocabulary TermFoundDefinitionDescription oron Page Exampleabsolute valuebetweennessbisectcongruent segments[con-GROO-unt]coordinate[co-OR-duh-net]coordinate planecoordinatesgreatest possible errormeasuremeasurements
  • 33. ©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY2Geometry: Concepts and Applications 25Vocabulary TermFoundDefinitionDescription oron Page Examplemidpointordered pairorigin[OR-a-jin]percent of errorprecision[pree-SI-zhun]quadrants[KWAH-druntz]theorem[THEE-uh-rem]unit of measurevectorx-axisx-coordinatey-axisy-coordinate
  • 34. Real Numbers and Number LinesFor each situation, write a real number with ten digitsto the right of the decimal point.a rational number between 6 and 8 with a 2-digitrepeating patternSample answer: 7.3232323232 . . .an irrational number greater than 5Sample answer: 5.4344334443 . . .For each situation, write a real numberwith ten digits to the right of the decimal point.a. a rational number between Ϫ4 and Ϫ1 with a 3-digitrepeating patternb. an irrational number less than Ϫ7Your Turn©Glencoe/McGraw-Hill26 Geometry: Concepts and Applications2–1• Find the distancebetween two points ona number line.WHAT YOU’LL LEARNOn the first tab of yourFoldable, write RationalNumbers and on thesecond tab, writeIrrational Numbers.Under each tab, describethe sets of rational andirrational numbers andgive several examplesof each.ORGANIZE ITPostulate 2-1 Number Line PostulateEach real number corresponds to exactly one point on anumber line. Each point on a number line corresponds toexactly one real number.Postulate 2-2 Distance PostulateFor any two points on a line and a given unit of measure,there is a unique positive real number called the measureof the distance between the points.Postulate 2-3 Ruler PostulateThe points on a line can be paired with the real numbers sothat the measure of the distance between correspondingpoints is the positive difference of the numbers.
  • 35. Use the number line below to find CE.The coordinate of C is , and the coordinate of E is .CE ϭ ͦϪ1 Ϫ ᎏ13ᎏͦ ϭ ͦϪ1ᎏ13ᎏͦϭ ͦϪ1ᎏ13ᎏͦ orErin traveled on I-85 from Durham, North Carolina, toCharlotte. The Durham entrance to I-85 that she used isat the 173-mile marker, and the Charlotte exit she usedis at the 39-mile marker. How far did Erin travel on I-85?ͦ173 Ϫ 39ͦ ϭ ͦ134ͦ ϭShe traveled miles on I-85.a. Refer to Example 3. Find AE.b. Rahmi’s drive starts at the 263-mile markerof I-35 and finishes at the 287-mile marker. How far didRahmi drive on I-35?Your Turn3210Ϫ1Ϫ2Ϫ3B C D EA F G©Glencoe/McGraw-Hill2–1Geometry: Concepts and Applications 27Page(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITXY represents themeasure of the distancebetween points X and Y.The number that corresponds to a point on a numberline is called the coordinate of the point.A point with coordinate is known as the origin.The absolute value of a number is the number of units anumber is from on the number line.BUILD YOUR VOCABULARY (pages 24–25)
  • 36. Segments and Properties of Real NumbersPoints K, L, and J are collinear. If KL ϭ 31, JL ϭ 16,and JK ϭ 47, determine which point is between theother two.Check to see which two measures add to equal the third.ϩ ϭKL ϩ JL ϭ JKTherefore, is between and .Points A, B, and C are collinear. If AB ϭ 54,BC ϭ 33, and AC ϭ 21, determine which point is between theother two.If FG ϭ 12 and FJ ϭ 47, find GJ.FG ϩ GJ ϭ FJ Definition of betweennessϩ GJ ϭ Substitution Property12 ϩ GJ ϭ 47 Substitution PropertyGJ ϭF JHGYour Turn©Glencoe/McGraw-Hill28 Geometry: Concepts and Applications2–2• Apply properties ofreal numbers to themeasure of segments.WHAT YOU’LL LEARN Point R is between points P and Q if and only if R, P and Qare and PR ϩ RQ ϭ PQ.BUILD YOUR VOCABULARY (page 24)On the third tab of yourFoldable write Measureand on the fourth tabwrite Unit of Measure.Under each tab, explainthe differences betweenthe terms and giveexamples of each.ORGANIZE IT
  • 37. If BE ϭ 17 and AE ϭ 25, find AB.Use a ruler to draw a segment 8 centimeters long. Thenfind the length of the segment in inches.Use a metric ruler to draw the segment. Mark a point and callit X. Then put the 0 point at point X and draw a line segmentextending to the 8 centimeter mark. Mark the endpoint Y.The length of XෆYෆ is centimeters.0 1 2 3 4 5 6 7 8 9 10centimeters (cm)X YA EC DBYour Turn©Glencoe/McGraw-Hill2–2Geometry: Concepts and Applications 29Measurements are composed of parts; a numbercalled the measure and the unit of measure.The precision of a measurement depends on theunit used to make the measurement.The greatest possible error is the smaller unit usedto make the measurement.The percent of error is the of thegreatest possible error with the measurement itself,multiplied by .BUILD YOUR VOCABULARY (pages 24–25)Properties of Equalityfor Real Numbers• Reflexive PropertyFor any number a,a ϭ a.• Symmetric PropertyFor any numbers a andb, if a ϭ b, then b ϭ a.• Transitive PropertyFor any numbers a, b,and c, if a ϭ b andb ϭ c, then a ϭ c.• Addition andSubtraction PropertiesFor any numbers a, b,and c, if a ϭ b, thena ϩ c ϭ b ϩ c, anda Ϫ c ϭ b Ϫ c.• Multiplication andDivision PropertiesFor any numbers a, b,and c, if a ϭ b, thena и c ϭ b и c, and ifc 0, then ᎏacᎏ ϭ ᎏbcᎏ.• Substitution PropertyFor any numbers a andb, if a ϭ b, then a maybe replaced by b in anyequation.KEY CONCEPTS
  • 38. Use a customary ruler to measure XෆYෆ in inches. Put the 0point at X and measure the distance to Y.The length of XෆYෆ is about inches.Use a ruler to draw a segment 3 centimeterslong. Then find the length of the segment in inches.Your Turn0 1 2 3 4inches (in.)X Y©Glencoe/McGraw-Hill2–230 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENT
  • 39. Use the figure below to determine whether eachstatement is true or false. Explain your reasoning.a. DෆEෆ ഡ GෆHෆBecause DE ϭ 4 and GH ϭ , ϭ .So, ഡ is a true statement.b. EෆFෆ ഡ FෆGෆBecause EF ϭ and FG ϭ , EF ≠ FG. So, EෆFෆ isnot congruent to FFෆGෆ, and the statement is false.Use the figure below to determine whethereach statement is true or false. Explain your reasoning.a. AෆEෆ ഡ BෆGෆb. DෆGෆ ഡ FෆJෆϪ8 10Ϫ1Ϫ2Ϫ3Ϫ4Ϫ7 Ϫ5Ϫ6A B C D E JGF IH2 3 4 5 6Your Turn8Ϫ8 10Ϫ1Ϫ2Ϫ3Ϫ4Ϫ7 Ϫ5Ϫ6D E HF G2 3 4 5 6 7©Glencoe/McGraw-Hill2–3Geometry: Concepts and Applications 31Congruent SegmentsTheorems are statements that can be justified by usingreasoning.BUILD YOUR VOCABULARY (pages 24–25)• Identify congruentsegments.• Find midpoints ofsegments.WHAT YOU’LL LEARNDefinition of CongruentSegments Two segmentsare congruent if andonly if they havethe same length.On the fifthtab of you Foldable,write CongruentSegments. Under thetab, write the definitionand draw examples ofcongruent segments.KEY CONCEPT
  • 40. Determine whether the statement is true or false.Explain your reasoning.CෆDෆ is congruent to CෆDෆ.Congruence of segments is , so ഡ .Therefore, the statement is .Determine whether the statement is true orfalse. Explain your reasoning.MෆNෆ is congruent to NෆMෆ.MෆNෆ is congruent to NෆMෆ.Your Turn©Glencoe/McGraw-Hill2–332 Geometry: Concepts and ApplicationsA unique point on every segment that separates thesegment into segments of lengthis known as the midpoint.To bisect something means to separate it into twoparts.BUILD YOUR VOCABULARY (pages 24–25)Theorem 2-1Congruence of segments is reflexive.Theorem 2-2Congruence of segments is symmetric.Theorem 2-3Congruence of segments is transitive.Write the converse ofTheorem 2-2. Is theconverse true?(Lesson 1-4)REVIEW IT
  • 41. In the figure, K is the midpoint of JෆLෆ.Find the value of d.You need to find the value of d. Since K is the midpoint of JෆLෆ,JK ϭ KL. Write and solve an equation involving d, and solvefor d.JK ϭ KL Definition ofϭ Substitutionϭ Subtraction Property of Equalityϭ dIn the figure, D is the midpoint of XෆYෆ.Find the value of a.X D Y7a Ϫ 8 5aYour TurnJ K Ld ϩ 5 2d©Glencoe/McGraw-Hill2–3Geometry: Concepts and Applications 33Page(s):Exercises:HOMEWORKASSIGNMENTDefinition of Midpoint Apoint M is the midpointof a segment SSෆTෆ if andonly if M is betweeen Sand T and SM ϭ MT.On the sixthtab of your Foldable,write Midpoint. Underthe tab, write thedefinition and draw anexample showing themidpoint of a linesegment.KEY CONCEPT
  • 42. Graph point K at (Ϫ4, 1).Start at the origin. Move unitsto the left. Then, move unit up.Label this point K.O xyK©Glencoe/McGraw-Hill2–434 Geometry: Concepts and ApplicationsThe Coordinate Plane• Name and graphordered pairs on acoordinate plane.WHAT YOU’LL LEARNThe of the grid used to locate points is known asthe coordinate plane.The number line is the y-axis.The x-axis is the number line.The two axes separate the coordinate plane intoregions known as quadrants.The two axes at a called theorigin.An ordered pair of real numbers, called the coordinates ofa point, locates a on the coordinate plane.The number of the ordered pair is called thex-coordinate.The y-coordinate is the number of the orderedpair.BUILD YOUR VOCABULARY (pages 24–25)On the seventh tab ofyour Foldable, writeCoordinate Plane.Under the tab, drawa coordinate plane,labeling the fourquadrants and thetwo axes.On the eighth tab ofyour Foldable, writeOrdered Pair andCoordinates. Under thetab, give an example ofan ordered pair. Labelthe x-coordinate andthe y-coordinate forthe pair.ORGANIZE IT
  • 43. ©Glencoe/McGraw-Hill2–4Geometry: Concepts and Applications 35Graph point L at (1, Ϫ4).Name the coordinates of points L and M.Point L is units to the right of the origin and unitbelow the origin. Its coordinates are .Point M is to the left of the origin and unitsabove the origin. Its coordinates are .Name the coordinates of points P and Q.O xyQPYour TurnxOyLMO xyLYour TurnPostulate 2-4Completeness Property for Points in the PlaneEach point in a coordinate plane corresponds to exactlyone ordered pair of real numbers. Each ordered pair ofreal numbers corresponds to exactly one point in acoordinate plane.Explain how to graphany ordered pair (x, y).Describe which directionyou move when x or yare either positive ornegative.WRITE IT
  • 44. Graph y ϭ Ϫ2.The graph of y ϭ Ϫ2 is a line that intersectsthe y-axis at .Graph x ϭ Ϫ1.xyOx ϭ Ϫ1Your TurnO xyy ϭ Ϫ2©Glencoe/McGraw-Hill2–436 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTTheorem 2-4If a and b are real numbers, a vertical line contains allpoints (x, y) such that x ϭ a, and a horizontal line containsall points (x, y) such that y ϭ b.
  • 45. Find the coordinate of the midpoint of AෆBෆ.Use the Midpoint Formula to find the coordinate of themidpoint of AෆBෆ.ᎏa ϩ2bᎏ ϭ ᎏϪ42ϩ 1ᎏϭ orThe coordinate of the midpoint is .Find the coordinate of the midpoint of OෆKෆ.810Ϫ1Ϫ2Ϫ3Ϫ4Ϫ5Ϫ6O K2 3 4 5 6 7Your TurnAϪ3Ϫ4 Ϫ2 Ϫ1 0 1B©Glencoe/McGraw-Hill2–5Geometry: Concepts and Applications 37Midpoints• Find the coordinates ofthe midpoint of asegment.WHAT YOU’LL LEARNOn the ninth tab ofyour Foldable, writeMidpoint for a NumberLine. Under the tab,explain how to find themidpoint of a segmenton a number line.ORGANIZE ITTheorem 2-5 Midpoint Formula for a Number LineOn a number line, the coordinate of the midpoint of asegment whose endpoints have coordinates a and b isᎏa ϩ2bᎏ.Theorem 2-6 Midpoint Formula for a Coordinate PlaneOn a coordinate plane, the coordinates of the midpoint ofa segment whose endpoints have coordinates (x1, y1) and(x2, y2) are΂ ,΃.y1 ϩ y2ᎏ2x1 ϩ x2ᎏ2SOL G.2.a The student will use . . . coordinate methods to solve problems involvingsymmetry and transformation. This will include investigating and using formulas forfinding distance, midpoint, and slope.
  • 46. Find the coordinates of D, the midpoint of CෆEෆ, givenendpoints C(2, 1) and E(16, 8).Use the Midpoint Formula to find the coordinates of D.΂ ,΃ ϭ΂ᎏ2ᎏ, ᎏ2ᎏ΃ϭ΂ᎏ2ᎏ, ᎏ2ᎏ΃ϭThe coordinates of D are .Find the coordinates of Y, the midpoint of XෆZෆ,given endpoints X (Ϫ3, 5) and Z (6, Ϫ1).Suppose L (2, ؊5) is the midpoint of KෆMෆ and thecoordinates of K are (؊4, ؊3). Find the coordinatesof M.Let (x1, y1) or (Ϫ4, Ϫ3) be the coordinates of K and let (x2, y2)be the coordinates of M. So, x1 ϭ and y1 ϭ .Use the Midpoint Formula.΂ ,΃ ϭy1 ϩ y2ᎏ2x1 ϩ x2ᎏ2Your Turny1 ϩ y2ᎏ2x1 ϩ x2ᎏ2©Glencoe/McGraw-Hill2–538 Geometry: Concepts and Applicationsϩ ϩ
  • 47. x-coordinate of Mϭ Replace x1 with Ϫ4.ϭ 2 Multiply each side by .ϭϭ Add to isolate the variable.x2 ϭy-coordinate of MᎏϪ32ϩ y2ᎏ ϭ Replace x1 with Ϫ3.ᎏϪ32ϩ y2ᎏ ϭ Ϫ5 Multiply each side by .ϭϭ Add to isolate the variable.y2 ϭThe coordinates of M are .Suppose S΂Ϫᎏ12ᎏ, Ϫᎏ32ᎏ΃ is the midpoint of RෆTෆ andthe coordinates of R are (Ϫ2, Ϫ5). Find the coordinates of T.Your TurnϪ4 ϩ x2ᎏ2Ϫ4 ϩ x2ᎏ2©Glencoe/McGraw-Hill2–5Geometry: Concepts and Applications 39Page(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITThe x-coordinateof the midpoint isthe average of thex-coordinates ofthe endpoints. They-coordinate of themidpoint is the averageof the y-coordinates ofthe endpoints.
  • 48. ©Glencoe/McGraw-HillBRINGING IT ALL TOGETHERChoose the term that best completes the statement.1. The set of non-negative integers is also called the set of[natural/whole] numbers.2. The quotient of two integers, where the denominator is notzero, is a(n) [rational/irrational] number.3. Decimals that do not repeat or terminate are called[rational/irrational] numbers.Find.4. ΗϪ4 Ϫ 1Η 5. ΗϪ(Ϫ12)Η 6. Η11 ϩ 2Η7. Points X, Y, and Z are collinear. If XY ϭ 10 and XZ ϭ 3, find YZ.8. Points A, B, and C are collinear. If AB ϭ 6, BC ϭ 8, andAC ϭ 14, which point is between the other two points?9. Points M, N, and P are collinear. If P lies between M and N,MP ϭ 2, and PN ϭ 1, find MN.BUILD YOURVOCABULARYUse your Chapter 2 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 2, go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder(pages 24–25) to help you solvethe puzzle.VOCABULARYPUZZLEMAKERC H A P T E R2STUDY GUIDE40 Geometry: Concepts and Applications2-1Real Numbers and Number Lines2-2Segments and Properties of Real Numbers
  • 49. Complete the statement.10. Two segments are if they are equal in length.11. When a segment is separated into two congruent segments,the segment is .12. Statements known as can be justified usinglogical reasoning.13. Points A, B, and C are collinear. If AෆCෆ ഡ CෆBෆ, then the point C isthe of AෆBෆ.Refer to the graph and name theordered pair for each point.14. point P15. point L16. point AGraph and label the following points on the abovecoordinate plane.17. point N (Ϫ4, 2) 18. point E (3, 1) 19. point S (1, Ϫ5)20. On a number line, if X ϭ Ϫ2 and Y ϭ 4, what is the coordinateof midpoint Z?21. Find the coordinates of the midpoint of a segment whoseendpoints are (Ϫ5, Ϫ1) and (Ϫ3, 3).22. Find the coordinates of the other endpoint of a segment whosemidpoint has coordinates (4, 5) and second endpoint at (2, Ϫ1).O xyLPNESA©Glencoe/McGraw-HillChapter BRINGING IT ALL TOGETHER2Geometry: Concepts and Applications 412-4The Coordinate Plane2-5Midpoints2-3Congruent Segments
  • 50. ©Glencoe/McGraw-HillCheck the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 2 Practice Test on page 85of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 2 Study Guide and Reviewon pages 82–84 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 2 Practice Test onpage 85 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 2 Foldable.• Then complete the Chapter 2 Study Guide and Review onpages 82–84 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 2 Practice Test onpage 85 of your textbook.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 2.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R2Checklist42 Geometry: Concepts and Applications
  • 51. Angles©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R3Geometry: Concepts and Applications 43NOTE-TAKING TIP: When you take notes, listen orread for main ideas. Then record those ideas insimplified form for future reference.FoldFold in half lengthwise.FoldFold again in thirds.OpenOpen and cut along thesecond fold to makethree tabs.LabelLabel as shown. Makeanother 3-tab fold andlabel as shown.Begin with a sheet of plain 8ᎏ12ᎏ" ϫ 11" paper.Chapter3Angles and Pointsinterior onanangleexterior
  • 52. ©Glencoe/McGraw-Hill44 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 3.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R3BUILD YOUR VOCABULARYVocabulary TermFoundDefinitionDescription oron Page Exampleacute angle[a-KYOOT]adjacent angles[uh-JAY-sent]angleangle bisectorcomplementary angles[kahm-pluh-MEN-tuh-ree]congruent anglesdegreesexteriorinteriorlinear pair[LIN-ee-ur]
  • 53. ©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY3Geometry: Concepts and Applications 45Vocabulary TermFoundDefinitionDescription oron Page Exampleobtuse angle[ob-TOOS]opposite raysperpendicular[PER-pun-DI-kyoo-lur]protractorquadrilateral[KWAD-ruh-LAT-er-ul]right anglesidesstraight anglesupplementary angles[SUP-luh-MEN-tuh-ree]trianglevertex[VER-teks]vertical angles
  • 54. Angles©Glencoe/McGraw-Hill46 Geometry: Concepts and Applications3–1Name the angle in four ways. Then identifyits vertex and its sides.The angle can be named in four ways:.Its vertex is . Its sides are and .Name the angle in fourways. Then identify its vertex and its sides.Name all angles having D as their vertex.There are distinct angles withvertex D: .Your TurnREVIEW ITName the sides ofЄABC. (Lesson 1-2)• Name and identify partsof an angle.WHAT YOU’LL LEARN Opposite rays are two rays that are part of the sameand have only their in common.The figure formed by is referred toas a straight angle.Any case where two rays have a commonis known as angle.The common is called the vertex.The two rays that make up the are called thesides of the angle.BUILD YOUR VOCABULARY (pages 44–45)REMEMBER ITRead the symbol Єas angle.HB 1ZYX Z5CB1 2DA
  • 55. ©Glencoe/McGraw-Hill3–1Geometry: Concepts and Applications 47Page(s):Exercises:HOMEWORKASSIGNMENTName all angles having Aas their vertex.Tell whether each point is in the interior,exterior, or on the angle.Point A: Point A is on the of the angle.Point B: Point B is on the of the angle.Point C: Point C is .Tell whether each point is in the interior,exterior, or on the angle.a. Point T b. Point Nc. Point DDmnNTYour TurnYour TurnAn angle separates a into parts: theinterior of the angle, the exterior of the angle, and theangle itself.BUILD YOUR VOCABULARYCBAIn your first Foldable,explain and drawexamples of interiorpoints, exterior points,and points on the angle.Include underappropriate tab.Angles and Pointsinterior onanangleexteriorORGANIZE ITAYXWZ543(page 44)
  • 56. Angle Measure©Glencoe/McGraw-Hill48 Geometry: Concepts and Applications3–2Use a protractor to measure ЄKLM.STEP 1 Place the center point of the protractor on vertex L.Align the straightedgewith side LM៮៬.STEP 2 Use the scale that beginswith 0 at LM៮៬. Read whereLK៮៬ crosses this scale.Angle KLM measures .Use a protractor to measure ЄXYZ.Find the measures of ЄDHE,ЄEHG, and ЄFHG.mЄDHE ϭ HD៮៬ is at 0° on the left.mЄEHG ϭ HG៮៬ is at 0° on the right.mЄFHG ϭ HG៮៬ is at 0° on the right.Your Turn• Measure, draw, andclassify angles.WHAT YOU’LL LEARN Angles are measured in units called degrees.A protractor is a tool used to measure angles and sketchangles of a given measure.BUILD YOUR VOCABULARY (pages 44–45)Postulate 3-1 Angle Measurement PostulateFor every angle, there is a unique positive number between0 and 180 called the degree measure of the angle.KLM908070605040302010100 110 12013014015016017080 7060504030201010011012013014015016017001801800XYZGD H908070605040302010100 110 12013014015016017080 7060504030201010011012013014015016017001801800FE
  • 57. ©Glencoe/McGraw-Hill3–2Geometry: Concepts and Applications 49Find mЄPQR, mЄRQS, and mЄSQT.Use a protractor to draw an angle having a measure of 35°.STEP 1 Draw BC៮៬.STEP 2 Place the center point of theprotractor on B. Align the marklabeled with the ray.STEP 3 Locate and draw point A at the mark labeled .Draw BA៮៬.Use aprotractor to draw anangle having ameasure of 78°.Your TurnYour TurnREMEMBER ITRead mЄPQR ϭ 75as the degree measureof angle PQR is 75.REMEMBER ITThe symbol isused to indicate a rightangle.Postulate 3-2 Protractor PostulateOn a plane, given AB៮៬ and a number r between 0 and 180,there is exactly one ray with endpoint A, extending on eachside of AB៮៬ such that the degree measure of the angleformed is r.A right angle has a degree measure of 90.The degree measure of an acute angle is greater than 0and less than 90.An obtuse angle has a degree measure greater than 90and less than 180.A three-sided closed figure with three interior angles isa triangle.A four-sided closed figure with four interior angles is aquadrilateral.BUILD YOUR VOCABULARY (pages 44–45)CAB35ЊR STQP
  • 58. 50 Geometry: Concepts and Applications©Glencoe/McGraw-Hill3–2Classify each angle as acute, obtuse, or right.Classify each angle as acute, obtuse, or right.a. b.The measure of ЄA is 100. Solve for x.You know that mЄA ϭ 100and mЄA ϭ ϩ 10.Write and solve an equation.ϭ 3x ϩ 10 Substitution100 Ϫ ϭ 3x ϩ 10 Ϫ Subtract fromeach side.ϭ 3xϭ Divide each side by .ϭ xThe measure of ЄN is 135. Solve for x.Your Turn127Њ41ЊYour Turn30Њ90Њ(7x Ϫ 5)ЊNIn your second Foldable,explain and drawexamples of rightangles, acute angles,and obtuse angles.Include underappropriate tab.Angles and Pointsinterior onanangleexteriorORGANIZE ITPage(s):Exercises:HOMEWORKASSIGNMENT90 3x3x ϩ 10A
  • 59. ©Glencoe/McGraw-HillThe Angle Addition Postulate3–3Geometry: Concepts and Applications 51If mЄKNL ϭ 110 and mЄLNM ϭ 25, find mЄKNM.mЄKNM ϭ mЄKNL ϩ mЄLNMϭ ϩ 25 Substitutionϭ So, mЄKNM ϭ .Find mЄ2 if mЄ1 ϭ 75 and mЄABC ϭ 140.mЄ2 ϭ mЄABC Ϫ mЄ1ϭ Ϫ Substitutionϭ So, mЄ2 ϭ .Find mЄJKL and mЄLKM if mЄJKM ϭ 140.mЄJKL ϩ mЄLKM ϭ mЄJKMϩ (2x Ϫ 10) ϭ 140 Substitutionϭ Combine like terms.6x Ϫ 10 ϩ ϭ 140 ϩ Add to each side.6x ϭx ϭ Divide each side by .• Find the measure andbisector of an angle.WHAT YOU’LL LEARNPostulate 3-3 Angle Addition PostulateFor any angle PQR, if A is in the interior of ЄPQR thenmЄPQA ϩ mЄAQR ϭ mЄPQR.K NLM110Њ25ЊA BCD21J KLM4xЊ(2x Ϫ 10)Њ
  • 60. 3–3Replace x with in each expression.mЄJKL ϭ 4x mЄLKM ϭ 2x Ϫ 10ϭ 4 ϭ 2 Ϫ 10ϭ ϭ Ϫ 10 ϭTherefore, mЄJKL ϭ and mЄLKM ϭ .a. If mЄABC ϭ 95 and mЄCBD ϭ 65, find mЄABD.b. If mЄXYZ ϭ 110 and mЄXYW ϭ 22, find mЄWYZ.c. Find mЄRSZ and mЄZST if mЄRST ϭ 135.Your Turn©Glencoe/McGraw-Hill52 Geometry: Concepts and Applications95Њ65ЊB ADCY XWZZRTS(4x ϩ 5)Њ9xЊ
  • 61. If FD៮៬ bisects ЄCFE and mЄCFE ϭ 70, findmЄ1 and mЄ2.Since FD៮៬ bisects ЄCFE, mЄ1 ϭ mЄ2.mЄ1 ϩ mЄ ϭ mЄCFE Postulate 3–3mЄ1 ϩ mЄ2 ϭ Replace mЄCFE with .mЄ1 ϩ ϭ 70 Replace mЄ2 with .2΂mЄ ΃ϭ 70 Combine like terms.ϭ Divide each side by .mЄ1 ϭSince mЄ1 ϭ mЄ2, mЄ2 ϭ .If EG៮៬ bisects ЄFEH and mЄFEH ϭ 98, findmЄ1 and mЄ2.Your Turn©Glencoe/McGraw-Hill3–3Geometry: Concepts and Applications 53A that divides an angle into angles ofequal is called the angle bisector.BUILD YOUR VOCABULARY (page 44)2(mЄ1) 70CDFE12E FGH12Page(s):Exercises:HOMEWORKASSIGNMENTREVIEW ITDෆFෆ is bisected at point E,and DF ϭ 8. What doyou know about thelengths DE and EF?(Lesson 2-3)
  • 62. Adjacent Angles and Linear Pairs of Angles54 Geometry: Concepts and ApplicationsDetermine whether Є1 and Є2 are adjacent angles.. They have the samebut no .. They have the sameand a common with no interiorpoints in common.. They have abut no common .Determine whether Є1 and Є2 are adjacentangles.a. b.c.121212Your Turn©Glencoe/McGraw-Hill3–4• Identify and useadjacent angles andlinear pairs of angles.WHAT YOU’LL LEARN Adjacent angles share a common side and a vertex, buthave no points in common.When the noncommon sides of adjacent angles form a, the angles are said to form a linear pair.BUILD YOUR VOCABULARY (page 44)1 21 212
  • 63. CM៮៮៬ and CE៮៬ are opposite rays.Name the angle that forms a linearpair with ЄTCM.ЄTCE and ЄTCM have a commonside , the samevertex , and opposite rays and .So, ЄTCE forms a linear pair with ЄTCM.Do Є1 and ЄTCE form a linear pair? Justify your answer., they are not angles.Refer to Examples 4 and 5.a. Name the angle that forms a linear pairwith ЄHCE.b. Determine if ЄTCA and ЄTCH form a linear pair. Justifyyour answer.List at least two models of linear pairs in yourclassroom or home.List at least two models of adjacent angles onyour school playground.Your TurnYour Turn©Glencoe/McGraw-Hill3–4Geometry: Concepts and Applications 55Page(s):Exercises:HOMEWORKASSIGNMENTList the differences andsimilarities betweenlinear pairs of anglesand adjacent angles.WRITE ITMTCEAH2134
  • 64. Complementary and Supplementary AnglesUse the figure to name a pair of nonadjacentsupplementary angles.mЄAGB ϩ ϭ , and they have the samevertex , but sides. Therefore, ЄAGBand are nonadjacent supplementary angles.Use the above figure to find the measure of an anglethat is supplementary to ЄBGC.Let x ϭ measure of angle supplementary to ЄBGC.mЄBGC ϩ x ϭ 180 Defn. of SupplementaryAnglesϩ x ϭ 180 mЄBGC ϭ35 ϩ x Ϫ ϭ 180 Ϫ Subtract from eachside.x ϭABCDFEG55Њ80Њ35Њ10Њ©Glencoe/McGraw-Hill56 Geometry: Concepts and Applications3–5• Identify and usecomplementary andsupplementary angles.WHAT YOU’LL LEARN Complementary angles are two angles whose degreemeasures total 90.Supplementary angles are two angles whose degreemeasures total 180.BUILD YOUR VOCABULARY (pages 44–45)SOL G.3 The student will solve practical problems involving complementary,supplementary, and congruent angles that include vertical angles, angles formedwhen parallel lines are cut by a transversal, and angles in polygons.
  • 65. a. In the figure, name apair of nonadjacentsupplementary angles.b. In the figure, find an angle with a measure supplementaryto ЄBAF.Angles C and D are supplementary. If mЄC ϭ 12x andmЄD ϭ 4(x ϩ 5), find x. Then find mЄC and mЄD.mЄC ϩ mЄD ϭ 180 Defn. of Supplementary Anglesϩ 4(x ϩ 5) ϭ 180 Substitution12x ϩ 4x ϩ ϭ 180 Distributive Propertyϭ 160 Combine like terms.ᎏ1166xᎏ ϭ ᎏ11660ᎏ Divide each side by 16.x ϭReplace x with in each expression.mЄC ϭ 12xϭ 12 ormЄD ϭ 4(x ϩ 5)ϭ 4΂ ϩ 5΃orAngles X and Y arecomplementary. If mЄX ϭ 2x andmЄY ϭ 8x, find x. Then find mЄXand mЄY.Your TurnYour Turn©Glencoe/McGraw-Hill3–5Geometry: Concepts and Applications 57Page(s):Exercises:HOMEWORKASSIGNMENT54Њ35Њ15Њ75Њ55Њ36Њ90ЊDEFGHBACREMEMBER ITSupplementaryangles must be adjacentangles to form a linearpair.
  • 66. 58 Geometry: Concepts and ApplicationsCongruent AnglesFind the value of x in each figure.The angles are angles.So, x ϭ .Since the angles are vertical angles, theyare congruent.x Ϫ 18 ϭx Ϫ 18 ϩ ϭ 75 ϩ Add to each side.x ϭFind the value of x in each figure.a. b.38Њ (3x Ϫ 10)Њ115ЊxЊYour Turn100ЊxЊ©Glencoe/McGraw-Hill3–6• Identify and usecongruent and verticalangles.WHAT YOU’LL LEARN Congruent angles have the same measure.When two lines , angles areformed. There are two pairs of nonadjacent angles.These pairs are vertical angles.BUILD YOUR VOCABULARY (pages 44–45)Theorem 3-1 Vertical Angle TheoremVertical angles are congruent.REMEMBER ITThe notationЄA Х ЄB is read asangle A is congruent toangle B.75Њ(x Ϫ 18)ЊSOL G.3 The student will solve practical problems involving complementary,supplementary, and congruent angles that include vertical angles, angles formedwhen parallel lines are cut by a transversal, and angles in polygons.
  • 67. ©Glencoe/McGraw-Hill3–6Geometry: Concepts and Applications 59Page(s):Exercises:HOMEWORKASSIGNMENTSuppose ЄA Х ЄB and mЄB ϭ 47. Find the measure ofan angle that is supplementary to ЄA.Since ЄA Х ЄB, their supplements are congruent.The supplement of ЄB is 180 Ϫ 47 or . So, themeasure of an angle that is supplementary to ЄA is .In the figure, Є1 is supplementary toЄ2, Є3 is supplementary to Є2, andmЄ2 is 105. Find mЄ1 and mЄ3.Є1 and Є2 are supplementary.So, mЄ1 ϭ Ϫ 105 or . Є3 and Є2 aresupplementary. So, mЄ3 ϭ Ϫ 105 or .a. Suppose ЄX Х ЄY and mЄY ϭ 82.Find the measure of an angle that issupplementary to ЄX.b. In the figure, Є1 is supplementary to Є2 and Є4.If mЄ4 ϭ 54, find mЄ1, mЄ2, and mЄ3.Your TurnTheorem 3–2 If two angles are congruent, then theircomplements are congruent.Theorem 3–3 If two angles are congruent, then theirsupplements are congruent.Theorem 3–4 If two angles are complementary to thesame angle, then they are congruent.Theorem 3–5 If two angles are supplementary to the sameangle, then they are congruent.Theorem 3–6 If two angles are congruent andsupplementary, then each is a right angle.Theorem 3–7 All right angles are congruent.1233412
  • 68. Perpendicular LinesRefer to the figure todetermine whether each ofthe following is true or false.QS៮៮ Ќ OP៮៮. QS៮៮ and OP៮៮ do notform angles.Therefore, they perpendicular.Є7 is an obtuse angle.. Є7 forms a with an acute angle.In the figureWෆYෆ Ќ ZෆTෆ. Determine whethereach of the following is trueor false.a. mЄWZU ϩ mЄUZT ϭ 90b. ЄSZY is obtuse.Your Turn©Glencoe/McGraw-Hill60 Geometry: Concepts and Applications3–7• Identify, use propertiesof, and constructperpendicular linesand segments.WHAT YOU’LL LEARNLines that at an angle of degreesare said to be perpendicular lines.BUILD YOUR VOCABULARY (page 45)REMEMBER ITRead p Ќ q as line pis perpendicular to line q.Theorem 3–8If two lines are perpendicular, then they form right angles.1 5 8 3472 6QONSRPMWUTSYRZX
  • 69. Find mЄ1 and mЄ2 if AC៮៮ Ќ BD៮៮,mЄ1 ϭ 8x Ϫ 2 and mЄ2 ϭ 16x Ϫ 4.Since AෆCෆ Ќ BෆDෆ, ЄAED is a right angle.mЄAED ϭ 90 Definition of perpendicularlinesЄ1 ϩ Є ϭ ЄAED Angle Addition PostulatemЄ1 ϩ mЄ ϭ mЄAEDmЄ1 ϩ mЄ2 ϭ Substitution(8x Ϫ 2) ϩ (16x Ϫ 4) ϭ 90 Substitution24x Ϫ 6 ϭ 90 Combine like terms.24x Ϫ 6 ϩ 6 ϭ 90 ϩ 6 Add 6 to each side.24x ϭ 96ᎏ2244xᎏ ϭ ᎏ9264ᎏ Divide each side by 24.x ϭReplace x with to find mЄ1 and mЄ2.mЄl ϭ 8x Ϫ 2 mЄ2 ϭ 16x Ϫ 4ϭ 8΂ ΃Ϫ 2 ϭ 16΂ ΃Ϫ 4ϭ 32 Ϫ 2 or 30 ϭ 64 Ϫ 4 or 60Therefore, mЄ1 ϭ 30 and mЄ2 ϭ 60.Find mЄ3 and mЄ4if AC៮៮ Ќ BF៮៮, mЄ3 ϭ 7x ϩ 6 andmЄ4 ϭ 12x ϩ 27.Your Turn©Glencoe/McGraw-Hill3–7Geometry: Concepts and Applications 61Page(s):Exercises:HOMEWORKASSIGNMENT21EDFA BCBD EFCA43
  • 70. ©Glencoe/McGraw-Hill62 Geometry: Concepts and ApplicationsBRINGING IT ALL TOGETHERIndicate whether the statement is true or false.1. XY៮៬ and YZ៮៬ are the sides of ЄXYZ.2. The vertex of an angle is a point where two rays intersect.3. A straight angle is also a line.Use a protractor to measure the specified angles. Then,classify them as acute, right, or obtuse angles.4. ЄBAC5. ЄCAE6. ЄDAE7. If mЄQPR ϭ 30 and mЄRPS ϭ 51,find mЄQPS.8. If mЄQPX ϭ 137 and mЄQPR ϭ 30,find mЄRPX.BUILD YOURVOCABULARYUse your Chapter 3 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 3, go to:www.glencoe.com/sec/math/t.resources/free/index.phpYou can use your completedVocabulary Builder (pages 44–45)to help you solve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R3STUDY GUIDE3-1Angles3-2Angle MeasureCDBAE3-3The Angle Addition PostulateTXYRQPS
  • 71. ©Glencoe/McGraw-Hill9. In the figure QN៮៬ and QP៮៬ are opposite rays.Name the angles that form a linear pair.10. If mЄ1 ϭ 36, what is the measure of itscomplement?11. What is the measure of an angle supplementaryto mЄ1 ϭ 36?Lines m and n intersect at point P. What is themeasure of each of the four angles formed?12. 13.14. 15.If WZ៮៮ is constructed perpendicular to XY៮៮,list six terms that describe ЄXWZ and ЄYWZ.16.17.18. 20.19. 21.Chapter BRINGING IT ALL TOGETHER3Geometry: Concepts and Applications 633-4Adjacent Angles and Linear Pairs of Angles3-5Complementary and Supplementary Angles3-6Congruent Angles3-7Perpendicular LinesZWX YPmn(3x ϩ 25)Њ(4x ϩ 5)ЊNPQ15432
  • 72. Check the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 3 Practice Test on page 137of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 3 Study Guide and Reviewon pages 134–136 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 3 Practice Test onpage 137.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 3 Foldable.• Then complete the Chapter 3 Study Guide and Review onpages 134–136 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 3 Practice Test onpage 137.Visit geoconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 3.Student SignatureTeacher SignatureC H A P T E R3ChecklistParent/Guardian SignatureChecklist©Glencoe/McGraw-Hill64 Geometry: Concepts and Application
  • 73. ©Glencoe/McGraw-HillC H A P T E R4Geometry: Concepts and Applications 65NOTE-TAKING TIP: When you take notes, listen orread for main ideas. Then record those ideas in asimplified form for future reference.Chapter4ParallelsUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.NOTE-TAKING TIP: When taking notes, it is oftena good idea to write in your own words a summaryof the lesson. Be sure to paraphrase key points.FoldFold in half along the width.OpenOpen and fold thebottom to form apocket. Glue edges.RepeatRepeat steps 1 and 2three times and glueall three pieces together.LabelLabel each pocket with thelesson names. Place anindex card in each pocket.Begin with three sheets of plain 8ᎏ12ᎏ" ϫ 11" paper.Parallels
  • 74. ©Glencoe/McGraw-Hill66 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 4.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R4BUILD YOUR VOCABULARYVocabulary TermFoundDefinitionDescription oron Page Examplealternate exterioranglesalternate interioranglesconsecutive interioranglescorresponding anglesexterior anglesfinitegreat circleinterior anglesline
  • 75. ©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY4Geometry: Concepts and Applications 67Vocabulary TermFoundDefinitionDescription oron Page Exampleline of latitudeline of longitudelinear equationparallel lines[PARE-uh-lel]parallel planesskew lines[SKYOO]slopeslope-intercept formtransversaly-intercept
  • 76. Parallel Lines and PlanesName the parts of the prism shown below. Assumesegments that look parallel are parallel.all planes parallel to plane SKLPlane is parallel to plane SKL.all segments that intersect MෆTෆintersect MෆTෆ.all segments parallel to MෆTෆis parallel to MෆTෆ.all segments skew to MෆTෆare skew to MෆTෆ.MS KQRNLT©Glencoe/McGraw-Hill68 Geometry: Concepts and Applications4–1Parallel lines are two lines in the same that donot intersect.Parallel planes are the same apart at allpoints and intersect.Lines that do not and are not in theplane are said to be skew lines.BUILD YOUR VOCABULARY (pages 66–67)• Describe relationshipsamong lines, parts oflines, and planes.WHAT YOU’LL LEARNUse the index cardlabeled Parallel Linesand Planes to record thedefinitions in this lesson,along with examples tohelp you remember themain idea.ParallelsORGANIZE IT
  • 77. Name the parts of the prism shown below.Assume segments that look parallel are parallel.a. all segments parallel to RෆSෆb. all segments that intersect RෆSෆc. a pair of parallel planesd. all segments skew to XෆTෆRSTZYXYour Turn©Glencoe/McGraw-Hill4–1Geometry: Concepts and Applications 69Page(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITA plane that passesthrough points A, B, Cand D can be namedusing any three of thepoints.
  • 78. Parallel Lines and TransversalsIdentify each pair of angles as alternate interior,alternate exterior, consecutive interior, or vertical.Є3 and Є5Є3 and Є5 are interior angles on the same side as thetransversal, so they are angles.Є1 and Є8Є1 and Є8 are exterior angles on opposite sides of thetransversal, so they are angles.1 23 45 67 8©Glencoe/McGraw-Hill70 Geometry: Concepts and Applications4–2• Identify therelationships amongpairs of interior andexterior angles formedby two parallel linesand a transversal.WHAT YOU’LL LEARNUse the index cardlabeled Parallel Linesand Transversals torecord the definitionsand theorems in thislesson. Draw picturesand examples to helpyou remember them.ParallelsORGANIZE ITA line, line segment, or ray that intersects two or morelines at different is known as a transversal.Interior angles lie in between the two lines.Alternate interior angles are on sides ofthe transversal.Consecutive interior angles are on the sideof the transversal.Exterior angles lie the two lines.Alternate exterior angles are on sides ofthe transversal.BUILD YOUR VOCABULARY (pages 66–67)SOL G.3 The student will solve practical problems involving complementary,supplementary, and congruent angles that include vertical angles, anglesformed when parallel lines are cut by a transversal, and angles in polygons.
  • 79. Identify each pair of angles as alternateinterior, alternate exterior, consecutive interior, orvertical.a. Є3 and Є5b. Є3 and Є6In the figure, pʈq, and r is atransversal. If mЄ6 ϭ 115,find mЄ7.Є6 and Є7 are alternateangles, so by Theorem 4-3, they are .Therefore, mЄ7 ϭ .If mЄ1 ϭ 50, find mЄ8.Your Turna14 35 6782nmYour Turn©Glencoe/McGraw-Hill4–2Geometry: Concepts and Applications 71Theorem 4-1 Alternate Interior AnglesIf two parallel lines are cut by a transversal, then each pairof alternate interior angles is congruent.Theorem 4-2 Consecutive Interior AnglesIf two parallel lines are cut by a transversal, then each pairof consecutive interior angles is supplementary.Theorem 4-3 Alternate Exterior AnglesIf two parallel lines are cut by a transversal, then each pairof alternate exterior angles is congruent.The sum of the degreemeasures of threeangles is 180. Are thethree anglessupplementary? Explain.(Lesson 3-5)REVIEW IT213 465 78rpq1435 67 82
  • 80. In the figure, AB៭៮៬ʈCD៭៮៬, and t isa transversal. If mЄ6 ϭ 128,find mЄ7, mЄ8, and mЄ9.Є6 and Є8 are consecutiveinterior angles, so by Theorem 4-2they are supplementary.mЄ6 ϩ mЄ8 ϭ 180ϩ mЄ8 ϭ 180 Replace mЄ6.128 ϩ mЄ8 Ϫ ϭ 180 Ϫ Subtract 128from each side.mЄ8 ϭЄ7 and Є8 are alternate interior angles, so by Theorem 4-1they are congruent. Therefore, mЄ7 ϭ .Є6 and Є9 are angles, so byTheorem 4-1 they are congruent. Therefore, mЄ9 ϭ .In the figure, nʈm, and a is a transversal.If mЄ6 ϭ 73, find mЄ1, mЄ4, and mЄ7.Your Turn©Glencoe/McGraw-Hill4–272 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTIf angles P and Q arevertical angles andmЄP ϭ 47, what ismЄQ? (Lesson 3-6)REVIEW ITa14 35 6782nm8769A BC DtREMEMBER ITIn figures with twopairs of parallel lines,arrowheads indicate thefirst pair and doublearrowheads indicate thesecond pair.
  • 81. Lines a and b are cut by transversal c. Name two pairsof corresponding angles.Corresponding angles lie on the same of thetransversal and have vertices. Two pairs ofcorresponding angles are .In the figure, aʈb, and k isa transversal.Which angle is congruent to Є1?Explain your answer.Є1 Х , since angles arecongruent ΂Postulate ΃.1 5 623 4 7 8acb©Glencoe/McGraw-HillTransversals and Corresponding Angles4–3Geometry: Concepts and Applications 73• Identify therelationships amongpairs of correspondingangles formed by twoparallel lines and atransversal.WHAT YOU’LL LEARNUse the index cardlabeled Transversals andCorresponding Angles torecord the postulates,theorems, and othermain ideas in this lesson.Draw pictures andexamples as needed.ParallelsORGANIZE ITWhen a crosses two lines, an interiorangle and an exterior angle that are on the sideof the transversal are called corresponding angles.BUILD YOUR VOCABULARY (page 66)Postulate 4-1 Corresponding AnglesIf two parallel lines are cut by a transversal, then each pairof corresponding angles is congruent.2134akbSOL G.3 The student will solve practical problems involving complementary,supplementary, and congruent angles that include vertical angles, anglesformed when parallel lines are cut by a transversal, and angles in polygons.
  • 82. Find the measure of Є1 if mЄ4 ϭ 60.mЄ1 ϭ mЄ3Є3 and Є4 are a linear pair, so they are supplementary.mЄ3 ϩ mЄ4 ϭ 180mЄ3 ϩ ϭ 180 Replace mЄ4with .mЄ3 ϩ 60 Ϫ ϭ 180 Ϫ Subtract 60 fromeach side.mЄ3 ϭmЄ1 ϭ Substitutiona. Refer to the figure in Example 1. Name two different pairsof corresponding angles.b. Refer to the figure in Example 2. Which angle is congruentto Є2? Explain your answer.c. Refer to the figure in Example 2. Find the measure of Є2 ifmЄ3 ϭ 145.Your Turn©Glencoe/McGraw-Hill4–374 Geometry: Concepts and ApplicationsTypes of angle pairsformed when atransversal cuts twoparallel lines.1. Congruenta. alternate interiorb. alternate exteriorc. corresponding2. Supplementarya. consecutive interiorKEY CONCEPTSTheorem 4-4 Perpendicular TransversalIf a transversal is perpendicular to one of two parallel lines,it is perpendicular to the other.
  • 83. In the figure, pʈq, and transversal ris perpendicular to q. IfmЄ2 ϭ 3(x ϩ 2), find x.pЌr Theorem 4-4Є2 is a right angle. Definition ofperpendicularlinesmЄ2 ϭDefinition ofright anglesmЄ2 ϭ Givenϭ 3(x ϩ 2) Replace mЄ2 with .ϭ Distributive Property90 Ϫ ϭ 3x ϩ 6 Ϫ Subtract 6 from each side.84 ϭ 3xᎏ834ᎏ ϭ ᎏ33xᎏ Divide each side by .ϭ xIn the figure, aʈb and r is a transversal.If mЄ1 ϭ 3x Ϫ 5 and mЄ2 ϭ 2x ϩ 35, find x.r12abYour Turnprq2©Glencoe/McGraw-Hill4–3Geometry: Concepts and Applications 75Page(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITThere are alwaysfour pairs ofcorresponding angleswhen two lines are cutby a transversal.
  • 84. Proving Lines ParallelIf mЄ1 ϭ 5x ϩ 10 and mЄ2 ϭ 6x Ϫ 4,find x so that aʈb.From the figure, you know that Є1 and Є2 are correspondingangles. According to Postulate 4-2, if mЄ1 ϭ mЄ2, then aʈb.mЄ1 ϭ mЄ2ϭ Substitution5x Ϫ 5x ϩ 10 ϭ 6x Ϫ 5x Ϫ 4 Subtract 5x from each side.10 ϭ x Ϫ 410 ϩ 4 ϭ x Ϫ 4 ϩ 4 Add 4 to each side.ϭ xFind c so that rʈs.Your Turn©Glencoe/McGraw-Hill4–4• Identify conditions thatproduce parallel linesand construct parallellines.WHAT YOU’LL LEARNUse the index cardlabeled Proving LinesParallel to record thepostulates, theorems,and important conceptsin this lesson. Recordexamples to help youremember the mainidea.ParallelsORGANIZE ITPostulate 4-2 In a plane, if two lines are cut by a transversalso that a pair of corresponding angles is congruent, then thelines are parallel.Theorem 4-5 In a plane, if two lines are cut by a transversalso that a pair of alternate interior angles is congruent, thenthe two lines are parallel.Theorem 4-6 In a plane, if two lines are cut by a transversalso that a pair of alternate exterior angles is congruent, thenthe two lines are parallel.Theorem 4-7 In a plane, if two lines are cut by a transversalso that a pair of consecutive interior angles is supplementary,then the two lines are parallel.Theorem 4-8 In a plane, if two lines are perpendicular tothe same line, then the two lines are parallel.12bacrs(4c Ϫ 10)Њ(3c ϩ 11)Њ76 Geometry: Concepts and ApplicationsSOL G.4 The student will use the relationships between angles formed by two linescut by a transversal to determine if two lines are parallel and verify, using algebraicand coordinate methods as well as deductive proofs.
  • 85. Identify the parallel segments inthe letter E.ЄFEC and ЄDCA are corresponding angles.mЄFEC ϭ mЄDCA Both angles measure 68°.EෆFෆʈCෆDෆ Postulate 4-2ЄBAC and ЄDCE are corresponding angles.mЄBAC ϭ mЄDCE Both angles measure 112°.AෆBෆʈCෆDෆ Postulate 4-2AෆBෆʈCෆDෆʈEෆFෆ Transitive PropertyIdentify the parallellines in the figure.Find the value of x so thatKL៭៮៬ ʈ MN៭៮៬.PQ៭៮៬ is a transversal for KL៭៮៬ and MN៭៮៬.If (9x)° ϭ (10x Ϫ 8)°, then KL៭៮៬ ʈ MN៭៮៬by Theorem 4-6.9x ϭ 10x Ϫ 89x Ϫ 9x ϭ 10x Ϫ 9x Ϫ8 Subtract 9x from each side.0 ϭ x Ϫ 80 ϩ 8 ϭ x Ϫ 8 ϩ 8 Add to each side.ϭ xThus, if x ϭ , then .Find c so that rʈs.Your TurnYour Turn©Glencoe/McGraw-Hill4–4Geometry: Concepts and Applications 77Page(s):Exercises:HOMEWORKASSIGNMENTWhat is the relationshipbetween Theorem 4-1and Theorem 4-5?(Lesson 4-2)REVIEW IT112ЊACEBDF68Њ112Њ68Њrspq122Њ58Њ59ЊKQP9xЊ(10x Ϫ 8)ЊLM Nr s132Њ(15c ϩ 12)Њ
  • 86. Slope©Glencoe/McGraw-Hill78 Geometry: Concepts and Applications4–5• Find the slopes of linesand use slope toidentify parallel andperpendicular lines.WHAT YOU’LL LEARN Slope is the ratio of the vertical change to the horizontalchange, or the to the , as you movefrom one point on the line to another.BUILD YOUR VOCABULARY (page 67)Find the slope of each line.m ϭ ᎏ02ϪϪ20ᎏ ϭ ᎏϪ22ᎏ ϭm ϭ ᎏϪ22ϪϪ(Ϫ(Ϫ32))ᎏ ϭ ᎏϪ25ϩ 2ᎏ ϭ ᎏ05ᎏ ϭFind the slope of each line.a. b. yxO(4, 2)(4, –3)yxO(4, 2)(–2, –3)Your TurnyxO(Ϫ3, Ϫ2) (2, Ϫ2)yxO(0, 2)(2, 0)Definition of SlopeThe slope m of a linecontaining two pointswith coordinates (x1, y1)and (x2, y2) is thedifference in they-coordinates dividedby the difference inthe x-coordinates.Use theindex card labeled Slopeto record the definitionsand postulates in thislesson.KEY CONCEPTSOL G.2.a The student will use pictorial representations, including computer software,constructions, and coordinate methods, to solve problems involving symmetry andtransformation. This will include investigating and using formulas for finding . . . slope.
  • 87. ©Glencoe/McGraw-Hill4–5Page(s):Exercises:HOMEWORKASSIGNMENTPostulate 4-3Two distinct nonvertical lines are parallel if and only if theyhave the same slope.Postulate 4-4Two nonvertical lines are perpendicular if and only if theproduct of their slopes is Ϫ1.Explain how you candetermine whether aline has a positive ornegative slope byobserving its graph.WRITE ITGiven A΂Ϫ2, Ϫᎏ12ᎏ΃, B΂2, ᎏ12ᎏ΃, C(5, 0), and D(4, 4), provethat AB៭៮៬ Ќ CD៭៮៬.First, find the slopes of AB៭៮៬ and CD៭៮៬.slope of AB៭៮៬ ϭ ϭ ϭslope of CD៭៮៬ ϭ ᎏ44ϪϪ05ᎏ ϭ ᎏϪ41ᎏ ϭThe product of the slopes for AB៭៮៬ and CD៭៮៬ is (Ϫ4)or . Therefore, Ќ .Given A(Ϫ3, Ϫ4), B(Ϫ1, 7), C(2, Ϫ5), and D(4, 6),prove that AB៭៮៬ʈCD៭៮៬.Your Turnᎏ12ᎏ ϩ ᎏ12ᎏᎏ2 ϩ 2ᎏ12ᎏ Ϫ ΂Ϫᎏ12ᎏ΃ᎏᎏ2 Ϫ (Ϫ2)Geometry: Concepts and Applications 79
  • 88. Equations of LinesName the slope and y-intercept of the graph of eachequation.y ϭ ᎏ23ᎏx ϩ 6 The slope is . The y-intercept .y ϭ 0 The slope is . The y-intercept .x ϭ 7 The graph is a line.The slope isundefined. There is no y-intercept.3y ϩ 12 ϭ 6xRewrite the equation in slope-intercept form by solving for y.3y ϩ 12 ϭ 6x3y ϩ 12 Ϫ 12 ϭ 6x Ϫ 12 Subtract 12 from each side.3y ϭ 6x Ϫ 12ᎏ33yᎏ ϭ ᎏ6x Ϫ312ᎏ Divide each side by 3.y ϭ 2x Ϫ 4 Simplify. This is written inslope-intercept form.The slope m ϭ . The y-intercept is .©Glencoe/McGraw-Hill80 Geometry: Concepts and Applications4–6• Write and graphequations of lines.WHAT YOU’LL LEARN The graph of a linear equation is a straight line.The y-value of the point where the line crosses theis called the y-intercept.The slope-intercept form of a linear equation is written as, where m is the slope and b is they-intercept.BUILD YOUR VOCABULARY (pages 66–67)Slope-Intercept FormAn equation of the linehaving slope m andy-intercept b isy ϭ mx ϩ b.KEY CONCEPT
  • 89. Name the slope and y-intercept of thegraph of each equation.a. y ϭ Ϫ6x ϩ 13b. y ϭ 8c. x ϭ 7d. 4x ϩ 3y ϭ 5Graph 2x Ϫ y ϭ 4 using the slope and y-intercept.First, rewrite the equation in slope-intercept form.2x Ϫy ϭ 42x Ϫ y Ϫ ϭ 4 Ϫ Subtract 2x from each side.Ϫy ϭᎏϪϪ1yᎏ ϭ ᎏ4ϪϪ12xᎏ Divide each side by Ϫ1.y ϭ Slope-intercept formThe y-intercept is Ϫ4. So, the point(0, Ϫ4) is on the line. Since the slopeis 2, or ᎏ21ᎏ, plot a point by using a riseof units (up) and a run ofunit (right). Draw a line through thetwo points.Your Turn©Glencoe/McGraw-Hill4–6Geometry: Concepts and Applications 81yxO(1, Ϫ2)(0, Ϫ4)Use the index cardlabeled Equations ofLines to recordimportant formulas andideas in this lesson. Giveexamples that show themost important ideas inthe lesson.ParallelsORGANIZE IT
  • 90. Graph 3x Ϫ 4y ϭ Ϫ8 using the slope andy-intercept.Write an equation of the line parallel to the graph ofy ϭ Ϫ2x ϩ 3 that passes through the point at (0, 1).Because the lines are parallel, they must have the sameslope. So, m ϭ .To find b, use the ordered pair (0, 1) and substitutefor m, x, and y in the slope-intercept form.y ϭ mx ϩ b1 ϭ (0) ϩ b m ϭ , (x, y) ϭ1 ϭ 0 ϩ bϭ bThe value of b is . So, the equation of the line is.a. Write an equation of the line parallel to the graph ofϪ5x ϩ y ϭ 6 that passes through the point (Ϫ1, 3).b. Write an equation of the line perpendicular to the graph ofy ϭ Ϫ2x ϩ 1 that passes through the point (4, Ϫ5).Your TurnyxO(0, 2)(Ϫ4, Ϫ1)Your Turn©Glencoe/McGraw-Hill4–682 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTExplain how you canfind the slope of a lineperpendicular to a givenline.WRITE IT
  • 91. BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillChoose the term that best completes each sentence.1. (Skew/Parallel) lines always lie on the same plane.2. (Perpendicular/Skew) lines never have any points in common.3. (Parallel/Perpendicular) lines never intersect.Refer to the figure and match theterm with the best representativeangle pair. Angle pairs cannot bematched more than once.4. consecutive interior angles5. exterior angles6. alternate interior angles7. alternate exterior anglesBUILD YOURVOCABULARYUse your Chapter 4 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 4, go to:www.glencoe.com/sec/math/t_resources/free/index.php.You can use your completedVocabulary Builder (pages 66–67)to help you solve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R4STUDY GUIDEGeometry: Concepts and Applications 834-1Parallel Lines and Planes4-2Parallel Lines and Transversalsr14 35 6872pqa. Є2 and Є7b. Є3 and Є6c. Є4 and Є6d. Є1 and Є7e. Є3 and Є4
  • 92. Chapter BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillIn the figure, ᐉʈm, andtransversal r is perpendicularto m. Name all angles congruentto the given angle.8. Є49. Є310. Є9Refer to the above figure to find the measure of thespecified angle if mЄ3 ϭ 40.11. Є4 12. Є513. Є8 14. Є2Find the values of a, b, andc so that ᐉʈmʈn.15. a ϭ16. b ϭ17. c ϭ18. Name the parallel lines.484 Geometry: Concepts and Applications4-3Transversals and Corresponding Angles4-4Proving Lines Parallelsrᐉm1 35 68 9247ᐉnm(5b Ϫ 6)Њ12cЊaЊ36Њ39Њ36Њ56Њ88Њᐉmnk
  • 93. ©Glencoe/McGraw-HillA wheelchair access ramp must be added to a home. Oneplan showed a ramp that started 30 feet away from theentrance. The entrance was 3 feet higher than ground level.The second plan started the ramp 15 feet from the same3-foot high entrance.19. What is the slope of each ramp?20. Which slope is steeper?21. Given A(0, 4), B(3, 6), C(1, 2), and D(3, Ϫ1), determinewhether AB៭៮៬ and CD៭៮៬ are parallel, perpendicular, or neither.Identify the slope and y-intercept of each equation.22. y ϭ Ϫ6x ϩ ᎏ12ᎏ23. 5x Ϫ 4y ϭ 724. y ϭ Ϫ225. x ϭ 526. Write an equation of a line parallel to y ϭ 3x ϩ 2 that passesthrough the point (Ϫ1, Ϫ4).Chapter BRINGING IT ALL TOGETHER4Geometry: Concepts and Applications 854-5Slope4-6Equations of Lines
  • 94. ©Glencoe/McGraw-HillCheck the one that applies. Suggestions to help you study are given witheach item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 4 Practice Test onpage 183 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 4 Study Guide and Reviewon pages 180–182 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 4 Practice Test onpage 183.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 4 Foldable.• Then complete the Chapter 4 Study Guide and Review onpages 180–182 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 4 Practice Test onpage 183.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 4.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R4Checklist86 Geometry: Concepts and Applications
  • 95. Triangles and Congruence©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R5Geometry: Concepts and Applications 87NOTE-TAKING TIP: When you take notes, definenew terms and write about the new concepts youare learning in your own words. Then, write yourown examples that use the new terms andconcepts.FoldFold in half lengthwise.FoldFold the top to the bottom.OpenOpen and cut along thesecond fold to make twotabs.LabelLabel each tab as shown.Begin with a sheet of plain 8ᎏ12ᎏ" ϫ 11" paper.Chapter5Trianglesclassifiedby AnglesTrianglesclassifiedby Sides
  • 96. acute trianglebasebase anglescongruent trianglescorresponding partsequiangular triangle[eh-kwee-AN-gyu-lur]equilateral triangle[EE-kwuh-LAT-ur-ul]imageincluded angleincluded sideisometry[eye-SAH-muh-tree]isosceles triangle[eye-SAHS-uh-LEEZ]©Glencoe/McGraw-Hill88 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 5.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R5BUILD YOUR VOCABULARYVocabulary TermFoundDefinitionDescription oron Page Example
  • 97. Vocabulary TermFoundDefinitionDescription oron Page Examplelegsmappingobtuse trianglepreimagereflectionright trianglerotationscalene triangle[SKAY-leen]transformationtranslationvertexvertex angle©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY5Geometry: Concepts and Applications 89
  • 98. Classifying Triangles©Glencoe/McGraw-Hill90 Geometry: Concepts and Applications5–1Classify each triangle by its angles and by its sides.The triangle is a triangle.The triangle is an triangle.30Њ30Њ120Њ• Identify the parts oftriangles and classifytriangles by their parts.WHAT YOU’LL LEARNDraw examples of acute,obtuse, right, scalene,isosceles, and equilateraltriangles in your notes.Trianglesclassifiedby AnglesTrianglesclassifiedby SidesORGANIZE ITThe side that is opposite the vertex angle in antriangle is called the base.In an isosceles triangle, the two angles formed by theand one of the congruent are calledbase angles.The congruent sides in an isosceles triangle are the legs.The vertex of each angle of a is a vertex ofthe triangle.The angle formed by the sides in antriangle is called the vertex angle.BUILD YOUR VOCABULARY (pages 88–89)
  • 99. Classify each triangle by its angles and byits sides.a. b.Find the measures of XY៮៮ and YZ៮៮ of isosceles triangleXYZ if ЄX is the vertex angle.Since ЄX is the vertex angle, Х .So, XY ϭ . Write and solve an equation.XY ϭϭ Substitution2n ϩ 2 Ϫ ϭ 10 Ϫ Subtract fromeach side.ϭDivide each sideby .n ϭThe value of n is .102n Ϫ 22n ϩ 2Y ZX140Њ20Њ20Њ44Њ74Њ62ЊYour Turn©Glencoe/McGraw-Hill5–1Geometry: Concepts and Applications 91ϭREMEMBER ITThe vertex of eachangle is a vertex of thetriangle.
  • 100. To find the measures of XෆYෆ and YෆZෆ, replace n with inthe expression for each measure.XY ϭ 2n ϩ 2ϭ 2΂ ΃ϩ 2ϭ ϩ 2ϭYZ ϭ 2n Ϫ 2ϭ 2΂ ΃Ϫ 2ϭ Ϫ 2ϭTherefore, XY ϭ and YZ ϭ .Triangle DEF is an isosceles triangle with baseෆEෆFෆ. Find DE and EF.DFEx Ϫ 23x Ϫ 87Your Turn©Glencoe/McGraw-Hill5–192 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENT
  • 101. Theorem 5-1 Angle Sum TheoremThe sum of the measures of the angles of a triangle is 180.Find mЄP in ᭝MNP if mЄM ϭ 80 and mЄN ϭ 45.mЄP ϩ mЄM ϩ mЄN ϭ 180 Angle Sum TheoremmЄP ϩ ϩ ϭ 180 SubstitutionmЄP ϩ 125 ϭ 180mЄP ϩ 125 Ϫ ϭ 180 Ϫ Subtract.mЄP ϭFind the value of each variable in ᭝ABC.ЄABC is a vertical angle to the given anglemeasure of 75. Since vertical angles arecongruent, mЄABC ϭ 75 ϭ x.mЄABC ϩ mЄBCA ϩ mЄCAB ϭ 180 Angle Sum Theoremϩ 58 ϩ ϭ 180 Substitution133 ϩ y ϭ 180133 ϩ y Ϫ 133 ϭ 180 Ϫ 133 Subtract.y ϭTherefore, x ϭ and y ϭ .Find the value of each variable.yЊxЊ45Њ65ЊYour Turn©Glencoe/McGraw-HillAngles of a Triangle5–2Geometry: Concepts and Applications 93• Use the Angle SumTheoremWHAT YOU’LL LEARNWhat does it meanwhen two angles arecomplementary?(Lesson 3-5)REVIEW IT75Њ58ЊyЊxЊBCA
  • 102. Find mЄJ and mЄK in righttriangle JKL.mЄJ ϩ mЄK ϭ 90 Theorem 5-2(x ϩ 15) ϩ (x ϩ 9) ϭ 90 Substitutionϩ 24 ϭ 90 Combine like terms.2x ϩ 24 Ϫ ϭ 90 Ϫ Subtract.2x ϭ 66Divide.x ϭReplace x with in each angle expression.mЄJ ϭ ϩ 15 ormЄK ϭ ϩ 9 orTherefore, mЄJ ϭ and mЄK ϭ .Find the value of a, b, and c.cЊbЊaЊ35ЊYour Turn©Glencoe/McGraw-Hill5–294 Geometry: Concepts and ApplicationsIs it possible to have tworight angles in atriangle? Justify youranswer.WRITE IT2xϭ66Page(s):Exercises:HOMEWORKASSIGNMENTTheorem 5-2The acute angles of a right triangle are complementary.Theorem 5-3The measure of each angle of an equiangular triangle is 60.When all three angles in a triangle are congruent, thetriangle is said to be equiangular.BUILD YOUR VOCABULARY (page 88)(x ϩ 15)Њ(x ϩ 9)ЊJKL
  • 103. Identify each motion as a translation, reflection, orrotation.Identify each motion as a translation,reflection, or rotation.a. b.Your Turn©Glencoe/McGraw-HillGeometry in Motion5–3Geometry: Concepts and Applications 95• Identify translations,reflections, androtations and theircorresponding parts.WHAT YOU’LL LEARNa figure from one position to anotherwithout turning it is called a translation.a figure over a line creates the mirror imageof the figure, or a reflection.a figure around a fixed point createsa rotation.BUILD YOUR VOCABULARY (page 89)SOL G.2.c The student will use pictorial representations, including computer software,constructions, and coordinate methods, to solve problems involving symmetry and trans-formation. This will include determining whether a figure has been translated, reflected, orrotated.
  • 104. In the figure, ᭝RST ᭝XYZ by a translation.Name the image of ЄT.᭝RST ᭝XYZ ЄT corresponds to.Name the side that corresponds to XY៮៮.Point R corresponds to point .᭝RST ᭝XYZPoint S corresponds to point .So, corresponds to .In the figure,᭝QRS ᭝DEF by a rotation.a. Name the angle that corresponds to ЄR.b. Name the side that corresponds to QෆRෆ.Your Turn©Glencoe/McGraw-Hill5–396 Geometry: Concepts and ApplicationsPairing each point on the original figure, or, with exactly one point on theis called mapping.The moving of each of a preimage to a newfigure called the image is a transformation.The new figure in a is called theimage.In a transformation, the figure is called thepreimage.BUILD YOUR VOCABULARY (pages 88–89)TRXYZSQSRFDE
  • 105. Identify the type(s) of transformations that were usedto complete the work below.Some figures can be moved to another withoutturning or flipping. Other figures have been turned arounda point with respect to the original.Therefore, the transformations are and.Identify the type(s) of transformations thatwere used to complete the work below.Your Turn©Glencoe/McGraw-Hill5–3Geometry: Concepts and Applications 97Page(s):Exercises:HOMEWORKASSIGNMENTTranslations, reflections, and rotations are all isometriesand do not change the or of thefigure being moved.BUILD YOUR VOCABULARY (page 88)
  • 106. Congruent TrianglesIf ᭝ABC Х ᭝FDE, name the congruent angles and sides.Then draw the triangles, using arcs and slash marks toshow congruent angles and sides.Name the three pairs of congruent angles by looking at theorder of the vertices in the statement ᭝ABC Х ᭝FDE.ЄA Х , ЄB Х ,and ЄC Х .Since A corresponds to , and B corresponds to ,Х .Since B corresponds to D, and C corresponds to E,Х .Since corresponds to F, and corresponds to E,Х FෆEෆ.A CBF ED©Glencoe/McGraw-Hill98 Geometry: Concepts and Applications5–4• Identify correspondingparts of congruenttriangles.WHAT YOU’LL LEARNDefinition of CongruentTriangles If thecorresponding parts oftwo triangles arecongruent, then the twotriangles are congruent.Likewise, if two trianglesare congruent, then thecorresponding parts ofthe two triangles arecongruent.KEY CONCEPTIf a triangle can be translated, rotated, or reflected ontoanother triangle so that all of thecorrespond, the triangles are said to be congruent.The parts of congruent triangles that are calledcorresponding parts.BUILD YOUR VOCABULARY (page 88)SOL G.5.a The student will investigate and identify congruence and similarityrelationships between triangles.
  • 107. The corresponding parts oftwo congruent trianglesare marked on the figure.Write a congruencestatement for the twotriangles.List the congruent angles and sides.ЄL Х ЄM Х ЄR ЄN ХLෆNෆ Х SෆTෆ Х TෆRෆ Х SෆRෆThe congruence statement can be written by matching theof the angles. Therefore,᭝LMN Х .a. If ᭝ACB Х ᭝ECD, name thecongruent angles and sides.Then draw the triangles,using arcs and slash marksto show congruent anglesand sides.b. Write another congruence statement for the two trianglesother than the one given above.Your TurnT RMSLN©Glencoe/McGraw-Hill5–4Geometry: Concepts and Applications 99Page(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITThe order of thevertices in a congruencestatement shows thecorresponding parts ofthe congruent triangles.
  • 108. SSS and SASIn two triangles, DF៮៮ Х UV៮៮, FE៮៮ Х VW៮៮, and DE៮៮ Х UW៮៮.Write a congruence statement for the two triangles.Draw a pair of triangles. Identify thecongruent parts with . Label the vertices ofone triangle.Use the given information to label the in thesecond triangle.By SSS, Х .In two triangles, CෆBෆ Х EෆFෆ, CෆAෆ Х EෆDෆ, andBෆAෆ Х FෆDෆ. Write a congruence statement for the two triangles.Your TurnFD EVU W©Glencoe/McGraw-Hill100 Geometry: Concepts and Applications5–5• Use the SSS and SAStests for congruence.WHAT YOU’LL LEARNPostulate 5-1 SSS PostulateIf three sides of one triangle are congruent to threecorresponding sides of another triangle, then the trianglesare congruent.REMEMBER ITThe letterdesignating theincluded angle appearsin the name of bothsegments that form theangle.SOL G.5.b The student will prove two triangles are congruent or similar, giveninformation in the form of a figure or statement, using algebraic and coordinate aswell as deductive proofs.In a triangle, the formed by two givenis the included angle.BUILD YOUR VOCABULARY (page 88)
  • 109. Determine whether thetriangles shown at the rightare congruent. If so, write acongruence statement andexplain why the trianglesare congruent. If not,explain why not.There are three pairs of sides,RෆSෆ Х , Х ZෆXෆ and RෆTෆ Х .Therefore, Х by .Determine whetherthe triangles to the right arecongruent. If so, write a congruencestatement and explain why thetriangles are congruent. If not,explain why not.AEDB CYour TurnR STX ZY©Glencoe/McGraw-Hill5–5Geometry: Concepts and Applications 101Page(s):Exercises:HOMEWORKASSIGNMENTPostulate 5-2 SAS PostulateIf two sides and the included angle of one triangle arecongruent to the corresponding sides and included angle ofanother triangle, then the triangles are congruent.Explain the SSS and SAStests for congruence inyour own words. Givean example of each.WRITE IT
  • 110. ASA and AASIn ᭝DEF and ᭝ABC, ЄD Х ЄC, ЄE Х ЄB, and DE៮៮ Х CB៮៮.Write a congruence statement for the two triangles.Draw a pair of triangles. Mark the congruentparts with and . Label the vertices ofone triangle D, E, and F.Locate C and B on the unlabeled triangle in the samepositions as and . The unassigned vertex is. Therefore, Х .In ᭝RST and ᭝XYZ, SෆTෆ Х XෆZෆ, ЄS Х ЄX, andЄT Х ЄZ. Write a congruence statement for the two triangles.Your TurnAC BFD E©Glencoe/McGraw-Hill102 Geometry: Concepts and Applications5–6• Use the ASA and AAStests for congruence.WHAT YOU’LL LEARNThe of the triangle that falls between two givenis called the included side and is the onecommon side to both angles.BUILD YOUR VOCABULARY (page 88)Postulate 5-3 ASA PostulateIf two angles and the included side of one triangle arecongruent to the corresponding angles and included side ofanother triangle, then the triangles are congruent.SOL G.5.b The student will prove two triangles are congruent or similar, giveninformation in the form of a figure or statement, using algebraic and coordinate aswell as deductive proofs.
  • 111. ᭝XYZ and ᭝QRS each have one pair of sides and onepair of angles marked to show congruence. What otherpair of angles needs to be marked so the two trianglesare congruent by AAS?If ЄQ and ЄX are marked , andХ , then and would haveto be congruent for the triangles to be congruent by .᭝ACB and ᭝FED each have one pair of sidesand one pair of angles marked to show congruence. What otherpair of angles needs to be marked so the two triangles arecongruent by AAS?AC BD EFYour TurnYX Z RQS©Glencoe/McGraw-Hill5–6Geometry: Concepts and Applications 103Page(s):Exercises:HOMEWORKASSIGNMENTTheorem 5-4 AAS TheoremIf two angles and a nonincluded side of one triangle arecongruent to the corresponding two angles andnonincluded side of another triangle, then the trianglesare congruent.
  • 112. ©Glencoe/McGraw-HillBRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 5 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 5, go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder (pages 88–89)to help you solve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R5STUDY GUIDE104 Geometry: Concepts and ApplicationsComplete each statement.1. The sum of the measures of a triangle’s interior angles is .2. The angle is the angle formed by two congruent sides of anisosceles triangle.3. The angles of a right triangle are complementary.4. A triangle with no congruent sides is .Find the value of each variable.5. 6. y61Њa aa5–1Classifying Triangles5–2Angles of a Triangle
  • 113. ©Glencoe/McGraw-HillSuppose ᭝SRN ᭝CDA.7. Which angle corresponds to ЄS?8. Name the preimage of AෆDෆ.9. Identify the transformation that occurred in the mapping.If ᭝ABC Х ᭝QRS, name the corresponding congruent parts.10. ЄB 11. AෆCෆ12. RෆQෆ 13. ЄC14. The pairs of triangles at the rightare congruent. Write a congruencestatement and the reason thetriangles are congruent.Underline the best term to make the statement true.15. [Mapping/Congruence] of triangles is explained by SSS, SAS, ASA,and AAS.16. AAS indicates two angles and their [included/nonincluded] side.Chapter BRINGING IT ALL TOGETHER5Geometry: Concepts and Applications 105S CN AR D5–4Congruent Triangles5–5SSS and SAS5–3Geometry in MotionFHG LJK5-6ASA and AAS
  • 114. ©Glencoe/McGraw-HillCheck the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 5 Practice Test onpage 223 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 5 Study Guide and Reviewon pages 220–222 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 5 Practice Test onpage 223 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 5 Foldable.• Then complete the Chapter 5 Study Guide and Review onpages 220–222 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 5 Practice Test onpage 223 of your textbook.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 5.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R5Checklist106 Geometry: Concepts and Applications
  • 115. More About Triangles©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R6Geometry: Concepts and Applications 107NOTE-TAKING TIP: As you read a lesson, takenotes on the materials. Include definitions,concepts, and examples. After you finish eachlesson, make an outline of what you learned.FoldFold each sheet of paper inhalf along the width. Thencut along the crease.StapleStaple the eight half-sheetstogether to form abooklet.CutCut seven lines from thebottom of the top sheet,six lines from the secondsheet, and so on.LabelLabel each tab with alesson number. The lasttab is for vocabulary.Begin with four sheets of lined 8ᎏ12ᎏ" ϫ 11" paper.6-16-26-36-46-56-66-7VocabularyChapter6
  • 116. Vocabulary TermFoundDefinitionDescription oron Page Examplealtitudeangle bisectorcentroidcircumcenter[SIR-kum-SEN-tur]concurrentEuler linehypotenuse[hi-PA-tin-oos]©Glencoe/McGraw-Hill108 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 6.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R6BUILD YOUR VOCABULARY
  • 117. ©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY6Geometry: Concepts and Applications 109Vocabulary TermFoundDefinitionDescription oron Page Exampleincenterlegmediannine-point circleorthocenter[OR-tho-SEN-tur]perpendicular bisectorPythagorean Theorem[puh-THA-guh-REE-uhn]Pythagorean triple
  • 118. 110 Geometry: Concepts and ApplicationsMediansIn ᭝ABC, CෆEෆ and AෆDෆ are medians.If CD ϭ 2x ϩ 5, BD ϭ 4x Ϫ 1, andAE ϭ 5x Ϫ 2, find BE.Since CෆEෆ and AෆDෆ are medians,D and E are midpoints. Solve for x.CD ϭ BD Definition of medianϭ Substitution2x ϩ 5 Ϫ ϭ 4x Ϫ 1 Ϫ Subtract.5 ϭ 2x Ϫ 15 ϩ ϭ 2x Ϫ 1 ϩ Add.6 ϭ 2x Divide.ϭ xUse the values for x and AE to find BE.AE ϭ BE Definition of median5x Ϫ 2 ϭ BE Substitution5( )Ϫ 2 ϭ BE Substitution15 Ϫ 2 ϭ BEϭ BEIn ᭝OPS, SෆTෆ and QෆPෆ aremedians. If PT ϭ 3x Ϫ 1, OT ϭ 2x ϩ 1, andOQ ϭ 4x Ϫ 2, find SQ.Your TurnABEDC©Glencoe/McGraw-Hill6–1• Identify and constructmedians in triangles.WHAT YOU’LL LEARN A median is a segment that joins a vertex of a triangle andthe midpoint of the side opposite that vertex.BUILD YOUR VOCABULARY (page 109)Under the tab forLesson 6-1, draw anexample of a median.Label the congruentparts. Under the tab forVocabulary, write thevocabulary words forthis lesson.6-16-26-36-46-56-66-7VocabularyORGANIZE ITSPOTQ
  • 119. Geometry: Concepts and Applications 111In ᭝XYZ, XෆPෆ, ZෆNෆ, and YෆMෆ aremedians.Find YQ if QM ϭ 4.Since QM ϭ , YQ ϭ 2 ؒ or .If QZ ϭ 18, what is ZN?Since QZ ϭ 18 and QZ ϭ ᎏ23ᎏ ؒ ZN, solve the equation18 ϭ ᎏ23ᎏ ؒ ZN for ZN.18 ϭ ᎏ23ᎏ ؒ ZNᎏ32ᎏ(18) ϭ ᎏ32ᎏ΂ᎏ23ᎏZN΃ Multiply each side by .ϭ ZNIn ᭝EFG, FෆAෆ, GෆBෆ,and EෆCෆ are medians.a. Find EO if CO ϭ 3.b. If FA ϭ 18, what are the measuresof AO and OF?A BCOEG FYour Turn©Glencoe/McGraw-Hill6–1The three of a triangle intersect at acommon point known as the centroid.When three or more lines or segments meet at the samepoint, they are said to be concurrent.BUILD YOUR VOCABULARY (page 108)Theorem 6-1The length of the segment from the vertex to the centroidis twice the length of the segment from the centroid to themidpoint.Page(s):Exercises:HOMEWORKASSIGNMENTMZPYNQX
  • 120. Altitudes and Perpendicular BisectorsIs AD៮៮ an altitude of the triangle?AෆDෆ is a perpendicularsegment. So, AෆDෆ analtitude of the triangle.Is GJ៮៮ an altitude of the triangle?GෆJෆ Ќ FෆHෆ, is a vertex, andis on the side opposite G.So, GෆJෆ an altitude of the triangle.a. Is BෆDෆ an altitude of the triangle?b. Is XෆYෆ an altitude of the triangle?YXZCBDAYour TurnF JG HADB C©Glencoe/McGraw-Hill112 Geometry: Concepts and Applications6–2• Identify and constructaltitudes andperpendicular bisectorsin triangles.WHAT YOU’LL LEARN An altitude of a triangle is a perpendicular segment withone endpoint at a and the other endpoint onthe opposite that vertex.BUILD YOUR VOCABULARY (page 108)REMEMBER ITEvery triangle hasthree altitudes—onethrough each vertex.Altitudes of TrianglesAcute Triangle Thealtitude is inside thetriangle.Right Triangle Thealtitude is a side of thetriangle.Obtuse Triangle Thealtitude is outside thethe triangle.KEY CONCEPT
  • 121. Is MN៮៮ a perpendicular bisector of aside of the triangle?Since N is the midpoint of KෆLෆ, MෆNෆ is abisector of side KෆLෆ. MෆNෆ perpendicular to KෆLෆ,so MෆNෆ is a perpendicular bisector in ᭝KLM.Is AD៮៮ a perpendicular bisector ofa side of the triangle?AෆDෆ Ќ BෆCෆ but D themidpoint of BෆCෆ. So, AෆDෆ a perpendicular bisector ofside BෆCෆ in ᭝ABC.a. Is BෆDෆ a perpendicular bisector ofthe triangle?b. Is LෆMෆ a perpendicular bisectorof the triangle?KHLJMCBDAYour TurnADB CL MKN©Glencoe/McGraw-Hill6–2Geometry: Concepts and Applications 113Under the tab forLesson 6-2, drawone example of analtitude and one of aperpendicular bisector.Label congruent partsand right angles.6-16-26-36-46-56-66-7VocabularyORGANIZE ITA line or segment thata side of a triangle is called the perpendicular bisector ofthat side.BUILD YOUR VOCABULARY (page 109)
  • 122. Tell whether MN៮៮ is an altitude, a perpendicularbisector, both, or neither.; MෆNෆ Ќ KෆLෆ but N the midpoint ofKෆLෆ. So, MෆNෆ aof side KෆLෆ in ᭝KLM.Tell whether XෆOෆ is an altitude, a perpendicularbisector, both, or neither.YZXOYour TurnKNML©Glencoe/McGraw-Hill6–2114 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTHow is a perpendicularbisector different from amedian?WRITE IT
  • 123. Geometry: Concepts and Applications 115In ᭝ABC, AC៮៮ bisects ЄBAD. If mЄ1 ϭ 41, find mЄ2.Since AෆCෆ bisects ЄBAD, mЄ1 ϭ .Since mЄ1 ϭ , mЄ2 ϭ .In ᭝KMN, NL៮៮ bisects ЄKNM. If ЄKNM is a right angle,find mЄ2.mЄ2 ϭ ᎏ12ᎏ(mЄKNM)mЄ2 ϭ ᎏ12ᎏ΂ ΃mЄ2 ϭIn ᭝WYZ, ZX៮៮ bisects ЄWZY. If mЄ1 ϭ 55, find mЄWZY.mЄWZY ϭ 2(mЄ1)mЄWZY ϭ 2΂ ΃mЄWZY ϭa. In ᭝XYZ, YෆWෆ bisects ЄXYZ.If mЄ2 ϭ 33, find mЄ1.YZXW12Your TurnWXYZ21NKLM12AC1 2B D©Glencoe/McGraw-HillAngle Bisectors of Triangles6–3• Identify and use anglebisectors in triangles.WHAT YOU’LL LEARNUnder the tab forLesson 6-3, draw anexample of an anglebisector. Label thecongruent parts.6-16-26-36-46-56-66-7VocabularyORGANIZE ITAn angle bisector of a triangle is a segment that separatesan angle of the triangle into two angles.BUILD YOUR VOCABULARY (page 108)
  • 124. b. In ᭝NOM, OෆPෆ bisects ЄNOM.If ЄNOM ϭ 85, find mЄ4.c. In ᭝RST, SෆUෆ bisects ЄRST.If mЄ6 ϭ 36.5, find mЄRST.In ᭝FHI, IG៮ is an angle bisector.Find mЄHIG.mЄHIGϭ mЄFIGϭ4x ϩ 1ϭ 5x Ϫ 5 Distributive Property4x ϩ 1 Ϫ 4x ϭ 5x Ϫ 5 Ϫ 4x Subtract.1ϭ x Ϫ 51 ϩ 5ϭ x Ϫ 5 ϩ 5 Add.ϭ xmЄHIG ϭ 4x ϩ 1 ϭ 4΂ ΃ϩ 1 ϭ ϩ 1 ϭIn ᭝JKL, KෆMෆ is anangle bisector. Find mЄJKM.JLK M5x Ϫ 24x ϩ 7Your TurnHIFG5(x Ϫ 1)Њ(4x ϩ 1)ЊTURS65OPNM3 4©Glencoe/McGraw-Hill6–3116 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENT
  • 125. Find the values of the variables.In the top triangle, find the value ofbase angle x. Since the triangle isisosceles, and one base angle ϭ 35,x ϭ .In the bottom triangle, find the value of base angle y. Since theother base angle ϭ 45, y ϭ .Find the valuesof the variables.KLJ12y60ЊxЊYour Turn45Њ35ЊxЊyЊ©Glencoe/McGraw-HillIsosceles Triangles6–4Geometry: Concepts and Applications 117• Identify and useproperties of isoscelestriangles.WHAT YOU’LL LEARN A leg of an isosceles triangle is one of the twosides.BUILD YOUR VOCABULARY (page 109)Theorem 6-2 Isosceles Triangle TheoremIf two sides of a triangle are congruent, then the anglesopposite those sides are congruent.Theorem 6-3The median from the vertex angle of an isosceles trianglelies on the perpendicular bisector of the base and the anglebisector of the vertex angle.Under the tab forLesson 6-4, draw anexample of an isoscelestriangle. Label thecongruent parts, andthe special names forsides and angles.6-16-26-36-46-56-66-7VocabularyORGANIZE ITTheorem 6-4 Converse of Isosceles Triangle TheoremIf two angles of a triangle are congruent, then the sidesopposite those angles are congruent.
  • 126. ©Glencoe/McGraw-Hill6–4118 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTCan an isosceles trianglebe an equiangulartriangle?WRITE ITIn ᭝DEF, Є1 Х Є2 and mЄ1 ϭ 28.Find mЄF, DF, and EF.First, find mЄF. You know thatmЄ1 ϭ 28. Since Є1 Х Є2,mЄ2 ϭ 28.mЄ1 ϩ mЄ2 ϩ mЄF ϭ 180 Angle Sum Theoremϩ ϩ mЄF ϭ 180 Replace mЄ1 andmЄ2.ϩ mЄF ϭ 18056 ϩ mЄF Ϫ 56 ϭ 180 Ϫ 56 Subtract.mЄF ϭNext, find DF. Since Є1 Х Є2, Theorem 6-4 states thatDෆFෆ Х EෆFෆ.DF ϭ EF Congruent segments2x ϩ 2 ϭ 3x Ϫ 3 Replace DF and EF.2x ϩ 2 Ϫ ϭ 3x Ϫ 3 Ϫ Subtract.2 ϭ x Ϫ 32 ϩ ϭ x Ϫ 3 ϩ Add.ϭ xBy replacing x with 5, you find that DF ϭ 2x ϩ 2 ϭ 2(5) ϩ 2 ϭ10 ϩ 2 or . EF ϭ 3x Ϫ 3 ϭ 3(5) Ϫ 3 ϭ 15 Ϫ 3 or .Find the values ofthe variables.XYZ5x Ϫ 23x ϩ 8yЊyЊYour Turn123x Ϫ 3D FE2x ϩ 2Theorem 6-5A triangle is equilateral if and only if it is equiangular.
  • 127. Determine whether each pair of right triangles iscongruent by LL, HA, LA, or HL. If it is not possible toprove that they are congruent, write not possible.There is one pair of congruentangles, ЄDFE Х ЄGFE. Thehypotenuses are congruent, DෆFෆ Х GෆFෆ.Therefore, ᭝DEF Х ᭝GEF by .FEDG©Glencoe/McGraw-HillRight Triangles6–5Geometry: Concepts and Applications 119• Use tests for congruenceof right triangles.WHAT YOU’LL LEARNIn a triangle the side opposite theangle is known as the hypotenuse.The two sides that form the angle are called legs.BUILD YOUR VOCABULARY (pages 108–109)Theorem 6-6 LL TheoremIf two legs of one right triangle are congruent to thecorresponding legs of another right triangle, then thetriangles are congruent.Theorem 6-7 HA TheoremIf the hypotenuse and an acute angle of one right triangleare congruent to the hypotenuse and corresponding angleof another right triangle, then the triangles are congruent.Theorem 6-8 LA TheoremIf one leg and an acute angle of one right triangle arecongruent to the corresponding leg and angle of anotherright triangle, then the triangles are congruent.Postulate 6-1 HL PostulateIf the hypotenuse and a leg of one right triangle arecongruent to the hypotenuse and corresponding leg ofanother right triangle, then the triangles are congruent.Under the tab forLesson 6-5, draw anexample of a righttriangle. Label thespecial names for thesides of the triangle.Under the tab forVocabulary, write thevocabulary words forthis lesson.6-16-26-36-46-56-66-7VocabularyORGANIZE IT
  • 128. 120 Geometry: Concepts and ApplicationsThere is one pair ofacute angles, ЄZ Х ЄL. There is one pair of, XෆZෆ Х KෆLෆ.Therefore, ᭝YXZ Х ᭝JKL by .Determine whether each pair of righttriangles is congruent by LL, HA, LA, or HL. If it isnot possible to prove that they are congruent, writenot possible.a.b.A BCRQPSTR UYour TurnLKJY ZX©Glencoe/McGraw-Hill6–5Page(s):Exercises:HOMEWORKASSIGNMENTWhich test forcongruence is used toestablish the LATheorem? Explain.WRITE IT
  • 129. Find the length of the hypotenuseof the right triangle.Use the Pythagorean Theorem to find thelength of the hypotenuse.c2ϭ a2ϩ b2c2ϭ2ϩ2Replace a and b.c2ϭ ϩc2ϭ 400 Take the square root ofeach side.c ϭ The length is .a. Find the length of the hypotenuseof the right triangle. c40 in.9 in.Your Turn16 ft12 ft©Glencoe/McGraw-HillThe Pythagorean Theorem6–6Geometry: Concepts and Applications 121• Use the PythagoreanTheorem and itsconverse.WHAT YOU’LL LEARNTheorem 6-9 Pythagorean TheoremIn a right triangle, the square of the length of thehypotenuse c is equal to the sum of the squares of thelengths of the legs a and b.Under the tab forLesson 6-5, write thePythagorean Theorem.Draw a right triangleand label the legs a andb, and the hypotenuse c.6-16-26-36-46-56-66-7VocabularyORGANIZE ITThe Pythagorean Theorem can be used to determine thelengths of the sides of a right triangle. It states that theof the squares of the of a right triangleequals the square of the hypotenuse.BUILD YOUR VOCABULARY (page 109)SOL G.7 The student will solve practical problems involving right triangles by using thePythagorean Theorem, properties of special right triangles, and right triangle trigonometry.Solutions will be expressed in radical form or as decimal approximations.
  • 130. The lengths of three sides of a triangle are 4, 5, and6 meters. Determine whether this triangle is a righttriangle.Since the longest side is meters, use as c, themeasure of the hypotenuse.c2ϭ a2ϩ b2Pythagorean Theorem62՘ 42ϩ 52Replace c with , a with, and b with .36 ՘ 16 ϩ 2536 41Since c2a2ϩ b2, the triangle a righttriangle.The lengths of three sides of a triangle are 5,12, and 13 yards. Determine whether this triangle is a righttriangle.Your Turnb. Find the length of one leg of a right triangle if the length ofthe hypotenuse is 25 cm and the length of the other leg is23 cm.©Glencoe/McGraw-Hill6–6122 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTTheorem 6-10 Converse of the Pythagorean TheoremIf c is the measure of the longest side of a triangle, a and bare the lengths of the other two sides, and c2ϭ a2ϩ b2,then the triangle is a right triangle.REMEMBER ITAlways check to seethat c represents thelength of the longestside.
  • 131. ©Glencoe/McGraw-Hill6–7Geometry: Concepts and Applications 123Use the Distance Formula to find the distance betweenA(6, 2) and B(4, Ϫ4). Round to the nearest tenth, ifnecessary.Use the Distance Formula. Replace (x1, y1) with (6, 2) and(x2, y2) with .d ϭ ͙(x2 Ϫෆx1)2ϩෆ (y2 Ϫෆ y1)2ෆ Distance FormulaAB ϭ΂4 Ϫ ΃2ϩ ΂ Ϫ 2΃2SubstitutionAB ϭ ͙(Ϫ2)2ෆϩ (Ϫ6ෆ)2ෆAB ϭ ϩAB ϭ ͙40ෆAB Ϸa. Use the Distance Formula to find the distance betweenM(2, 2) and N(Ϫ6, Ϫ4). Round to the nearest tenth, ifnecessary.Your TurnDistance on the Coordinate Plane• Find the distancebetween two points onthe coordinate plane.WHAT YOU’LL LEARNUnder the tab forLesson 6-7, write theDistance Formula. Thenshow an example tohelp you remember themain idea.6-16-26-36-46-56-66-7VocabularyTheorem 6-11 Distance FormulaIf d is the measure of the distance between two points withcoordinates (x1, y1) and (x2, y2), thend ϭ ͙(x2 Ϫxෆ1)2ϩ (ෆy2 Ϫ yෆ1)2ෆ.ORGANIZE ITSOL G.2.a The student will use pictorial representations,. . ., constructions, and coordinatemethods, to solve problems involving symmetry and transformation. This will includeinvestigating and using formulas for finding distance, midpoint, and slope.
  • 132. ©Glencoe/McGraw-Hill124 Geometry: Concepts and Applicationsb. Determine whether ᭝TRI with vertices T(Ϫ4, 1), R(2, 5),and I(2, Ϫ2) is isosceles.Akio took a ride in a hot-air balloon. The flight began4 miles north of his house. The balloon landed 3 milessouth and 2 miles east of his house. If the balloontraveled in a straight line between the starting andending points of the flight, what was the length ofAkio’s balloon ride?Suppose Akio’s house is located at the origin (0, 0). If theballoon ride began 4 miles north of his house, it began at(x1, y1), or (0, 4). Since the balloon landed 3 miles south and2 miles east of his house, it landed at (x2, y2) at (2, Ϫ3). Usethe Distance Formula to find the length of the balloon ride.d ϭ ͙(x2 Ϫෆx1)2ϩෆ (y2 Ϫෆ y1)2ෆϭ ͙(2 Ϫ 0ෆ)2ϩ (ෆϪ3 Ϫෆ 4)2ෆϭ ͙22ϩ (ෆϪ7)2ෆϭ ͙4 ϩ 4ෆ9ෆ ϭ ϷAkio’s balloon ride was approximately miles.Marcelle went to a friend’s house to complete ahomework project after school instead of going directly home.The school lies 2 blocks north of her home. Her friend’s houseis located 3 blocks west and 1 block north of her home. IfMarcelle traveled in a straight path from school to her friend’shome, what was the length of her walk?Your TurnRAkio’s houseStart(0, 4)End(2, Ϫ3)TRI6–7REMEMBER ITOnly use thepositive square rootssince distance is notnegative.Page(s):Exercises:HOMEWORKASSIGNMENT
  • 133. BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillC H A P T E R6STUDY GUIDEGeometry: Concepts and Applications 125Complete the sentence.1. The midpoint of a side of a triangle and the vertex of the oppositeangle are endpoints of a .2. A triangle’s medians are at the centroid.3. In ᭝ABC, BෆDෆ is a median and BD ϭ 6.What is BE?For the triangles shown, state whether AB is an altitude, aperpendicular bisector, both, or neither.4. 5. 6.BABABAEBA DCBUILD YOURVOCABULARYUse your Chapter 6 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 6, go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder(pages 108–109) to help yousolve the puzzle.VOCABULARYPUZZLEMAKER6-1Medians6-2Altitudes and Perpendicular Bisectors
  • 134. Chapter BRINGING IT ALL TOGETHER©Glencoe/McGraw-Hill6126 Geometry: Concepts and Applications7. In ᭝JKL, KෆHෆ bisects ЄJKL. If mЄ1 ϭ 12, find mЄJKL.8. What is the value of x so that BD is an angle bisector?Indicate whether the statement is true or false.9. The vertex angle of an isosceles triangle is opposite one of thecongruent sides.10. An isosceles triangle must be equiangular.For each triangle, find the values of the variables.11. 12.28Њ28Њ5 yxЊF HGABC63ЊxЊyЊ6-3Angle Bisectors of Triangles6-4Isosceles TrianglesJHL K12BA(5x ϩ 2)Њ(6x Ϫ 5)ЊDC
  • 135. ©Glencoe/McGraw-HillChapter BRINGING IT ALL TOGETHER6Geometry: Concepts and Applications 1276-6The Pythagorean Theorem6-7Distance on the Coordinate PlaneDetermine whether each pair of right triangles iscongruent by LL, HA, LA, or HL. If it is not possible to provethat they are congruent, write not possible.13. 14.Find the missing measure in each right triangle. Round tothe nearest tenth, if necessary.15. 16.Use the Distance Formula to find the distance between eachpair of points. Round to the nearest tenth, if necessary.17. G(Ϫ3, 1), H(4, 5) 18. R(Ϫ1, 2), S(5, Ϫ6) 19. A(12, 0), B(0, 5)20. Andre walked 2 blocks west of his home to school. After school,he walked to the store which is 1 block east and 1 block northof his home. About how far apart are the school and the store?12 10a cm58c ft6-5Right Triangles
  • 136. ©Glencoe/McGraw-HillStudent Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R6Checklist128 Geometry: Concepts and ApplicationsCheck the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 6.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 6 Practice Test onpage 271 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 6 Study Guide and Reviewon pages 268–270 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 6 Practice Test onpage 271.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 6 Foldable.• Then complete the Chapter 6 Study Guide and Review onpages 268–270 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 6 Practice Test onpage 271.
  • 137. Triangle Inequalities©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R7Geometry: Concepts and Applications 129NOTE-TAKING TIP: When you take notes, definenew vocabulary words, describe new ideas, andwrite examples that help you remember themeanings of the words and ideas.FoldFold lengthwise to the holes.CutCut along the top line andthen cut 4 tabs.LabelLabel each tab withinequality symbols. Store theFoldable in a 3-ring binder.Begin with a sheet of sheet of notebook paper.Chapter7ϾՆϽՅ
  • 138. ©Glencoe/McGraw-Hill130 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 7.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R7BUILD YOUR VOCABULARYVocabulary TermFoundDefinitionDescription oron Page Exampleexterior angleinequality[IN-ee-KWAL-a-tee]remote interior angles
  • 139. Refer to the number line and replace in DR LN withϽ, Ͼ, or ϭ to make a true sentence.DR LNΈ Ϫ 0Έ ΈϪ2 Ϫ Έ͉Ϫ3͉ ͉Ϫ4͉3 43 4Refer to the number line and replace inPR QS with Ͻ, Ͼ, or ϭ to make a true sentence.4 5 6321P Q R S0Ϫ1Ϫ2Ϫ3Ϫ4Ϫ5Your TurnϪ1 4321S R N D L0Ϫ2Ϫ3Ϫ4Ϫ5Ϫ6©Glencoe/McGraw-HillSegments, Angles, and InequalitiesGeometry: Concepts and Applications 131• Apply inequalities tosegment and anglemeasurements.WHAT YOU’LL LEARNWrite words under eachtab to describe eachsymbol on your foldable.ϾՆϽՅORGANIZE ITStatements that contain the symbols orcompare quantities or measures that do not have the samevalue and are called inequalities.BUILD YOUR VOCABULARY (page 130)Postulate 7-1 Comparison PropertyFor any two real numbers a and b, exactly one of thefollowing statements is true: a Ͻ b, a ϭ b, or a Ͼ b.Theorem 7-1If point C is between points A and B, and A, C, and B arecollinear, then AB Ͼ AC and AB Ͼ CB.Theorem 7-2If EP៮៬ is between ED៮៬ and EF៮៬, then mЄDEF Ͼ mЄDEP andmЄDEF Ͼ mЄPEF.7–1
  • 140. Refer to the figure. Determinewhether each statement istrue or false.AB Ͼ JKAB ϭ and JK ϭ48 Ͼ SubstitutionThis is because isgreater than .mЄAHC mЄHKLmЄAHC ϭ and mЄHKL ϭ45 SubstitutionThis is because is not greater thanor equal to .Refer to the figure. Determine whethereach statement is true or false.a. XY Ͻ XZb. mЄXYZ Ͻ mЄZXYXY ZYour TurnBQ135˚LJ45˚12 in.48 in.36 in.HNKAC12 in.©Glencoe/McGraw-Hill7–1132 Geometry: Concepts and Applications
  • 141. In the figure, mЄC Ͼ mЄA. If each of these measureswere divided by 5, would the inequality still be true?mЄC Ͼ mЄA47 Ͼ Replace mЄC withand mЄA with .47 Ϭ Ͼ 43 Ϭ Divide each side by .ϾThe inequality still holds because is greaterthan .In ᭝XYZ, mЄX Ͼ mЄZ. If each of thesemeasures doubled, would this inequality still hold true?50Њ 55Њ75ЊZ YXYour Turn43Њ47Њ39 cm42 cm57 cmBAC©Glencoe/McGraw-Hill7–1Geometry: Concepts and Applications 133Page(s):Exercises:HOMEWORKASSIGNMENTTransitive PropertyFor any numbers a, b,and c,1. If a Ͻ b and b Ͻ c, thena Ͻ c.2. If a Ͼ b and b Ͼ c, thena Ͼ c.Addition and SubtractionProperties For anynumbers a, b, and c,1. If a Ͻ b, then a ϩ c Ͻb ϩ c and a Ϫ c Ͻ b Ϫ c.2. If a Ͼ b, then a ϩ c Ͼb ϩ c and a Ϫ c Ͼ b Ϫ c.Multiplication andDivision PropertiesFor any numbers a, b,and c,1. If c Ͼ 0, and a Ͻ b thenac Ͻ bc and ᎏacᎏ Ͻ ᎏbcᎏ.2. If c Ͼ 0 and a Ͼ b thenac Ͼ bc and ᎏacᎏ Ͼ ᎏbcᎏ.KEY CONCEPTS
  • 142. Exterior Angle TheoremName the remote interior angleswith respect to Є4.Angle forms awith Є2. Therefore, and Є3 areremote angles with respect to Є4.Name the remote interior angles with respectto Є2.1351211 6789102 4Your Turn12 34©Glencoe/McGraw-Hill134 Geometry: Concepts and Applications7–2• Identify exterior anglesand remote interiorangles of a triangle anduse the Exterior AngleTheorem.WHAT YOU’LL LEARNIn your notes, recordexamples of each typeof inequality under theappropriate tab. Be sureto write about therelationships betweensides and angles of atriangle.ϾՆϽՅORGANIZE ITAn exterior angle of a triangle is an angle that forms apair with one of the angles of the triangle.Remote interior angles of a triangle are theangles that do not form a linear pair with theangle.BUILD YOUR VOCABULARY (page 130)Theorem 7-3 Exterior Angle TheoremThe measure of an exterior angle of a triangle is equal tothe sum of the measures of its two remote interior angles.SOL G.9 The student will use measures of interior and exterior angles of polygonsto solve problems. Tessellations and tiling problems will be used to make connectionsto art, construction, and nature.
  • 143. In the figure, if mЄ1 ϭ 145 andmЄ5 ϭ 82, what is mЄ3?mЄ1 ϭ mЄ5 ϩ Exterior Angle Theorem145 ϭ ϩ mЄ3 Replace mЄ1 with 145and mЄ5 with 82.145 Ϫ ϭ 82 ϩ mЄ3 Ϫ Subtractfrom each side.ϭ mЄ3In the figure, if mЄ6 ϭ 8x, mЄ3 ϭ 12, and mЄ2 ϭ 4(x ϩ 5),find the value of x.mЄ6 ϭ mЄ3 ϩ Exterior Angle Theoremϭ ϩ 4(x ϩ 5) Replace mЄ6 with 8x,mЄ3 with 12 and mЄ2with 4(x ϩ 5).8x ϭ 12 ϩ ϩ8x ϭ ϩ 4x Combineterms.8x Ϫ ϭ 32 ϩ 4x Ϫ Subtractfrom each side.ϭ 32Divide each sideby .x ϭ1 2 35 64©Glencoe/McGraw-Hill7–2Geometry: Concepts and Applications 135Theorem 7-4 Exterior Angle Inequality TheoremThe measure of an exterior angle of a triangle is greaterthan the measure of either of its two remote interior angles.4xϭ32Under the tab labeledwith a greater than sign,summarize Theorem 7-4.ϾՆϽՅORGANIZE IT
  • 144. a. Find the measure of Є1in the figure.b. In the figure, if mЄ6 ϭ10x ϩ 3, mЄ3 ϭ 6x Ϫ 6,and mЄ12 ϭ 49, find thevalue of x.Name the two angles in ᭝CDEthat have measures less than 82.The measure of the exterior angle withrespect to Є1 is . Angles and are itsremote interior angles. By Theorem , 82 Ͼ mЄand 82 Ͼ mЄ . Therefore, and havemeasures less than 82.Name the two angles in ᭝JKL that havemeasures less than 117.117LKJYour Turn13 2C DE 82Њ1351211 6789102 440170S RQYour Turn©Glencoe/McGraw-Hill7–2136 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTTheorem 7-5If a triangle has one right angle, then the other two anglesmust be acute.REMEMBER ITThe measures of theangles in any trianglehave a sum of 180degrees.
  • 145. In ᭝KLM, list the angles in order from least to greatestmeasure.Write the segment measures in order from togreatest. Then, use Theorem to write the measuresof the angles opposite those sides in the same order.KM Ͻ KL Ͻ LMmЄ Ͻ mЄ Ͻ mЄTherefore, the angles in order from least to greatest areЄ , Є , and Є .In ᭝QPS, list the anglesin order from least to greatest measure.9.5 1011Q SPYour Turn15 m10 m21 mKL M©Glencoe/McGraw-HillInequalities Within a Triangle7–3Geometry: Concepts and Applications 137• Identify the relationshipsbetween the sides andangles of a triangle.WHAT YOU’LL LEARNTheorem 7-6If the measures of three sides of a triangle are unequal,then the measures of the angles opposite those sides areunequal in the same order.Theorem 7-7If the measures of three angles of a triangle are unequal,then the measures of the sides opposite those angles areunequal in the same order.SOL G.6 The student, given information concerning the lengths of sides and/or measures ofangles, will apply the triangle inequality properties to determine whether a triangle exists and toorder sides and angles. These concepts will be considered in the context of practical situations.
  • 146. Identify the side of ᭝KLM with the greatest measure.Write the angle measures in order from least to .Then, use Theorem to write the measures of the sidesopposite those angles in the same order.mЄM Ͻ mЄL Ͻ mЄKϽ ϽTherefore, has the greatest measure.In ᭝XYZ, list the sidesin order from least to greatest measure.59 6160X ZYYour Turn59Њ 54Њ67ЊL MK©Glencoe/McGraw-Hill7–3138 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTUnder the tab labeledwith a greater than sign,summarize Theorem 7-8using the words “greaterthan”.ϾՆϽՅORGANIZE ITTheorem 7-8In a right triangle, the hypotenuse is the side with thegreatest measure.
  • 147. Determine if the three numbers can be the measures ofthe sides of a triangle.6, 7, 96 ϩ 7 Ͼ 96 ϩ 9 Ͼ 77 ϩ 9 Ͼ 6All possible cases true. Sides with these measuresform a triangle.1, 7, 87 ϩ 8 Ͼ 18 ϩ 1 Ͼ 71 ϩ 7 Ͼ 8All possible cases true. Sides with thesemeasures form a triangle.Determine if the three numbers can be themeasures of the sides of a triangle.a. 15, 40, 19b. 4, 18, 21Your Turn©Glencoe/McGraw-HillTriangle Inequality Theorem7–4Geometry: Concepts and Applications 139• Identify and use theTriangle InequalityTheorem.WHAT YOU’LL LEARNUnder the tab labeledwith a greater thansign, summarizeTheorem 7-9.ϾՆϽՅORGANIZE ITTheorem 7-9 Triangle Inequality TheoremThe sum of the measures of any two sides of a triangleis greater than the measure of the third side.SOL G.6 The student, given information concerning the lengths of sides and/or measures ofangles, will apply the triangle inequality properties to determine whether a triangle exists and toorder sides and angles. These concepts will be considered in the context of practical situations.
  • 148. What are the greatest and least possible whole-numbermeasures for a side of a triangle whose other two sidesmeasure 4 feet and 6 feet?Let x be the measure of the third side of the triangle. x isgreater than the difference of the measures of the twoother sides.x Ͼ 6 Ϫx Ͼx is less than the sum of the measures of the two other sides.x Ͻ 6 ϩx ϽTherefore, Ͻ x Ͻ .If the measures of two sides of a triangle are 12 metersand 14 meters, find the range of possible measures ofthe third side.Let x be the measure of the third side of the triangle. x isgreater than the difference of the measures of the twoother sides.x Ͼ 14 Ϫx Ͼx is less than the sum of the measures of the two other sides.x Ͻ 14 ϩx ϽTherefore, Ͻ x Ͻ .©Glencoe/McGraw-Hill7–4140 Geometry: Concepts and ApplicationsIn your own words,explain why two sides ofa triangle, when addedtogether, cannot equalthe length of the thirdside.WRITE IT
  • 149. a. What are the greatest and least possible whole-numbermeasures for a side of a triangle whose other two sidesmeasure 23 cm and 29 cm?b. If the measures of two sides of a triangle are 11 inchesand 3 inches, find the range of possible measures of thethird side.Your Turn©Glencoe/McGraw-Hill7–4Geometry: Concepts and Applications 141Page(s):Exercises:HOMEWORKASSIGNMENT
  • 150. ©Glencoe/McGraw-Hill142 Geometry: Concepts and ApplicationsBRINGING IT ALL TOGETHERReplace with Ͻ, Ͼ, or ϭ to make a true sentence.1. JK KX 2. LM JL3. KM JL 4. KL XM5. mЄBCD mЄBDE6. mЄCBE mЄEDC7. Name the remote interior angles of ᭝ABCwith respect to Є5.8. BෆDෆ Ќ AෆCෆ and mЄ15 ϭ 139.What is mЄ10?9. If mЄ1 ϭ 19x, mЄ16 ϭ 6x, andmЄDAB ϭ 91, find the value of x.603850 164779EABCD4 5321J K X L M0Ϫ2 Ϫ1Ϫ3Ϫ4Ϫ5Ϫ61359 10 111615171812 13 146 7 842A CEBDBUILD YOURVOCABULARYUse your Chapter 7 Foldableto help you study for yourchapter test.To make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 7, go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder (page 130)to help you solve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R7STUDY GUIDE7-1Segments, Angles, and Inequalities7-2Exterior Angle Theorem
  • 151. ©Glencoe/McGraw-HillIn each triangle, list the angles from least to greatest.10. 11.In each triangle, list the sides measuring least to greatest.12. 13.Determine if the numbers given can be measures of the sides ofa triangle.14. 7.7, 16.8, 11.3 15. 36, 12, 2816. 7, 9, 16Find the range of possible values for the third side of the triangle.17. 16, 7 18. 12, 1019. 5, 91696ZYX4682CAB7.610.18.5YXZ8914CBAChapter BRINGING IT ALL TOGETHER7Geometry: Concepts and Applications 1437-3Inequalities Within a Triangle7-4Triangle Inequality Theorem
  • 152. ©Glencoe/McGraw-HillCheck the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 7 Practice Test on page305 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 7 Study Guide and Reviewon pages 302–304 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 7 Practice Test onpage 305.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 7 Foldable.• Then complete the Chapter 7 Study Guide and Review onpages 302–304 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 7 Practice Test onpage 305.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 7.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R7Checklist144 Geometry: Concepts and Applications
  • 153. Quadrilaterals©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R8Geometry: Concepts and Applications 145NOTE-TAKING TIP: When you read and learn newconcepts, help yourself remember these conceptsby taking notes, writing definitions andexplanations, and drawing models as needed.FoldFold each sheet of paperin half from top tobottom.CutCut along the fold. Staplethe six sheets together toform a booklet.CutCut five tabs. The top tabis 3 lines wide, the next tabis 6 lines wide, and so on.LabelLabel each of the tabswith a lesson number.Begin with three sheets of lined 8ᎏ12ᎏ" ϫ 11"paper.Chapter88–28–1Quadrilaterals
  • 154. base anglesbasesconsecutive[con-SEK-yoo-tiv]diagonalsisosceles trapeziodkitelegsmedian©Glencoe/McGraw-Hill146 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 8.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in thesecond column for reference when you study.C H A P T E R8BUILD YOUR VOCABULARYVocabulary TermFoundDefinitionDescription oron Page Example
  • 155. midsegmentnonconsecutiveparallelogramquadrilateralrectanglerhombus[ROM-bus]squaretrapezoid[TRAP-a-ZOYD]©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY8Geometry: Concepts and Applications 147Vocabulary TermFoundDefinitionDescription oron Page Example
  • 156. ©Glencoe/McGraw-Hill148 Geometry: Concepts and Applications8–1Refer to quadrilateral DEFG.Name all pairs of consecutive angles.and ЄG, and ЄF, and ЄE, andand ЄD are consecutive angles.Name all pairs of nonconsecutive vertices.D and are nonconsecutive vertices. and G arenonconsecutive vertices.Name all pairs of consecutive sides.DෆGෆ and , FෆGෆ and , EෆFෆ and , and DෆEෆand are pairs of consecutive sides.GEDFQuadrilaterals• Identify parts ofquadrilaterals andfind the sum of themeasures of theinterior angles of aquadrilateral.WHAT YOU’LL LEARNUnder the tab forLesson 8-1, write therules for classifyingquadrilaterals. Draw aquadrilateral and labelthe consecutive andnonconsecutive sides, aswell as the diagonals.8–28–1QuadrilateralsORGANIZE ITA quadrilateral is a geometric figure withsides and vertices.Any two sides, vertices, or angles of a quadrilateral areeither consecutive or nonconsecutive.Segments that join vertices arecalled diagonals.BUILD YOUR VOCABULARY (pages 146–147)SOL G.8.a The student will investigate and identify properties of quadrilateralsinvolving opposite sides and angles, consecutive sides and angles, and diagonals.
  • 157. ©Glencoe/McGraw-Hill8–1Geometry: Concepts and Applications 149Find the missing measure if three of the four anglemeasures in quadrilateral ABCD are 90, 120, and 40.mЄA ϩ mЄB ϩ mЄC ϩ mЄD ϭ 360 Theorem 8-1ϩ ϩ ϩ mЄD ϭ 360 Substitutionϩ mЄD ϭ 360250 ϩ mЄD Ϫ 250 ϭ 360 Ϫ 250 Subtract.mЄD ϭFind the missing measure if three of the fourangle measures in quadrilateral RMSQ are 115, 75, and 50.Your TurnREMEMBER ITIn a quadrilateral,nonconsecutive sides,vertices, or angles arealso called oppositesides, vertices, or angles.Page(s):Exercises:HOMEWORKASSIGNMENTRefer to quadrilateralWXYZ.a. Name all pairs of consecutive angles.b. Name all pairs of nonconsecutive vertices.c. Name all pairs of consecutive sides.Z YW XYour TurnTheorem 8-1The sum of the measures of the angles of a quadrilateralis 360.REMEMBER ITConsecutive sidesshare a vertex;nonconsecutive sidesdo not.Consecutive vertices arethe endpoints of a sidewhile nonconsecutivevertices are not.Consecutive anglesshare a side of thequadrilateral whilenonconsecutive anglesdo not.
  • 158. ©Glencoe/McGraw-Hill150 Geometry: Concepts and Applications8–2 ParallelogramsSOL G.8.a The student will investigate and identify properties of quadrilateralsinvolving opposite sides and angles, consecutive sides and angles, and diagonals.In parallelogram KLMN,KL ϭ 23, KN ϭ 15, andmЄK ϭ 105.Find LM and MN.KෆLෆ Х MෆNෆ and KෆNෆ Х LෆMෆ Theorem 8-3KL ϭ and KN ϭ Definition of congruentsegmentsϭ MN and ϭ LM Replace KL withand KN with .Find mЄM.ЄM Х ЄK Theorem 8-2mЄM ϭ Definition of congruent anglesmЄM ϭ Replace mЄK with .1523K LMN105Њ• Identify and use theproperties ofparallelograms.WHAT YOU’LL LEARNUnder the tab forLesson 8-2, write thedefinitions andtheorems to help youclassify parallelograms.Draw a parallelogramand label the congruentsides and angles, as wellas properties of thediagonals.8–28–1QuadrilateralsORGANIZE ITA parallelogram is a with two pairs ofsides.BUILD YOUR VOCABULARY (page 147)Theorem 8-2Opposite angles of a parallelogram are congruent.Theorem 8-3Opposite sides of a parallelogram are congruent.Theorem 8-4The consecutive angles of a parallelogram aresupplementary.
  • 159. ©Glencoe/McGraw-Hill8–2Geometry: Concepts and Applications 151Page(s):Exercises:HOMEWORKASSIGNMENTFind mЄL.mЄL ϩ mЄK ϭ 180 Theorem 8-4mЄL ϩ ϭ 180 Replace mЄK with .mЄL ϩ 105 – 105 ϭ 180 – 105 Subtract.mЄL ϭIn parallelogramABCD, AB ϭ 8, BC ϭ 3, andmЄC ϭ 115.a. Find AD and CD.b. Find mЄA.c. Find mЄB.D CA B115Њ38Your TurnTheorem 8-5The diagonals of a parallelogram bisect each other.Theorem 8-6A diagonal of a parallelogram separates it into twocongruent triangles.In parallelogram PQRS,if PR ϭ 32, find PL.Theorem 8-5 states that the diagonalsof a parallelogram bisect each other.Therefore, PෆLෆ Х LෆRෆ or PL ϭ ᎏ12ᎏ(PR).PL ϭ ᎏ12ᎏ(PR)PL ϭ ᎏ12ᎏ(32) or Replace PR with 32.In parallelogramPARL, if LA ϭ 48, find LO.PALROYour TurnP QRSL
  • 160. ©Glencoe/McGraw-Hill152 Geometry: Concepts and Applications8–3In quadrilateral WXYZ, if᭝WYZ Х ᭝YWX, how couldyou prove that WXYZ is aparallelogram?Show that both pairs of opposite sides are congruent.Statement Reason1. ᭝WYX Х ᭝YWX 1. Given2. WෆXෆ Х YෆZෆ 2.3. WෆZෆ Х YෆXෆ 3. CPCTC4. WXYZ is a parallelogram. 4.In quadrilateralABCD, ЄCAB Х ЄACD andAෆBෆ Х CෆDෆ. Show that ABCD is aparallelogram by providinga reason for each step.Statement Reason1. ЄCAB Х ЄACD 1. Given2. AෆBෆ Х CෆDෆ 2. Given3. AAෆCෆ Х AෆCෆ 3.4. ᭝CAB Х ᭝ACD 4. SAS5. AෆDෆ Х BෆCෆ 5.6. ABCD is a parallelogram. 6.A BD CYour TurnW XYZTests for Parallelograms• Identify and use tests toshow that aquadrilateral is aparallelogram.WHAT YOU’LL LEARNUnder the tab forLesson 8-3, write thetests for parallelograms.Remember to includethe definition of aparallelogram. Drawpictures to accompanyeach theorem.8–28–1QuadrilateralsORGANIZE ITREVIEW ITTheorem 8-7If both pairs of opposite sides of a quadrilateral arecongruent, then the quadrilateral is a parallelogram.What does CPCTCrepresent? (Lesson 5-4)SOL G.8.b The student will prove these properties of quadrilaterals, using algebraicand coordinate methods as well as deductive reasoning.
  • 161. ©Glencoe/McGraw-Hill8–3Geometry: Concepts and Applications 153Page(s):Exercises:HOMEWORKASSIGNMENTDetermine whether each quadrilateral is aparallelogram. If the figure is a parallelogram, give areason for your answer.The figure has one pair of opposite sidesthat are and congruent.Therefore, the quadrilateral is aby Theorem 8-8.One pair of opposite sides is congruentbut . The other pair ofopposite sides is but not. Therefore, thequadrilateral a parallelogram.Determine whether the figure is aparallelogram. Justify your answer.a.b.30 m15 m15 m30 mYour TurnExplain how alternateinterior angles could beused to show thatopposite angles in aparallelogram arecongruent.WRITE ITREMEMBER ITThere is usuallymore than one way toprove a statement.Theorem 8-8If one pair of opposite sides of a quadrilateral is paralleland congruent, then the quadrilateral is a parallelogram.Theorem 8-9If the diagonals of a quadrilateral bisect each other, thenthe quadrilateral is a parallelogram.
  • 162. ©Glencoe/McGraw-Hill154 Geometry: Concepts and Applications8–4Identify the parallelogram shown.The parallelogram has foursides and right angles. It is a.Identify theparallelogram shown.Your TurnRectangles, Rhombi, and Squares• Identify and useproperties ofrectangles, rhombi, andsquares.WHAT YOU’LL LEARNUnder the tab forLesson 8-4, draw thediagram for classifyingrectangles, rhombi, andsquares. Write notes andtheorems to help youremember the mainidea.8–28–1QuadrilateralsORGANIZE ITREMEMBER ITRhombi is the pluralof rhombus.A rectangle is a with fourangles.A parallelogram with congruent sides is arhombus.A parallelogram with sides and fourangles is a square.BUILD YOUR VOCABULARY (page 147)Theorem 8-10The diagonals of a rectangle are congruent.Theorem 8-11The diagonals of a rhombus are perpendicular.Theorem 8-12Each diagonal of a rhombus bisects a pair ofopposite angles.SOL G.8.c The student will use properties of quadrilaterals to solve practicalproblems.
  • 163. ©Glencoe/McGraw-Hill8–4Geometry: Concepts and Applications 155Page(s):Exercises:HOMEWORKASSIGNMENTExplain how squares canbe rhombi, rectangles,and parallelograms.WRITE ITRefer to rhombus ABCD.Which angles are congruent to Є1?Theorem 8-12 states the diagonals of a rhombusopposite . Therefore, is congruent toЄ2, , and Є6.If mЄ7 ϭ 35, find mЄADC.Theorem 8-12 states the diagonals of a bisectangles.Therefore, mЄ7 ϭ ᎏ12ᎏ(mЄADC).ϭ ᎏ12ᎏ(mЄADC) mЄ7 ϭ .ؒ ϭ ؒ ᎏ12ᎏ(mЄADC) Multiply each side.ϭ mЄADCRefer to the figure.a. Which angles are congruent to ЄPSQ in square PQRS?b. If AP ϭ 7, find QS.S RP QAYour Turn1A BCD2 345678
  • 164. ©Glencoe/McGraw-Hill156 Geometry: Concepts and Applications8–5In trapezoid ABCD, name the bases,legs, and the base angles.Bases: AෆBෆ and areparallel segments.Legs: AෆDෆ and are nonparallel segments.Base Angles: ЄA and form one pair of base angles,while ЄC and are the other pair of baseangles.In trapezoid WXYZ, name the bases, legs, andthe base angles.Z YW XYour TurnABCDTrapezoids• Identify and useproperties of trapezoidsand isoscelestrapezoids.WHAT YOU’LL LEARNA trapezoid is a quadrilateral with exactly pair ofsides.The sides are the bases of the trapezoid.The sides of the trapezoid are known aslegs.Each trapezoid has pairs of base angles.BUILD YOUR VOCABULARY (pages 146–147)Under the tab forLesson 8-5, write thedefinition of a trapezoidand of an isoscelestrapezoid. Draw atrapezoid and label thebases, legs, and baseangles. Label thecongruent angles, anddraw the median.8–28–1QuadrilateralsORGANIZE ITSOL G.8.c The student will use properties of quadrilaterals to solve practicalproblems.
  • 165. ©Glencoe/McGraw-Hill8–5Geometry: Concepts and Applications 157Find the length of the median KLin trapezoid EFGH if EF ϭ 35 andGH ϭ 40.KL ϭ ᎏ12ᎏ(EF ϩ GH) Theorem 8-13KL ϭ ᎏ12ᎏ( ϩ ) Replace EF and GH.KL ϭ ᎏ12ᎏ( )orFind the length ofthe median NO in trapezoid JKLM ifJK ϭ 22 and LM ϭ 26.J KMN OL2226Your Turn3540H GK LE FREVIEW ITThe median of a trapezoid is the segment that joins theof the legs.Another name for the median is the midsegment.If the legs of the trapezoid are , then thetrapezoid is an isosceles trapezoid.BUILD YOUR VOCABULARYTheorem 8-13The median of a trapezoid is parallel to the bases, and thelength of the median equals one-half the sum of thelengths of the bases.Theorem 8-14Each pair of base angles in an isosceles trapezoid iscongruent.(pages 146–147)The word “isosceles” isused for classifyingtriangles and trapezoids.What similarities doisosceles triangles andisosceles trapezoidshave? (Lesson 6-4)
  • 166. ©Glencoe/McGraw-Hill8–5158 Geometry: Concepts and ApplicationsThe measure of one angle in an isosceles trapezoidis 55. Find the measures of the other three angles.Let Є1 be the given angle, and let Є2 be the base anglecongruent to Є1.Є2 Х Є1 Theorem 8-14mЄ2 ϭmЄ2 ϭ Replace mЄ1 by 55.Let Є3 and Є4 be the other pair of base angles, with Є3adjacent to Є1.mЄ3 ϩ mЄ1 ϭ Consecutive interiorangles aresupplementary.mЄ3 ϩ ϭ 180 Replace mЄ1with .mЄ3 ϩ 55 Ϫ ϭ 180 – Subtract 55 from eachside.mЄ3 ϭSince Є3 and Є4 are congruent base angles, mЄ4 ϭ .The measures of the three missing angles are , ,and .The measure of one angle in an isoscelestrapezoid is 76. Find the measures of the other three angles.Your TurnPage(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITTrapezoids andparallelograms are bothquadrilaterals, but noquadrilateral can beboth a trapezoid and aparallelogram.
  • 167. BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillBUILD YOURVOCABULARYUse your Chapter 8 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 8, go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder(pages 146–147) to helpyou solve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R8STUDY GUIDEGeometry: Concepts and Applications 1591. Name the side opposite WZ៮៮.2. Name two diagonals.3. Name the vertex opposite Z.4. Name all consecutive sides.5. Find mЄX and mЄY.Given that JKLM is a parallelogram, find the missingmeasures.6. mЄL 7. mЄJ8. LM 9. mЄK10. If the measure of one angle of parallelogram PQRS is 79, whatare the measures of the other three interior angles?8-1QuadrilateralsMLJK66Њ108-2ParallelogramsZYXW91Њ95Њ2aЊaЊ
  • 168. Chapter BRINGING IT ALL TOGETHER©Glencoe/McGraw-Hill8160 Geometry: Concepts and ApplicationsState whether each figure is a parallelogram. Justify yourreason.11.12.13. Explain why quadrilateral ABCDis a parallelogram.Underline the best term to complete the statement.14. A parallelogram with four congruent sides is a[rhombus/rectangle].Identify each figure with as many terms as possible.Indicate if no term applies.Quadrilateral Parallelogram Square Rhombus Rectangle15. 16.D CA B8-3Tests for Parallelograms8-4Rectangles, Rhombi, and Squares
  • 169. ©Glencoe/McGraw-HillChapter BRINGING IT ALL TOGETHER8Geometry: Concepts and Applications 161Complete each statement.17. The segment that joins the midpoints of each leg ofa trapezoid is the .18. A is a quadrilateral with exactly one pair ofparallel sides.19. The nonparallel sides of a trapezoid are its .20. The parallel sides of a trapezoid are its .Refer to trapezoid ABCD with medianJK៮៮. Name each of the following.21. bases22. legs23. base angle pairs24. If AB ϭ 29 and DC ϭ 23, what is JK?25. If AD ϭ 18, find JD.26. If WXYZ is an isosceles trapezoid and one base anglemeasures 66, what are the remaining angle measures?A BDJ KC29238-5Trapezoids
  • 170. I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 8 Practice Test onpage 345 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 8 Study Guide and Reviewon pages 342–344 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 8 Practice Test onpage 345.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 8 Foldable.• Then complete the Chapter 8 Study Guide and Review onpages 342–344 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 8 Practice Test onpage 345.©Glencoe/McGraw-HillVisit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 8.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R8Checklist162 Geometry: Concepts and ApplicationsARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.
  • 171. Proportions and Similarity©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R9Geometry: Concepts and Applications 163NOTE-TAKING TIP: You can design visuals such asgraphs, diagrams, pictures, charts, and conceptmaps to help you organize information so thatyou can remember what you are learning.FoldFold lengthwise tothe holes.CutCut along the top lineand then cut 10 tabs.LabelLabel each tab withimportant terms.Store the Foldablein a 3-ring binder.Begin with a sheet of notebook paper.Chapter9
  • 172. Vocabulary TermFoundDefinitionDescription oron Page Example©Glencoe/McGraw-Hill164 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 9.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R9BUILD YOUR VOCABULARYcross productsextremesgolden ratiomeanspolygon[PA-lee-gon]
  • 173. Vocabulary TermFoundDefinitionDescription oron Page Example©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY9Geometry: Concepts and Applications 165proportion[pro-POR-shun]ratio[RAY-she-oh]scale drawingscale factorsidessimilar polygons
  • 174. Using Ratios and Proportions©Glencoe/McGraw-Hill166 Geometry: Concepts and Applications9–1Label the first tab ratio.Under the tab, writethe definition and givean example.Write each ratio in simplest form.ᎏ47050ᎏϭDivide the numerator anddenominator by .24 inches to 3 feetThe units of measure must be the same in a ratio. There areinches in one foot, so 24 inches equals feet.The ratio is .Write each ratio in simplest form.a. ᎏ13699ᎏb. 30 minutes to 2ᎏ12ᎏ hoursYour Turn• Use ratios andproportions tosolve problems.WHAT YOU’LL LEARNA comparison of numbers by division is called aratio.BUILD YOUR VOCABULARY (page 165)75 Ϭ400 ϬORGANIZE IT
  • 175. ©Glencoe/McGraw-Hill9–1Geometry: Concepts and Applications 167Solve ᎏ2340ᎏ ϭ ᎏ6x3ϩ54ᎏ.ᎏ2340ᎏ ϭ ᎏ6x3ϩ54ᎏ(35) ϭ (6x ϩ 4)840 ϭ 180x ϩ 120 Distributive Property840 Ϫ ϭ 180x ϩ 120 Ϫ Subtractfrom each side.720 ϭ 180xϭ Divide each sideby .ϭ xAn equation that shows two equivalent ratios is aproportion.The cross products are the product of theand the product of the .In a proportion, the of the first ratio andthe of the second ratio are theextremes.In a proportion, the of the first ratioand the of the second ratio are themeans.BUILD YOUR VOCABULARY (pages 164–165)Label the next four tabsproportion, crossproducts, extremes, andmeans. Under each tab,write the definition andgive an example.ORGANIZE IT720 180xTheorem 9-1 Property of ProportionsFor numbers a and c and any nonzero numbers b and d, ifᎏbaᎏ ϭ ᎏdcᎏ, then ad ϭ bc. Conversely, if ad ϭ bc, then ᎏbaᎏ ϭ ᎏdcᎏ.
  • 176. REMEMBER ITThe denominatorcan never equal zero.Solve ᎏx1Ϫ51ᎏ ϭ ᎏ45ᎏ.The ratio of children to adults at a holiday parade is2.5 to 1. If there are 1440 adults at the parade, howmany children are there?2.5(1440) ϭ Cross productsϭ xThe ratio of Republicans to Democrats castingtheir votes in the local election was 73 to 27. If 135 Democratsvoted, how many Republicans cast their votes?Your TurnYour Turn©Glencoe/McGraw-Hill9–1168 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTchildren 2.5 x1 1440ϭadults
  • 177. ©Glencoe/McGraw-HillSimilar Polygons9–2Geometry: Concepts and Applications 169Determine if the polygons are similar. Justifyyour answer.The polygons are . The corresponding angles arecongruent andDetermine if thepolygons are similar. Justify youranswer.Your Turn33 352 24.57.5• Identify similarpolygons.WHAT YOU’LL LEARNA polygon is a figure in a plane formed bysegments called sides.Similar polygons are the same but notnecessarily the same .BUILD YOUR VOCABULARY (pages 164–165)2ϭ23ϭ3ϭ7.5.40 4050 4030 2040 40SOL G.14.a The student will use proportional reasoning to solve practical problems,given similar geometric objects.Label the next threetabs polygon, sides,and similar polygons.Under each tab, writethe definition and givean example.Similar Polygons Twopolygons are similarif and only if theircorresponding anglesare congruent and themeasures of theircorresponding sidesare proportional.KEY CONCEPT
  • 178. ©Glencoe/McGraw-Hill9–2170 Geometry: Concepts and ApplicationsFind the values of x and y if ᭝ABC ϳ ᭝FED.Use the corresponding order of the vertices to write proportions.Definition of similar polygonsSubstitutionWrite the proportion to solve for x.ϭ (8)ϭᎏ1122ᎏx ϭ ᎏ11220ᎏ Divide each side by .x ϭNow write the proportion that can be solved for y.(12) ϭ y(8) Cross products60 ϭ 8yᎏ680ᎏ ϭ ᎏ88yᎏ Divide each side by .ϭ ySo, x ϭ and y ϭ .AB581215CxyD EFϭBCϭABFE FDϭ8ϭ5yxϭxϭ51215y8
  • 179. The triangles aresimilar. Find the values of x and y.In the blueprint, 1 inchrepresents an actuallength of 16 feet. Usethe blueprint to findthe actual dimensionsof the dining room.(y) ϭ 16΂ᎏ34ᎏ΃ Cross productsy ϭThe dimensions of the dining room are ft by ft.Refer to Example 3. Find the dimensions ofthe kitchen.Your TurnYour Turn815176xy©Glencoe/McGraw-Hill9–2Geometry: Concepts and Applications 171Scale drawings are used to represent something either tooor too to be drawn at its actual size.BUILD YOUR VOCABULARY (page 165)1 in.DININGROOM1 in.141 in.121 in.14in.34in.12KITCHENUTILITYROOMGARAGELIVING ROOMϭx ftin.blueprintactualblueprintactual1 in.16 ftx ϭϭy ftin.1 in.16 ftPage(s):Exercises:HOMEWORKASSIGNMENTLabel the next tab scaledrawings. Under thetab, write the definitionand give an example.ORGANIZE IT
  • 180. Similar TrianglesDetermine whether thetriangles are similar, If so, tellwhich similarity test is usedand write a similaritystatement.Since mЄF ϭ and mЄH ϭ ,᭝FGH ϳ ᭝MLK by .Determine whether the triangles are similar,If so, tell which similarity test is used and write a similaritystatement.Your Turn ©Glencoe/McGraw-Hill172 Geometry: Concepts and Applications9–3• Use AA, SSS, and SASsimilarity tests fortriangles.WHAT YOU’LL LEARNPostulate 9-1 AA SimilarityIf two angles of one triangle are congruent to twocorresponding angles of another triangle, then the twotriangles are similar.Theorem 9-2 SSS SimilarityIf the measures of the sides of a triangle are proportionalto the measures of the corresponding sides of anothertriangle, then the triangles are congruent.Theorem 9-3 SAS SimilarityIf the measures of two sides of a triangle are proportionalto the measures of two corresponding sides of anothertriangle and their included angles are congruent, then thetriangles are similar.37Њ37Њ43Њ43ЊKLMHFGWhy must only two pairsof corresponding anglesbe congruent for twotriangles to be similarrather than three?WRITE ITSOL G.5.b The student will prove two triangles are congruent or similar, giveninformation in the form of a figure or statement, using algebraic and coordinateas well as deductive proofs.
  • 181. ©Glencoe/McGraw-Hill9–3Geometry: Concepts and Applications 173Find the value of x.Since ᎏ182ᎏ ϭ ᎏ1128ᎏ, the triangles aresimilar by SAS similarity.ᎏ182ᎏ ϭ ᎏ1x4ᎏ Definition of similar polygonsx ϭ (12)(14) Cross products8x ϭ 168Divide each side by .x ϭFind the value of x.x2175Your Turn812121814x8x 168ϭ
  • 182. The shadow of a flagpole is 2 meters long at the sametime that a person’s shadow is 0.4 meters long. If theperson is 1.5 meters tall, how tall is the flagpole?x ϭ (2) Cross products0.4x ϭDivide.x ϭThe flagpole is meters tall.A diseased tree must be cut down before it falls.Which direction the fall is directed depends on the height ofthe tree. The man who will cut the tree down is 74-in. tall andcasts a shadow 60-in. long. If the tree’s shadow measures20 feet from its base, how tall is the tree?Your Turn©Glencoe/McGraw-Hill9–3174 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTflagpole’s shadowperson’s shadow20.40.4x 3ϭflagpole’s heightperson’s heightϭx
  • 183. Using the figure, complete theproportion ᎏV?Wᎏ ϭ ᎏSSWTᎏ.Since VෆWෆ ʈ RෆTෆ, ᭝SVW ϳ ᭝SRT.Therefore,Use the figure to completethe proportion ᎏXYYAᎏ ϭ ᎏY?Bᎏ.In the figure, MN៮៮ ʈ KL៮៮. Find the value of x.᭝JMN ϳ ᭝JKLDefinition ofsimilar polygonsSubstitution9x ϭ (6) Cross products9x ϭx ϭ Divide each side by 9.XZBAYYour Turnϭ6x©Glencoe/McGraw-HillProportional Parts and Triangles9–4Geometry: Concepts and Applications 175• Identify and use therelationships betweenproportional parts oftriangles.WHAT YOU’LL LEARNTheorem 9-4If a line is parallel to one side of a triangle and intersectsthe other two sides, then the triangle formed is similar tothe original triangle.S TWVRSTSWϭ .VW1263JM NLKxϭMNKL JLJNSOL G.5.a The student will investigate and identify congruence and similarityrelationships between triangles.
  • 184. Find the value of b.In the figure, AB៮៮ ʈ DE៮៮.Find the value of x.Theorem 9.5CE ϭ x, EB ϭ 6,CD ϭ x ϩ 5, DA ϭ 8x(8) ϭ (x ϩ 5) Cross products8x ϭ 6x ϩ Distributive Property8x Ϫ 6x ϭ 6x ϩ 30 Ϫ 6x Subtract 6x from each side.2x ϭ 30ᎏ22xᎏ ϭ ᎏ320ᎏ Divide each side by 2.x ϭFind the value of a.TSAR108 2a Ϫ 3aWYour TurnD EHFG25156bYour Turn©Glencoe/McGraw-Hill9–4176 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTTheorem 9-5If a line is parallel to one side of a triangle and intersectsthe other two sides, then it separates the sides intosegments of proportional lengths.ϭCEEBCDϭx ϩx68AB CDE xx + 5
  • 185. ©Glencoe/McGraw-Hill9–5Geometry: Concepts and Applications 177Determine whether DE៮៮ ʈ BC៮៮.Determine whether ᎏBDDAᎏ andᎏCEEAᎏ form a proportion.՘(8) ՘ (4) Cross products24 ϭ ✔Therefore, DෆEෆ ʈ BෆCෆ by Theorem 9-6.Determine whether HෆJෆ ʈ KෆMෆ.23x6KMHJLxYour TurnADB CE6384Triangles and Parallel Lines• Use proportions todetermine whetherlines are parallel tosides of triangles.WHAT YOU’LL LEARNBD6 8Theorem 9-6If a line intersects two sides of a triangle and separates thesides into corresponding segments of proportional lengths,then the line is parallel to the third side.CEEA՘SOL G.5.a The student will investigate and identify congruence and similarityrelationships between triangles.
  • 186. ©Glencoe/McGraw-Hill178 Geometry: Concepts and ApplicationsFor Examples 2 and 3, refer to thefigure shown.In the figure, X, Y, and Z aremidpoints of the sides of ᭝UVW.If XZ ϭ 7c, then what does UW equal?XZ ϭ ᎏ12ᎏ Theorem 9-7ϭ ᎏ12ᎏUW Replace XZ with .7c ϭ ΂ᎏ12ᎏUW΃ Multiply each side by .ϭ UWIn the figure, if mЄUYX ϭ d, then what is mЄYWZ?By Theorem 9-6, XෆYෆ ʈ VෆWෆ. Since XෆYෆ and VෆWෆ are parallelsegments cut by transversal UෆWෆ, ЄUYX and arecongruent angles.Therefore, mЄYWZ ϭ .N, O, and Pare the midpoints of thesides of ᭝EFG.a. If EF ϭ 25, then what does NP equal?b. If mЄEGF ϭ 85, then what is mЄENO?Your TurnV WZUYXEOFPNG9–5Theorem 9-7If a segment joins the midpoint of two sides of a triangle,then it is parallel to the third side, and its measure equalsone-half the measure of the third side.REMEMBER ITThe midsegment’sendpoints are themidpoints of the legs oftwo sides of a triangle.Page(s):Exercises:HOMEWORKASSIGNMENT
  • 187. ©Glencoe/McGraw-Hill9–6Geometry: Concepts and Applications 179Complete the proportion ᎏRSTTᎏ ‫؍‬ ᎏN?Pᎏ.Since ʈ NS៭៮៬ ʈ PT៭៮៬, the transversals are divided. Therefore, ᎏRSTTᎏ ϭ .In the figure, a ʈ b ʈ c.Find the value of x.ᎏSTRSᎏ ϭᎏ195ᎏ ϭ TS ϭ 9, SR ϭ 15,PN ϭ , NM ϭ x9(x) ϭ 15΂ ΃ Cross products9x ϭx ϭ Divide each side by .R Sa b cMNx1215 9PQTProportional Parts and Parallel Lines• Identify and use therelationships betweenparallel lines andproportional parts.WHAT YOU’LL LEARNTheorem 9-8If three or more parallel lines intersect two transversals, thelines divide the transversals proportionally.NPREVIEW ITWhat is the definitionof a transversal?(Lesson 4-2)NMxSOL G.14.a The student will use proportional reasoning to solve practical problems,given similar geometric objects.MRS TN P
  • 188. ©Glencoe/McGraw-Hill180 Geometry: Concepts and Applicationsa. Complete the proportion ᎏQZPPᎏ ϭ ᎏN?Bᎏ.b. In the figure, a ʈ b ʈ c. Find the value of x.188.5 6DE KJFxLacbP BZ NQ AYour Turn9–6Page(s):Exercises:HOMEWORKASSIGNMENTREVIEW ITExplain how to constructparallel lines.(Lesson 4-4)Theorem 9-9If three or more parallel lines cut off congruent segmentson one transversal, then they cut off congruent segmentson every transversal.
  • 189. ©Glencoe/McGraw-Hill9–7Geometry: Concepts and Applications 181The perimeter of ᭝DEF is 90 units, and ᭝ABC ϳ ᭝DEF.Find the value of each variable.ᎏDABEᎏ ϭ Theorem 9-10ᎏ2x6ᎏ ϭ ᎏ9600ᎏ 26 ϩ 10 ϩ 24 ϭ(60) ϭ (90) Cross products60x ϭ 2340 Divide.x ϭBecause the triangles are similar, find y and z.ᎏDDEFᎏ ϭ ᎏAACBᎏᎏ3y9ᎏ ϭ26y ϭ 390y ϭ2624yB E FDCA xz10Perimeters and Similarity• Identify and useproportionalrelationships ofsimilar triangles.WHAT YOU’LL LEARNTheorem 9-10If two triangles are similar, then the measures of thecorresponding perimeters are proportional to the measuresof the corresponding sides.perimeter of ᭝DEFperimeter of ᭝ABC• Write each vocabularyword from the lesson.• Explain its meaning inyour own words.• Use diagrams toclarify.ORGANIZE IT26ᎏDEFEᎏ ϭ ᎏBABCᎏϭ ᎏ2246ᎏ26z ϭz ϭzSOL G.14.a The student will use proportional reasoning to solve practical problems,given similar geometric objects.
  • 190. ©Glencoe/McGraw-Hill182 Geometry: Concepts and ApplicationsDetermine the scale factorof ᭝ABC to ᭝DEF.ᎏDABEᎏ ϭ ϭᎏBECFᎏ ϭ ϭᎏDACFᎏ ϭ ϭ The scale factor is .Determine the scale factor of ᭝RST to ᭝XYZ.ZYX162012.8 TSR453.2Your TurnB C E FDA24 243620 30169–7The scale factor, also known as the constant of, is the found bycomparing the measures of corresponding sides ofsimilar triangles.BUILD YOUR VOCABULARY (page 165)Label the next tab scalefactor. Under the tab,write the definition andgive an example.ORGANIZE IT324302324Page(s):Exercises:HOMEWORKASSIGNMENTThe perimeter of ᭝ABC is 20 units, and᭝ABC ϳ ᭝XYZ. Find the value of each variable.61014Z YdgfCABXYour Turn
  • 191. BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillC H A P T E R9STUDY GUIDEGeometry: Concepts and Applications 183Indicate whether the statement is true or false.1. Every proportion has two cross products.2. A ratio is a comparison of two numbers by division.3. The two cross products of a ratio are the extremesand the means.4. Cross products are always equal in a proportion.5. Simplify ᎏ27200ᎏ. 6. Solve: ᎏ8643ᎏ ϭ ᎏ111Ϫ2xᎏComplete the sentence.7. In measures of corresponding sides areproportional, and corresponding angles are congruent.8. represent something either too large ortoo small to be drawn at actual size.BUILD YOURVOCABULARYUse your Chapter 9 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 9, go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder(pages 164–165)to help you solve the puzzle.VOCABULARYPUZZLEMAKER9-1Using Ratios and Proportions9-2Similar Polygons
  • 192. Chapter BRINGING IT ALL TOGETHER©Glencoe/McGraw-Hill9184 Geometry: Concepts and Applications9. Given that the rectangles are similar, find the values of x andy to show similarity.Determine whether the pair of triangles is similar. Justifyyour reasons.10. 11.Complete the proportions.12. ᎏDADEᎏ ϭ13. ᎏEAEBᎏ ϭVertices A, B, and C are midpoints.14. AෆCෆ ʈ .15. If BC ϭ 6, then RT ϭ .16. If SB ϭ 4, AC ϭ .SCABTRBDECA3926x ϩ 7y Ϫ 43 35 515159-3Similar Triangles9-4Proportional Parts and TrianglesADCB9-5Triangles and Parallel Lines
  • 193. ©Glencoe/McGraw-HillChapter BRINGING IT ALL TOGETHER9Geometry: Concepts and Applications 185A tract of land bordering school property was divided intosections for five biology classes to plant gardens. The fencesseparating the plots are parallel, and the plots’ frontmeasures are shown. The entire back border measures254 feet. What are the individual border lengths, to thenearest tenth of a foot?17. A ϭ 18. D ϭ19. E ϭComplete the sentence.20. The scale factor is also called the constant of.21. Find the scale factor.᭝JKL ϳ ᭝MNO. The perimeter of ᭝JKL is 54. What are thevalues for the variables?22. a ϭ23. b ϭ24. c ϭCBA1510ZYX2416CBA20 ft.22 ft.frontbackED28 ft. 16 ft.25 ft.9-6Proportional Parts and Parallel Lines9-7Perimeters and SimilarityLKJ15a bcONM129
  • 194. I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 9 Practice Test onpage 397 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 9 Study Guide and Reviewon pages 394–396 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 9 Practice Test onpage 397.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 9 Foldable.• Then complete the Chapter 9 Study Guide and Review onpages 394–396 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 9 Practice Test onpage 397.©Glencoe/McGraw-HillC H A P T E R9Checklist186 Geometry: Concepts and ApplicationsCheck the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 9.Student Signature Parent/Guardian SignatureTeacher Signature
  • 195. Polygons and Area©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notesC H A P T E R10Geometry: Concepts and Applications 187NOTE-TAKING TIP: When you take notes, it isimportant to record major concepts and ideas.Refer to your journal when reviewing for tests.FoldFold the short side infourths.DrawDraw lines along the foldsand label each columnPrefix, Number of Sides,Polygon Name, and Figure.Begin with a sheet of 8ᎏ12ᎏ" ϫ 11" paper.Prefix Numberof SidesPolygonNameFigureChapter10
  • 196. ©Glencoe/McGraw-Hill188 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 10.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R10BUILD YOUR VOCABULARYVocabulary TermFoundDefinitionDescription oron Page Examplealtitudeapothem[a-pa-thum]centercomposite figure[kahm-PA-sit]concaveconvexirregular figureline of symmetry[SIH-muh-tree]
  • 197. ©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY10Geometry: Concepts and Applications 189Vocabulary TermFoundDefinitionDescription oron Page Exampleline symmetrypolygonal regionregular polygonregular tessellationrotational symmetrysemi-regular tessellationsignificant digitssymmetrytessellation[tes-a-LAY-shun]turn symmetry
  • 198. Naming PolygonsRefer to the figure for Examples 1–2.a. Identify polygon VWXYZ.The polygon has sides. It is a .b. Determine whether the polygon VWXYZ appears tobe regular or not regular. If not regular, explain why.The appear to be the same length, and theappear to have the same measure. The polygon is regular.Name two nonconsecutive vertices of polygon VWXYZ.W and Z, W and Y, V and X, V and Y, X and Z are examplesof vertices.Refer to the figure forparts a, b, and c.a. Identify polygon DEFGHIJ by its sides.b. Determine whether the polygon DEFGHIJ appears to beregular or not regular. If not regular, explain why.c. Name two nonconsecutive vertices of polygon DEFGHIJ.Your Turn©Glencoe/McGraw-Hill190 Geometry: Concepts and Applications10–1• Name polygonsaccording to the numberof sides and angles.WHAT YOU’LL LEARNA regular polygon has all congruent and allcongruent.BUILD YOUR VOCABULARY (page 189)Under the tabs labeledPrefix, Number of Sides,and Polygon Name,write the informationgiven in the table onpage 402. Under the tablabeled Figure, draw apicture of each polygon.Include regular andirregular polygons, aswell as convex andconcave polygons.Prefix Numberof SidesPolygonNameFigureORGANIZE ITD HE GJ IFZ YXVW
  • 199. Classify each polygon as convex or concave.a. When all the diagonals are drawn,points lie outside of thepolygon. So polygon ABCDEFis .b. Diagonal QෆSෆ lies outside the polygon,so PQRSTU is .Classify each polygon as convex or concave.a. b.YZXUVKJIHL MYour TurnQ R STP UA BCFE D©Glencoe/McGraw-Hill10–1Geometry: Concepts and Applications 191Page(s):Exercises:HOMEWORKASSIGNMENTAll of the diagonals of a convex polygon lie in theof the polygon.If any part of a diagonal lies of the polygon,the polygon is concave.BUILD YOUR VOCABULARY (page 188)REMEMBER ITMost polygons havemore than one diagonal.As the number of sidesincreases, so does thenumber of diagonals.
  • 200. Diagonals and Angle MeasureRefer to the regular pentagon for Examples 1–2.Find the sum of the measures of the interior angles.Sum of measures of interior anglesϭ (n Ϫ 2)180 Theorem 10-1ϭ ( Ϫ 2)180 Substitutionϭ ( )180ϭThe sum of the measures of the interior angles of a pentagonis .Find the measure of one interior angle.Each interior angle of a regular polygon has the same measure.Divide the of the measures by theof angles.measure of one interior angle ϭ orThe measure of one interior angle of a regular pentagon is.©Glencoe/McGraw-Hill192 Geometry: Concepts and Applications10–2• Find measures ofinterior and exteriorangles of polygons.WHAT YOU’LL LEARNOn the back of yourFoldable, you may wishto write the interiorangle sum for each ofthe different polygonslisted on your Foldable.Prefix Numberof SidesPolygonNameFigureORGANIZE ITTheorem 10-1If a convex polygon has n sides, then the sum of themeasures of the interior angles is (n Ϫ 2)180.N MLJKSOL G.9 The student will use measures of interior and exterior angles of polygonsto solve problems. Tessellations and tiling problems will be used to make connectionsto art, construction, and nature.
  • 201. a. Find the sum of the measures of the interior angles of aregular 15-sided polygon.b. Find the measure of one interior angle of a regular15-sided polygon.Your Turn©Glencoe/McGraw-Hill10–2Geometry: Concepts and Applications 193Page(s):Exercises:HOMEWORKASSIGNMENTTheorem 10-2In any convex polygon, the sum of the measures of theexterior angles, one at each vertex, is 360.Find the measure of one exterior angle of a regularoctagon.By Theorem 10-2, the sum of the measures of exterior anglesis . An octagon has exterior angles.measure of one exterior angle ϭ ᎏ3680ᎏ ϭFind the measure of one exterior angle of aregular 15-sided polygon.Your TurnREMEMBER ITTheorems 10-1 and10-2 only apply toconvex polygons.How do you find themeasure of an interiorangle of an n-sidedregular polygon?WRITE IT
  • 202. Areas of PolygonsFind the area of the polygon.Each square represents1 square centimeter.Since the area of each square represents one squarecentimeter, the area of each triangular half square represents0.5 square centimeter. There are 8 squares and 4 half squares.A ϭ 8(1) cm2ϩ 4(0.5) cm2A ϭ cm2ϩ cm2A ϭ cm2Find the area of the polygon. Eachsquare represents 1 square inch.Your Turn©Glencoe/McGraw-Hill194 Geometry: Concepts and Applications10–3• Estimate the areas ofpolygons.WHAT YOU’LL LEARNPostulate 10-1 Area PostulateFor any polygon and a given unit of measure, there is aunique number A called the measure of the area of thepolygon.Postulate 10-2Congruent polygons have equal areas.Postulate 10-3 Area Addition PostulateThe area of a given polygon equals the sum of the areas ofthe nonoverlapping polygons that form the given polygon.Any polygon and its are called a polygonalregion.A composite figure is a figure made fromthat have been placed together.BUILD YOUR VOCABULARY (pages 188–189)What formulas for areahave you learned?(Lesson 1-6)REVIEW IT
  • 203. Estimate the area of the polygon.Each square represents 20 squaremiles.Count each square as one unit and eachpartial square as a half unit regardlessof size. There are whole squaresand partial squares.number of squares ϭ (1) ϩ (0.5)ϭ ϩϭArea Ϸ 20 ϫ Each square represents20 square miles.ϭThe area of the polygon is about square miles, or.A swimming pool ata resort is shaped as shown on thegrid. Each square on the gridrepresents 16 square meters.Estimate the area of the pool.Your Turn©Glencoe/McGraw-Hill10–3Geometry: Concepts and Applications 195Page(s):Exercises:HOMEWORKASSIGNMENTHow can you determinethe area of a polygon bydividing it into familiarshapes?WRITE ITIrregular figures are not polygons and cannot be madefrom combinations of polygons. Their areas can beapproximated using combinations of polygons.BUILD YOUR VOCABULARY (page 188)
  • 204. Areas of Triangles and TrapezoidsFind the area of each triangle.A ϭ ᎏ12ᎏbh Theorem 10-3ϭ ᎏ12ᎏ΂ ΃΂ ΃ Replace b with and h with .ϭ ᎏ12ᎏ΂ ΃ϭA ϭ ᎏ12ᎏbh Theorem 10-3ϭ ᎏ12ᎏ΂ ΃΂ ΃ Replace b with and h with .ϭ ᎏ12ᎏ΂ ΃ϭFind the area of each triangle.a.114Your Turn8 cm9 cm6 ft15 ft©Glencoe/McGraw-Hill196 Geometry: Concepts and Applications10–4• Find the areas oftriangles andtrapezoids.WHAT YOU’LL LEARNWhat is an altitude of atriangle? (Lesson 6-2)REVIEW ITTheorem 10-3 Area of a TriangleIf a triangle has an area of A square units, a base of b units,and a corresponding altitude of h units, then A ϭ ᎏ12ᎏbh.
  • 205. b.c.5 ft342 ft125 m12 m©Glencoe/McGraw-Hill10–4Geometry: Concepts and Applications 197Page(s):Exercises:HOMEWORKASSIGNMENTDraw and label the baseand altitude for eachtriangle on yourFoldable. In addition,draw a trapezoidas an example of aquadrilateral. Drawand label the basesand altitude for thetrapezoid.Prefix Numberof SidesPolygonNameFigureORGANIZE IT The altitude of a trapezoid is a segment perpendicular toeach .BUILD YOUR VOCABULARY (page 188)Theorem 10-4 Area of a TrapezoidIf a trapezoid has an area of A square units, bases of b1 andb2 units, and an altitude of h units, then A ϭ ᎏ12ᎏh(b1 ϩ b2).Find the area of the trapezoid.A ϭ ᎏ12ᎏh(b1 ϩ b2) Theorem 10-4A ϭ ᎏ12ᎏ΂ ΃΂ ϩ ΃ Replace h with 6, b1 with 4,and b2 with 18.A ϭ ᎏ12ᎏ΂ ΃΂ ΃A ϭ ΂ ΃΂ ΃ orFind the area of the trapezoid.1.5 cm2 cm4.5 cmYour Turn6 m4 m18 m
  • 206. Areas of Regular PolygonsA regular octagon has a side lengthof 9 inches and an apothem that isabout 10.9 inches long. Find the areaof the octagon.First, find the perimeter of the octagon.P ϭ 8s All sides of a regular octagon are congruent.ϭ 8(9) or 72 Replace s with 9.Now find the area.A ϭ ᎏ12ᎏaP Theorem 10-5ϭ ᎏ12ᎏ(10.9)(72) or Replace a with 10.9 andP with 72.The area of the octagon is about in2.A regular pentagon has a sidelength of 8 inches and an apothem that is about5.5 inches long. Find the area of the pentagon.Your Turn©Glencoe/McGraw-Hill198 Geometry: Concepts and Applications10–5• Find the areas ofregular polygons.WHAT YOU’LL LEARNREMEMBER ITOnly regular polygonshave apothems.The center of a regular polygon is an interior point that isequidistant from all .The segment drawn from the center andto a side of a regular polygon is an apothem.BUILD YOUR VOCABULARY (page 188)Theorem 10-5 Area of a Regular PolygonIf a regular polygon has an area of A square units, anapothem of a units, and a perimeter of P units, thenA ϭ ᎏ12ᎏaP.Draw an apothem foreach of the regularpolygons drawn onyour Foldable.Prefix Numberof SidesPolygonNameFigureORGANIZE IT9 in.10.9 in.5.5 in.8 in.
  • 207. A regular octagon has a side lengthof 12 inches and an apothem that isabout 14.5 inches long. Find the areaof the shaded region of the octagon.Find the area of the octagon minus thearea of the unshaded region.Area of an octagon:A ϭ ᎏ12ᎏaP Theorem 10-5ϭ ᎏ12ᎏ΂ ΃΂ ΃ Replace a with 14.5 andP with 96.ϭ in2Area of a Triangle:A ϭ ᎏ12ᎏbh Theorem 10-3ϭ ᎏ12ᎏ(12)(14.5) Replace b with 12 andh with 14.5.ϭ in2The area of one triangular section is 87 in2. There are5 triangular sections in the unshaded region.The area of the unshaded region is 5΂ ΃ ϭ in2.Subtract the area of the unshaded region from the area of theoctagon.Area of shaded region ϭ Ϫ or in2Find the area of theshaded region of the regular hexagon.17.3 cm20 cmYour Turn©Glencoe/McGraw-Hill10–5Geometry: Concepts and Applications 199Page(s):Exercises:HOMEWORKASSIGNMENT12 in.14.5 in.Significant digits represent the precision of a.BUILD YOUR VOCABULARY (page 189)
  • 208. SymmetryFind all lines of symmetry for equilateral triangle ABC.Fold along all possible lines to see if the sides match. Thereare lines of symmetry along the lines shown inthe figure.Draw all lines of symmetry for regular pentagonJKXYZ.JKY XZJKZY XYour TurnAB C©Glencoe/McGraw-Hill200 Geometry: Concepts and Applications10–6• Identify figures withline symmetry androtational symmetry.WHAT YOU’LL LEARNREMEMBER ITFigures that haverotational symmetry donot necessarily haveline symmetry.Symmetry is when a figure has balanced proportions acrossa reference , line, or plane.When a line is drawn through the of a figureand one half is the image of the other, thefigure is said to have line symmetry.The reference line is known as the line of symmetry.BUILD YOUR VOCABULARY (pages 188–189)SOL G.2.b The student will use pictorial representations, . . ., to solve problemsinvolving symmetry and transformation. This will include investigating symmetry anddetermining whether a figure is symmetric with respect to a line or a point.
  • 209. Which of the figures have rotational symmetry?a.The figure can be turned 120° and 240° to look like theoriginal. The figure has symmetry.b.The figure must be turned 360° about its center to look likethe original. Therefore, it have rotationalsymmetry.Which of the figures has rotationalsymmetry?a.b.Your Turn©Glencoe/McGraw-Hill10–6Geometry: Concepts and Applications 201Page(s):Exercises:HOMEWORKASSIGNMENTA figure that can be turned or rotated less than 360ºabout a fixed point and that looks exactly as it does inthe is said to have turn symmetryor rotational symmetry.BUILD YOUR VOCABULARY (page 189)Draw a polygon that hasline symmetry but doesnot have rotationalsymmetry. Do you thinkit is possible to drawa figure with more than1 line of symmetry, butthat does not haverotational symmetry?Explain.WRITE IT
  • 210. TessellationsIdentify the figures used to create each tessellation.Then identify the tessellation as regular, semi-regular,or neither.Only squares are used. A square is a regularpolygon. The tessellation is .Hexagons are used and there are no gaps in thepattern, but the hexagons are not .The tessellation is a regular nor asemi-regular tessellation.Identify the tessellation as regular, semi-regular, or neither.a. b.Your Turn©Glencoe/McGraw-Hill202 Geometry: Concepts and Applications10–7• Identify tessellationsand create them byusing transformations.WHAT YOU’LL LEARNREMEMBER ITRegular and semi-regular tessellationsare created using onlyregular polygons.Tessellations are tiled patterns created byfigures to fill a plane without gaps or overlaps. They can bemade by translating, rotating, or reflecting polygons.A pattern is a regular tessellation when onlytype of regular polygon is used to form the pattern.When two or more regular polygons are used in thesame order at every vertex to form a pattern, it is asemi-regular tessellation.BUILD YOUR VOCABULARY (page 189)Page(s):Exercises:HOMEWORKASSIGNMENTSOL G.9 The student will use measures of interior and exterior angles of polygonsto solve problems. Tessellations and tiling problems will be used to make connectionsto art, construction, and nature.
  • 211. BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillIndicate whether the statement is true or false.1. All the diagonals of a concave polygon lieon the interior.2. A regular polygon is both equilateral andequiangular.Identify each figure by its sides. Indicate if the polygon appearsto be regular or not regular. If not regular, justify your reason.3. 4.Find the sum of the measures of the interior angles.5. 6.BUILD YOURVOCABULARYUse your Chapter 10 Foldableto help you study for yourchapter test.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 10, go to:www.glencoe.com/sec/math/t_resources/free/index.php.You can use your completedVocabulary Builder(pages 188–189) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R10STUDY GUIDEGeometry: Concepts and Applications 20310-1Naming Polygons10-2Diagonals and Angle Measure
  • 212. Chapter BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillFind the measure of one interior angle and oneexterior angle of the regular polygon.7. dodecagon 8. decagon9. The sum of the measures of four exterior angles of a pentagonis 280. What is the measure of the fifth exterior angle?Indicate whether the statement is true or false.10. A polygon and its interior are known as apolygonal region.Find the area of the polygon in square units.11. 12.Indicate whether the statement is true or false.13. The segment perpendicular to the parallelbases of a trapezoid is a median.Find the area of the triangle or trapezoid.14. 15.91825512.210204 Geometry: Concepts and Applications10-3Areas of Polygons10-4Areas of Triangles and Trapezoids
  • 213. ©Glencoe/McGraw-Hill16. Find the area of a trapezoid whosealtitude measures 4.5 cm and hasbases measuring 6.2 and 8.8 cm.17. What is the area of a triangle withbase length 6ᎏ13ᎏ in. and height 2 in.?18. Find the area of a regular 11-sided polygon with each sidemeasuring 7 cm and an apothem length of 11.9 cm.19. Find the area of the shaded region.Underline the best term to make the statement true.20. When a line is drawn through a figure and makes each half a mirrorimage of the other, the figure has [line/rotational] symmetry.21. When a figure looks exactly as it does in its original position after beingturned less than 360º around a fixed point, it has [line/rotational] symmetry.Determine whether the figure has line symmetry, rotational symmetry,both, or neither.22. 23.Identify the tessellation as regular, semi-regular, or neither.24. 25.65.2Chapter BRINGING IT ALL TOGETHER10Geometry: Concepts and Applications 20510-5Areas of Regular Polygons10-6Symmetry10-7Tessellations
  • 214. ©Glencoe/McGraw-HillCheck the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 10 Practice Test onpage 449 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 10 Study Guide and Reviewon pages 446–448 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 10 Practice Test onpage 449 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 10 Foldable.• Then complete the Chapter 10 Study Guide and Review onpages 446–448 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 10 Practice Test onpage 449 of your textbook.Visit geomconcepts.com toaccess your textbook, moreexamples, self-check quizzes,and practice tests to helpyou study the concepts inChapter 10.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R10Checklist206 Geometry: Concepts and Applications
  • 215. Circles©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R11Geometry: Concepts and Applications 207NOTE-TAKING TIP: When you take notes, writeconcise definitions in your own words. Addexamples that illustrate the concepts.Begin with seven sheets of plain paper.Chapter11DrawDraw and cut a circle fromeach sheet. Use a small plateor a CD to outline the circle.StapleStaple the circles together toform a booklet.LabelLabel the chapter name onthe front. Label the inside sixpages with the lesson titles.Circles
  • 216. Vocabulary TermFoundDefinitionDescription oron Page Exampleadjacent arcsarcscentercentral anglechordcirclecircumference[sir-KUM-fur-ents]circumscribedconcentricdiameter©Glencoe/McGraw-Hill208 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 11.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R11BUILD YOUR VOCABULARY
  • 217. Vocabulary TermFoundDefinitionDescription oron Page Example©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY11Geometry: Concepts and Applications 209experimental probability[ek-speer-uh-MEN-tul]inscribedlocilocusmajor arcminor arcpi (␲)radius[RAY-dee-us]sectorsemicircletheoretical probability[thee-uh-RET-i-kul]
  • 218. Parts of a Circle©Glencoe/McGraw-Hill210 Geometry: Concepts and Applications11–1Use circle P to determine whether eachstatement is true or false.RT៮៮ is a diameter of circle P.; RෆTෆ go through thecenter P. Therefore, RෆTෆ is not a diameter.PෆSෆ is a radius of circle P.; the endpoints of PෆSෆ are on the P anda point on the circle S. Therefore, PෆSෆ is a radius.PQRST• Identify and use partsof circles.WHAT YOU’LL LEARNUnder the tab forLesson 11-1, draw acircle with a radius, achord and a diameter.Label each specialsegment.CirclesORGANIZE ITA circle is the set of all points in a plane that are a givendistance from a given point in the plane, called theof the circle.In a circle, all points are from the center.A radius is a segment whose endpoints are theof the circle and a on the circle.A chord is a segment whose are on thecircle.A diameter is a that contains theof the circle.Two circles are concentric if they lie in the same plane,have the same , and have ofdifferent lengths.BUILD YOUR VOCABULARY (pages 208–209)SOL G.10 The student will investigate and solve practical problems involving circles,using properties of . . . , chords, . . . . Problems will include finding arc length and thearea of a sector, and may be drawn from applications of architecture, art, andconstruction.
  • 219. ©Glencoe/McGraw-Hill11–1Geometry: Concepts and Applications 211Page(s):Exercises:HOMEWORKASSIGNMENTUse circle T to determine whether eachstatement is true or false.a. AෆBෆ is not a diameter.b. TෆDෆ is not a radius.In circle R, QT៮៮ is a diameter.If QR ϭ 7, find QT.QෆRෆ is a radius, and d ϭ 2r.QT ϭ 2(QR) Replace d and r.QT ϭ 2΂ ΃ Replace QR with .QT ϭIn circle A, FෆCෆ is a diameter. If FC ϭ 25, find AB.FBDCAEYour TurnYour TurnTABCEDTheorem 11-1All radii of a circle are congruent.Theorem 11-2The measure of the diameter d of a circle is twice themeasure of the radius r of the circle.RQ7STDescribe the differencesbetween a radius, adiameter, and a chord.WRITE IT
  • 220. Arcs and Central Angles©Glencoe/McGraw-Hill212 Geometry: Concepts and Applications11–2In circle J, find mLM២, mЄKJM,and mLK២.mLM២ϭ mЄLJM Measure of minor arcmLM២ϭ 125mЄKJM ϭ mKM២Measure of central anglemЄKJM ϭmLK២ϭ 360 Ϫ mЄLJM Ϫ mЄKJM Measure of major arcmLK២ϭ 360 Ϫ 125 Ϫ 130 SubstitutionmLK២ϭ• Identify major arcs,minor arcs, andsemicircles and find themeasures of arcs andcentral angles.WHAT YOU’LL LEARNWhen two sides of an angle meet at the center of a circle,a central angle is formed.Each side of the central angle intersects a point on thecircle, dividing it into lines called arcs.A minor arc is formed by the intersection of the circle andsides of a central angle with interior degree measure lessthan 180.A major arc is the part of the circle in the ofthe central angle that measures greater than 180.Semicircles are arcs whose endpoints lieon the diameter of the circle.Adjacent arcs are arcs of a circle with exactly one pointin common.BUILD YOUR VOCABULARY (pages 208–209)KM130Њ125ЊJLSOL G.10 The student will investigate and solve practical problems involving circles,using properties of angles, arcs, chords, tangents, and secants. Problems . . . may bedrawn from applications of architecture, art, and construction.The degree measure ofa minor arc is the degreemeasure of its centralangle.The degree measure ofa major arc is 360 minusthe degree measure ofits central angle.The degree measure ofa semicircle is 180.Under the tabfor Lesson 11-2, draw acircle with a centralangle. Label the centralangle, the major andminor arcs and giveexamples of degreemeasurements for each.KEY CONCEPTS
  • 221. ©Glencoe/McGraw-Hill11–2Geometry: Concepts and Applications 213Page(s):Exercises:HOMEWORKASSIGNMENTIn circle A, CE៮៮ is a diameter.Find mBC២, mBE២, and mBDE២.mBC២ϭ mЄBAC Measure of minor arcmBC២ϭ Substitutionϩ mBC២ϭ mEBC២Arc AdditionPostulatemBE២ϩ ϭ 180 SubstitutionmBE២ϭ 132 Subtract.mBDE២ϭ Ϫ mBE២Measure major arcmBDE២ϭ Ϫ 132 SubstitutionmBDE២ϭIn circle X,mЄAXB ϭ 70, mDC២ϭ 45,and BෆEෆ and AෆDෆ are diameters.a. Find mEA២, mЄBXC, b. Find mAC២, mDAE២,and mED២. and mABE២.Your TurnDB82Њ48Њ AECEADBXCTheorem 11-3 In a circle or in congruent circles, twominor arcs are congruent if and only if their correspondingcentral angles are congruent.Postulate 11-1 Arc Addition PostulateThe sum of the measures of two adjacent arcs is themeasure of the arc formed by the adjacent arcs.REMEMBER ITA circle contains360°.
  • 222. In circle R, if PR៮៮ Ќ QT៮៮, find PQ.ЄPSQ is a right angle. Definition of perpendicular᭝PSQ is a triangle. Definition of right triangle΂ ΃2ϩ (SQ)2ϭ (PQ)2Pythagorean TheoremSQ ϭ 24 Theorem 11-52ϩ 242ϭ (PQ)2Replace PS withand SQ with 24.100 ϩ 576 ϭ (PQ)2͙676ෆ ϭ ͙(PQ)2ෆ Take the square rootof each side.ϭ PQIn circle P, AB ϭ 8 and PD ϭ 3.Find PC.Your TurnRQPS2410T• Identify and use therelationships amongarcs, chords, anddiameters.WHAT YOU’LL LEARNTheorem 11-4In a circle or in congruent circles, two minor arcs arecongruent if and only if their corresponding chordsare congruent.Theorem 11-5In a circle, a diameter bisects a chord and its arc if andonly if it is perpendicular to the chord.Arcs and Chords©Glencoe/McGraw-Hill214 Geometry: Concepts and Applications11–3SOL G.10 The student will investigate and solve practical problems involving circles,using properties of angles, arcs, chords, tangents, and secants. Problems will includefinding arc length and the area of a sector, . . . .Under the tab forLesson 11-3, drawdiagrams and givedescriptions tosummarizeTheorems 11-4 and 11-5.CirclesORGANIZE ITP BDAC
  • 223. ©Glencoe/McGraw-Hill11–3Geometry: Concepts and Applications 215Page(s):Exercises:HOMEWORKASSIGNMENTIn circle W, find XV if UW៮៮ Ќ XV៮៮, VW ϭ 35, and WY ϭ 21.ЄVYW is a angle. Definition of perpendicular᭝VYW is a right triangle. Definition of right triangle(WY)2ϩ (YV)2ϭ ΂ ΃2Pythagorean Theorem212ϩ (YV)2ϭ 352Replace WY and VW.ϩ (YV)2ϭ 1225(YV)2ϭ Subtract.͙(YV)2ෆ ϭ Take the square root ofeach side.YV ϭ ϭ XY Theorem 11-5XV ϭ YV ϩ XY Segment additionXV ϭ ϩ 28 SubstitutionXV ϭIn circle G, if CෆGෆ Ќ AෆEෆ, EG ϭ 20, CG ϭ 12, find AE.Your TurnWUX Y2135VA CGE1220
  • 224. Inscribed Polygons©Glencoe/McGraw-Hill216 Geometry: Concepts and Applications11–4Construct a regular octagon.Construct a quadrilateralby connecting the consecutiveof two diameters.Bisect adjacent . Extend the bisectors through theof the circle to the edges of the circle. The otherfour are where the other two perpendicularintersect the circle. Connect all of theconsecutive to form the regular .Construct a regular hexagon.Your Turn• Inscribe regularpolygons in circles andexplore the relationshipbetween the length ofa chord and its distancefrom the center of thecircle.WHAT YOU’LL LEARNUnder the tab forLesson 11-4, draw apolygon inscribed ina circle and anothercircumscribed aboutthe circle. Label eachdrawing appropriately.CirclesORGANIZE ITA polygon is inscribed in a circle if and only if everyof the polygon lies on the circle.A circumscribed polygon is a polygon with each sideto a .BUILD YOUR VOCABULARY (pages 208–209)
  • 225. ©Glencoe/McGraw-Hill11–4Geometry: Concepts and Applications 217Page(s):Exercises:HOMEWORKASSIGNMENTIn circle O, point O is the midpoint of AϪBϪ.If CR ϭ 2x Ϫ 1 and ST ϭ x ϩ 10, find x.OA ϭ Definition of midpointϭ TS Theorem 11-62x Ϫ 1 ϭ x ϩ Substitution2x ϭ x ϩ Add to each side.x ϭ Subtract fromeach side.In circle Y, NY ϭ YO. If AX ϭ 2x ϩ 15 andBZ ϭ 3x ϩ 6, what is the value of x?AXBNOYZYour TurnRSCABOTExplain Theorem 11-6 inyour own words.WRITE ITTheorem 11-6In a circle or in congruent circles, two chords are congruentif and only if they are equidistant from the center.
  • 226. Circumference of a Circle©Glencoe/McGraw-Hill218 Geometry: Concepts and Applications11–5The radius of a circle is 8 feet. Find the circumferenceof the circle to the nearest tenth.C ϭ 2␲ Theorem 11-7C ϭ 2␲ ΂ ΃ Replace r with .C ϭ 16␲ Ϸ feetThe diameter of a plastic pipe is 5 cm. Find thecircumference of the pipe to the nearest centimeter.C ϭ ␲ Theorem 11-7C ϭ ␲΂ ΃ SubstitutionC ϭ 5␲ Ϸ cm• Solve problemsinvolving circumferenceof circles.WHAT YOU’LL LEARNUnder the tab forLesson 11-5, give theformulas for finding thecircumference of a circleand give an example ofhow they are used.CirclesORGANIZE ITThe perimeter of a is known as thecircumference. It is the around the circle.The ratio of the of a circle to itsis always equal to the irrational numbercalled pi.BUILD YOUR VOCABULARY (pages 208–209)Theorem 11-7 Circumference of a CircleIf a circle has a circumference of C units and a radius ofr units, then C ϭ 2␲r or C ϭ ␲d.
  • 227. ©Glencoe/McGraw-Hill11–5Geometry: Concepts and Applications 219Page(s):Exercises:HOMEWORKASSIGNMENTa. Find the circumference of circleA to the nearest tenth.b. The diameter of a CD is 4.5 inches.Find its circumference to the nearest tenth.A circular garden has a radius of 20 feet. There is apath around the garden that is 3 feet wide. Jasminestands on the inside edge of the path, and Hitesh standson the outside edge. They each walk around the gardenexactly once while staying along their edge of the path.To the nearest foot, how much farther does Hitesh walkthan Jasmine?Jasmine: Hitesh:C ϭ 2␲r Theorem 11-7 C ϭ 2␲rC ϭ 2␲΂ ΃ Substitution C ϭ 2␲΂ ΃C ϭ C ϭSo, Hitesh walked Ϫ or approximatelyfeet more than Jasmine.A circle has a circumference of 20.5 meters.Find the radius of the circle to the nearest tenth.Your TurnYour Turn5 ftAREMEMBER ITPi (␲) is an exactconstant. The decimalapproximation 3.14… isonly an estimate.REVIEW ITHow do you find theperimeter of a polygon?(Lesson 1-6)
  • 228. Area of a Circle©Glencoe/McGraw-Hill220 Geometry: Concepts and Applications11–6Find the area of circle G.A ϭ ␲r2Theorem 11-8A ϭ ␲2Replace r.A ϭ 100␲ Ϸ cm2Find the area of a circle to the nearest tenthwhose diameter is 10 cm.If circle S has a circumference of 16␲ inches, find thearea of the circle to the nearest hundredth.C ϭ 2␲r Theorem 11-7ϭ 2␲r Replace C with .ϭ Divide each side by .ϭ rA ϭ ␲r2Theorem 11-8A ϭ ␲2Replace r with .A ϭ 64␲ Ϸ in2Your Turn• Solve problemsinvolving areas andsectors of circles.WHAT YOU’LL LEARNUnder the tab forLesson 11-6, give theformula for finding thearea of a circle and givean example of how itis used.CirclesORGANIZE ITTheorem 11-8 Area of a CircleIf a circle has an area of A square units and a radius ofr units, then A ϭ ␲r2.2␲16␲2␲2␲r10 cmGSOL G.10 The student will investigate and solve practical problems involving circles,using properties of angles, arcs, chords, tangents, and secants. Problems will includefinding arc length and the area of a sector, . . . .
  • 229. ©Glencoe/McGraw-Hill11–6Geometry: Concepts and Applications 221Find the area of the circle to the nearesthundredth whose circumference is 84␲ cm.A pond has a radius of 10 meters. In the center of thepond is a square island with a side length of 5 meters.The seeds of a nearby maple tree float down randomlyover the pond. What is the probability that a randomly-chosen seed will land in the water rather than on theisland? Assume that the seed will land somewherewithin the circular edge of the pond.A of pond ϭ ␲2ϭ Ϸ m2A of island ϭ2ϭ m2P(landing in pond) ϭϭ ϭϷA of pond Ϫ A of islandᎏᎏᎏA of pondYour TurnTheoretical probability is the chance for a successfuloutcome based on .Experimental probability is calculated from actualobservations and recording . It is the chance fora successful outcome based on observing patterns ofoccurrences.BUILD YOUR VOCABULARY (pages 208–209)314.2314.2 Ϫ314.2
  • 230. ©Glencoe/McGraw-Hill11–6222 Geometry: Concepts and ApplicationsAssume that all dartswill land on the dartboard. Find theprobability that a randomly-throwndart will land in the shaded region.Find the area of a 45° sector of a circle whose radius is8 in. Round to the nearest hundredth.A ϭ ΂ᎏ3N60ᎏ΃␲r2Theorem 11-9A ϭ ΂ᎏ34650ᎏ΃␲82SubstitutionA ϭ (0.125)(64)␲A ϭ Ϸ in2Find the area of a 30° sector of a circle whoseradius is 7.75 feet. Round to the nearest hundredth.Your Turn420204 2Your TurnPage(s):Exercises:HOMEWORKASSIGNMENTA sector of a circle is a region bounded by a centraland its corresponding .BUILD YOUR VOCABULARY (page 209)Theorem 11-9 Area of a Sector of a CircleIf a sector of a circle has an area of A square units, a centralangle measurement of N degrees, and a radius of r units,then A ϭ ΂ᎏ3N60ᎏ΃␲r2.
  • 231. BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillBUILD YOURVOCABULARYUse your Chapter 11 Foldable tohelp you study for your chaptertest.To make a crossword puzzle, wordsearch, or jumble puzzle of thevocabulary words in Chapter 11,go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder(pages 208–209) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R11STUDY GUIDEGeometry: Concepts and Applications 223Underline the term that best completes the statement.1. A chord that contains the center of the circle is the[diameter/radius].2. A [chord/radius] is a segment with endpoints of the circle.3. Two circles are [circumscribed/concentric] if they lie on thesame plane, have the same center, and have radii of differentlengths.In circle C, BD៮៮ is a diameter and mЄGCF ϭ 63. Find eachmeasure.4. mFG២5. mAD២6. mAB២7. mGEF២BGED AFC11-1Parts of a Circle11-2Arcs and Central Angles
  • 232. Chapter BRINGING IT ALL TOGETHER©Glencoe/McGraw-Hill11224 Geometry: Concepts and ApplicationsComplete each statement.8. If two chords are congruent in the same circle, the interceptedare also congruent.9. When the diameter of the circle bisects a chord of the circle,then it is to the chord andthe corresponding arc.10. In a circle, if two arcs are , theirare congruent.11. Construct an equilateraltriangle inscribed in acircle with radius 1 inch.12. Draw a circle inscribedin the triangle from theprevious problem.Which segment of thetriangle equals the radiusof the inscribed circle?13. What is the approximate length of the segment in Exercise 12?11-3Arcs and Chords11-4Inscribed Polygons
  • 233. ©Glencoe/McGraw-HillChapter BRINGING IT ALL TOGETHER11Geometry: Concepts and Applications 225Find the circumference of each circle.14. r ϭ ᎏ12ᎏ yd15. d ϭ 4.2 in.Find the radius of the circle whose circumference is given.16. 47 ft17. 22.7 in.Underline the term that best completes the statement.18. A region of a circle bounded by a central angle and itscorresponding arc is a(n) [arc/sector].19. The segment with endpoints at the center and on the circle isa [sector/radius].20. Find the area of the shaded region incircle B to the nearest hundredth.57Њ6 ft.B11-5Circumference of a Circle11-6Area of a Circle
  • 234. ©Glencoe/McGraw-HillCheck the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 11 Practice Test onpage 491 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 11 Study Guide and Reviewon pages 488–490 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 11 Practice Test onpage 491.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 11 Foldable.• Then complete the Chapter 11 Study Guide and Review onpages 488–490 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 11 Practice Test onpage 491.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 11.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R11Checklist226 Geometry: Concepts and Applications
  • 235. Surface Area and Volume©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R12Geometry: Concepts and Applications 227NOTE-TAKING TIP: When taking notes, explaineach new idea or concept in words and give oneor more examples.Chapter12FoldFold the paper inthirds lengthwise.OpenOpen and fold a 2"tab along the shortside. Then fold therest in fifths.DrawDraw lines alongfolds and label asshown.Begin with a plain piece of 11" ϫ 17" paper.SurfaceAreaVolumeCh.12PrismsCylindersPyramidsConesSpheres
  • 236. Vocabulary TermFoundDefinitionDescription oron Page Example©Glencoe/McGraw-Hill228 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 12. Asyou complete the study notes for the chapter, you will see Build Your Vocabularyreminders to complete each term’s definition or description on these pages.Remember to add the textbook page number in the second column for referencewhen you study.C H A P T E R12BUILD YOUR VOCABULARYaxiscomposite solidconecubecylinder[SIL-in-dur]edgefacelateral area[LAT-er-ul]lateral edgelateral facenetoblique cone[oh-BLEEK]oblique cylinderoblique prism
  • 237. ©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY12Geometry: Concepts and Applications 229Vocabulary TermFoundDefinitionDescription oron Page Exampleoblique pyramidPlatonic solidpolyhedron[pa-lee-HEE-drun]prism[PRIZ-um]pyramid[PEER-a-MID]regular pyramidright coneright cylinderright prismright pyramidsimilar solidsslant heightsolid figuressphere[SFEER]surface areatetrahedronvolume
  • 238. Solid FiguresName the faces, edges, and vertices ofthe polyhedron.The faces are quadrilaterals ABCD,, DCGH, ADHE, ABFE, .The edges are , BෆCෆ, CෆDෆ, , BෆFෆ, AෆEෆ, DෆHෆ, CෆGෆ,EෆFෆ, FෆGෆ, GෆHෆ, EෆHෆ.The vertices are A, B, , D, E, F, , H.©Glencoe/McGraw-Hill230 Geometry: Concepts and ApplicationsSolid figures enclose a part of space.Solids with flat surfaces that are are knownas polyhedrons.The two-dimensional polygonal surfaces of a polyhedronare its faces.Two faces of a polyhedron in a segmentcalled an edge.A prism is a with two faces, called bases, whichare formed by congruent polygons that lie in parallel planes.Faces in a prism that are not bases are parallelograms andare called lateral faces.The intersection of two lateral faces in aprism are called lateral edges and are parallel segments.A pyramid is a solid with all faces but one intersecting at acommon point called the vertex. The face not intersectingat the vertex is the base. The base of a pyramid is apolygon. The faces meeting at the vertex are lateral facesand are triangles.BUILD YOUR VOCABULARY (pages 228–229)REMEMBER ITEuclidean solids arealso called solid figures.A BCDEFGH• Identify solid figures.WHAT YOU’LL LEARN12–1
  • 239. ©Glencoe/McGraw-Hill12–1Geometry: Concepts and Applications 231Name the faces, edges, and vertices of thepolyhedron.Your TurnGive three real-worldexamples ofpolyhedrons.WRITE ITA Platonic solid is a polyhedron.A cube is a special rectangular prism where all the facesare .A triangular pyramid is known as a tetrahedron because allof its faces are .A cylinder is a solid that is not a . Itsbases are two congruent in parallel planes,and its lateral surface is curved.A cone is a solid that is not a . Its base isa , and the lateral surface is curved.A composite solid is a solid made by twoor more solids.BUILD YOUR VOCABULARY (pages 228–229)YWZX
  • 240. Is the pyramid in the figure atetrahedron or a rectangularpyramid?The pyramid has a baseand lateral faces. It is apyramid.Describe the Washington Monument in termsof solid figures.Your Turn©Glencoe/McGraw-Hill12–1232 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITCylinders and conesare terms referring tocircular cylinders andcircular cones.MRNQP
  • 241. ©Glencoe/McGraw-Hill12–2Geometry: Concepts and Applications 233Surface Areas of Prisms and CylindersFind the lateral area and total surface area of a cubewith side length 6 inches.Perimeter of Base Area of BaseP ϭ 4s B ϭ s2ϭ 4(6) or ϭ 62orLateral Area Surface AreaL ϭ Ph S ϭ L ϩ 2Bϭ (24)(6) or ϭ 144 ϩ 2(36)ϭ 144 ϩ 72 orThe lateral area of the cube is in2, and the surfacearea is in2In a right prism, a lateral edge is also an altitude.In an oblique prism, a lateral edge is not an altitude.The lateral area of a solid figure is the of all theareas of its lateral faces.The surface area of a solid figure is the of theareas of all its surfaces.A net is a two-dimensional figure that to forma solid.BUILD YOUR VOCABULARYIn the box for SurfaceArea of Prisms, make asketch of a prism. Thenwrite the formula forfinding the surface areaof a prism.SurfaceAreaVolumeCh.12PrismsCylindersPyramidsConesSpheresORGANIZE IT(pages 228–229)Theorem 12-1 Lateral Area of a PrismIf a prism has a lateral area of L square units and a heightof h units and each base has a perimeter of P units, thenL ϭ Ph.Theorem 12-2 Surface Area of a PrismIf a prism has a surface area of S square units and a heightof h units and each base has a perimeter of P units and anarea of B square units, then S ϭ Ph ϩ 2B.• Find the lateral areasand surface areas ofprisms and cylinders.WHAT YOU’LL LEARNSOL G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve practical problems. Calculators will be used to finddecimal approximations for results.
  • 242. Find the lateral area and the surface area ofthe rectangular prism.Find the lateral area and the surfacearea of the triangular prism.Use the Pythagorean Theorem to find thelength of side b.Perimeter of Base Area of Basec2ϭ a2ϩ b2P ϭ 10 ϩ 6 ϩ b B ϭ ᎏ12ᎏbh102ϭ 62ϩ b2ϭ 10 ϩ 6 ϩ 8 ϭ ᎏ12ᎏ(6)(8)100 ϭ 36 ϩ b2ϭ ϭϭ b2͙64ෆ ϭ ͙b2ෆϭ bFind the lateral and surface areas.L ϭ Ph S ϭ L ϩ 2Bϭ (24)(8) ϭ 192 ϩ 2(24)ϭ cm2ϭ 192 ϩ 48ϭ cm2Find the lateral area andthe surface area of the triangular prism.1512135Your TurnYour Turn©Glencoe/McGraw-Hill12–2234 Geometry: Concepts and ApplicationsWhat is the differencebetween lateral areaand surface area?WRITE IT10 cm6 cm8 cmWhat is the length ofthe hypotenuse of aright triangle with legs5 cm and 12 cm long?(Lesson 6-6)REVIEW IT9 ft7 ft15 ft
  • 243. ©Glencoe/McGraw-Hill12–2Geometry: Concepts and Applications 235Page(s):Exercises:HOMEWORKASSIGNMENTIn the box for SurfaceArea of Cylinders, makea sketch of a cylinder.Then write the formulafor finding the surfacearea of a cylinder.SurfaceAreaVolumeCh.12PrismsCylindersPyramidsConesSpheresORGANIZE ITFind the lateral area and surface area ofthe cylinder to the nearest hundredth.L ϭ 2␲rh S ϭ 2␲rh ϩ 2␲r2ϭ 2␲(8)(14) ϭ 703.72 ϩ 2␲(8)2Ϸ Ϸ ϩϷThe lateral area is about in2, and the surfacearea is about in2.Find the lateral area and surface area of thecylinder to the nearest hundredth.Your TurnThe axis of a cylinder is the segment whoseare centers of the circular bases.In a right cylinder, the axis is also an .In an oblique cylinder, the axis is not an altitude.BUILD YOUR VOCABULARY (pages 228–229)Theorem 12-3 Lateral Area of a CylinderIf a cylinder has a lateral area of L square units and aheight of h units and the bases have radii of r units, thenL ϭ 2␲rh.Theorem 12-4 Surface Area of a CylinderIf a cylinder has a surface area of S square units and aheight of h units and the bases have radii of r units, thenS ϭ 2␲rh ϩ 2␲r2.8 in.14 in.15 m34 m
  • 244. Volumes of Prisms and CylindersFind the volume of the triangular prism.Area of triangular baseB ϭ ᎏ12ᎏ(10)(24) or .V ϭ Bh Theorem 12-5ϭ ΂ ΃΂ ΃ Substitutionϭ m3Find the volume of the rectangular prism.Area of base B ϭ (2)(5) or .V ϭ Bh Theorem 12-5ϭ ΂ ΃΂ ΃ Substitutionϭ ft3a. Find the volume of the triangular prism.b. Find the volume of a rectangular prism with basedimensions of 8 cm by 9 cm and height 4.1 cm.Your Turn©Glencoe/McGraw-Hill236 Geometry: Concepts and ApplicationsUse the box for Volumeof Prisms. Sketch andlabel a prism. Thenwrite the formula forfinding the volume ofa prism.SurfaceAreaVolumeCh.12PrismsCylindersPyramidsConesSpheresORGANIZE ITVolume measures the space contained within a solid.BUILD YOUR VOCABULARY (page 229)Theorem 12-5 Volume of a PrismIf a prism has a volume of V cubic units, a base with an areaof B square units, and a height of h units, then V ϭ Bh.24 m36 m10 m5 ft2 ft20 ft20 in.16 in.12 in.• Find the volumes ofprisms and cylinders.WHAT YOU’LL LEARN12–3SOL G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve practical problems. Calculators will be used to finddecimal approximations for results.
  • 245. Find the volume of the cylinderto the nearest hundredth.V ϭ ␲r2h Theorem 12-6ϭ ␲(5)2(12) Substitutionϭ 300␲Ϸ cm3Find the volume ofthe cylinder to the nearest hundredth.Leticia is making a sand sculpture by filling a glasstube with layers of different-colored sand. The tube is24 inches high and 1 inch in diameter. How many cubicinches of sand will Leticia use to fill the tube?V ϭ ␲r2h Theorem 12-6ϭ ␲(0.5)2(24) Substitutionϭ (0.25)(24)␲ϭ 6␲ϷLeticia will need about in3of sand.Sam fills the cylindrical coffee grind containers.One bag has 32␲ cubic inches of grinds. How many cylindricalcontainers can Sam fill with two bags of grinds if each cylinderis 4 inches wide and 4 inches high?Your TurnYour Turn©Glencoe/McGraw-Hill12–3Geometry: Concepts and Applications 237Use the box for Volumeof Cylinders. Sketch andlabel a cylinder. Thenwrite the formula forfinding the volume of acylinder.SurfaceAreaVolumeCh.12PrismsCylindersPyramidsConesSpheresORGANIZE ITTheorem 12-6 Volume of a CylinderIf a cylinder has a volume of V cubic units, a radius ofr units, and a height of h units, then V ϭ ␲r2h.5 cm12 cm21 ft45 ftPage(s):Exercises:HOMEWORKASSIGNMENT
  • 246. Surface Areas of Pyramids and ConesFind the lateral area andthe surface area of thesquare pyramid.First, find the perimeter and the area ofthe base.P ϭ 4s B ϭ s2ϭ 4(15) or ϭ 152or©Glencoe/McGraw-Hill238 Geometry: Concepts and ApplicationsIn a right pyramid or a right cone, the isperpendicular to the base at its center.In a oblique pyramid or a oblique cone, the altitude isto the base at a point other thanits center.A pyramid is a regular pyramid if and only if it is apyramid and its base is a polygon.The height of each face of a regular pyramidis called the slant height of the pyramid.BUILD YOUR VOCABULARY (page 229)Theorem 12-7 Lateral Area of a Regular PyramidIf a regular pyramid has a lateral area of L square units, abase with a perimeter of P units, and a slant height ofഞ units, then L ϭ ᎏ12ᎏPഞ.Theorem 12-8 Surface Area of a Regular PyramidIf a regular pyramid has a surface area of S square units, aslant height of ഞ units, and a base with perimeter of P unitsand an area of B square units, then S ϭ ᎏ12ᎏPഞ ϩ B.Use the box for SurfaceArea of Pyramids. Sketchand label a pyramid.Then write the formulafor finding the surfacearea of a pyramid.SurfaceAreaVolumeCh.12PrismsCylindersPyramidsConesSpheresORGANIZE IT25 cm15 cm15 cm• Find the lateral areasand surface areas ofregular pyramidsand cones.WHAT YOU’LL LEARN12–4SOL G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve practical problems. Calculators will be used to finddecimal approximations for results.
  • 247. Lateral Area Surface AreaL ϭ ᎏ12ᎏPഞ S ϭ L ϩ Bϭ ᎏ12ᎏ(60)(25) ϭ 750 ϩ 225ϭ cm2ϭ cm2Find the lateral area andsurface area of the square pyramid.Find the lateral area and the surface area of a regulartriangular pyramid with a base perimeter of 24 inches,a base area of 27.7 square inches, and a slant height of8 inches.L ϭ ᎏ12ᎏPഞ Theorem 12-7ϭ ᎏ12ᎏ(24)(8) Substitutionϭ in2S ϭ L ϩ B Theorem 12-7ϭ 96 ϩ 27.7 Substitutionϭ in2Find the lateral area and the surface area of aregular triangular pyramid with a base perimeter of 18 inches,a base area of 15.6 square inches, and a slant height of11 inches.Your TurnYour Turn©Glencoe/McGraw-Hill12–4Geometry: Concepts and Applications 2391066
  • 248. Find the lateral area and the surfacearea of the cone to the nearesthundredth.L ϭ ␲rഞ Theorem 12-9ϭ ␲(20)(35) Substitutionϭ 700␲Ϸ cm2S ϭ ␲rഞ ϩ ␲r2Theorem 12-10ϭ ␲΂ ΃(35) ϩ ␲(20)2Substitutionϭ 2199.11 ϩ 400␲Ϸ 2199.11 ϩ 1256.64ϷThe lateral area is about cm2, and the surfacearea is about cm2.Find the lateral area and the surface area ofthe cone to the nearest hundredth.Your Turn©Glencoe/McGraw-Hill12–4240 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENT35 cm20 cm5.2 yd3.2 ydTheorem 12-9 Lateral Area of a ConeIf a cone has a lateral area of L square units, a slant heightof ഞ units, and a base with a radius of r units, thenL ϭ ᎏ12ᎏ(2␲rഞ) or ␲rഞ.Theorem 12-10 Surface Area of a ConeIf a cone has a surface area of S square units, a slant heightof ഞ units, and a base with a radius of r units, thenS ϭ ␲rഞ ϩ ␲r2.Use the box for SurfaceArea of Cones. Sketchand label a cone. Thenwrite the formula forfinding the surface areaof a cone.SurfaceAreaVolumeCh.12PrismsCylindersPyramidsConesSpheresORGANIZE IT
  • 249. Find the volume of the rectangularpyramid.B ϭ ഞwϭ (10)(4) orV ϭ ᎏ13ᎏBh Theorem 12-11ϭ ᎏ13ᎏ(40)(12) Substitutionϭ cm3Find the volume of the cone to the nearest hundredth.Find the height hh2ϩ 212ϭ 352h2ϩ 441 ϭ 1225h2ϭ 784͙h2ෆ ϭ ͙784ෆh ϭV ϭ ᎏ13ᎏ␲r2h Theorem 12-11ϭ ᎏ13ᎏ␲(21)2(28) SubstitutionϷ in342 in.35 in.©Glencoe/McGraw-HillVolumes of Pyramids and Cones12–5Geometry: Concepts and Applications 241Use the boxes forVolume of Pyramids andCones. Sketch and labela pyramid and a cone.Then write the formulafor finding the volumesof a pyramid and acone.SurfaceAreaVolumeCh.12PrismsCylindersPyramidsConesSpheresORGANIZE ITTheorem 12-11 Volume of a PyramidIf a pyramid has a volume of V cubic units and a height ofh units and the area of the base is B square units, thenV ϭ ᎏ13ᎏBh.12 cm10 cm4 cmTheorem 12-12 Volume of a ConeIf a cone has a volume of V cubic units, a radius of r units,and a height of h units, then V ϭ ᎏ13ᎏ␲r2h.• Find the volumes ofpyramids and cones.WHAT YOU’LL LEARNSOL G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve practical problems. Calculators will be used to finddecimal approximations for results.
  • 250. a. Find the volume of the triangularpyramid.b. Find the volume of the cone to thenearest hundredth.The sand in a cone with radius 3 cm and height 10 cm ispoured into a square prism with height of 29.5 cm andbase area of 4 cm2. How far up the side of the prism willthe sand reach when leveled?Volume of Cone Volume of PrismV ϭ ᎏ13ᎏ␲r2h V ϭ Bhϭ ᎏ13ᎏ␲(3)2(10) 94.25 ϭ 4hϭ 30␲ h ϷϷThe sand will level off at a height of about cm in theprism.The salt in a cone with radius 6 cm andheight 8 cm is poured into a square prism with height of20 cm and base area of 12 cm2. Will the prism be able tohold all of the salt?Your TurnYour Turn©Glencoe/McGraw-Hill12–5242 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENT80 in.42 in.REMEMBER ITUse the altitude ofa solid, not the slantheight, to find thevolume of the solid.25 m35 m30 m
  • 251. ©Glencoe/McGraw-HillSpheres12–6Geometry: Concepts and Applications 243Find the surface area and volume of a sphere withradius 5 cm.Surface Area VolumeS ϭ 4␲r2V ϭ ΂ᎏ43ᎏ΃␲r3ϭ 4␲(5)2ϭ ΂ᎏ43ᎏ΃␲(5)3ϭ 100␲ ϭ ΂ᎏ5030ᎏ΃␲Ϸ cm2Ϸ cm3Find the surface area and volume of a spherewith diameter 15 in.Your TurnA sphere is the set of all points that are a fixedfrom a given called the center.BUILD YOUR VOCABULARY (page 229)Theorem 12-13 Surface Area of a SphereIf a sphere has a surface area of S square units and a radiusof r units, then S ϭ 4␲r2.Theorem 12-14 Volume of a SphereIf a sphere has a volume of V cubic units and a radius ofr units, then V ϭ ΂ᎏ43ᎏ΃␲r3.• Find the surface areasand volumes of spheres.WHAT YOU’LL LEARNUse the boxes forVolume and SurfaceArea of Spheres. Sketchand label a sphere. Thenwrite the formulas forfinding the surface areaand volume of a sphere.SurfaceAreaVolumeCh.12PrismsCylindersPyramidsConesSpheresORGANIZE ITSOL G.13 The student will use formulas for surface area and volume of three-dimensional objects to solve practical problems. Calculators will be used to finddecimal approximations for results.
  • 252. ©Glencoe/McGraw-Hill12–6244 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTSome students build a snow sculpturefrom a cylinder and a sphere of snow.Both the sphere and the cylinder havea radius of 1 ft. and the height of thecylinder is 4 ft. Find the volume of thesnow used to build the sculpture.Volume of Cylinder Volume of SphereV ϭ ␲r2h V ϭ ΂ᎏ43ᎏ΃␲r3ϭ ␲(1)2(4) ϭ ΂ᎏ43ᎏ΃␲(1)3ϭ 4␲ ϭ ΂ᎏ43ᎏ΃␲Ϸ ϷThe volume of the snow used for the sculpture is about12.57 ϩ 4.19, or ft3.Felix and Brenda want to share an ice creamcone. Brenda wants half the scoop of ice cream on top, whileFelix wants the ice cream inside the cone. Assuming the halfscoop of ice cream on top is a perfect sphere, who will havemore ice cream? The cone and scoop both have radii of 1.5 inch;the cone is 3.25 inches long.Your Turn1 ft1 ft4 ft
  • 253. REMEMBER ITA scale factor is aone-dimensionalmeasure. Surface area isa two-dimensionalmeasure. Volume is athree-dimensionalmeasure.©Glencoe/McGraw-HillSimilarity of Solid Figures12–7Geometry: Concepts and Applications 245Determine whether the pairof solids is similar.ᎏ297ᎏ ՘ ᎏ1584ᎏ Definition of similarity(9)(54) ՘ (27)(18) Cross products486 ϭ 486The corresponding lengths are in , so thesolids similar.Determine whether each pair of solidsis similar.a. b.Your TurnSimilar solids are solids that have the same butnot necessarily the same .BUILD YOUR VOCABULARY (page 229)Characteristics of SimilarSolids For similar solids,the correspondinglengths are proportional,and the correspondingfaces are similar.KEY CONCEPT• Identify and use therelationship betweensimilar solid figures.WHAT YOU’LL LEARN9 cm18 cm27 cm54 cmSOL G.14.a The student will use proportional reasoning to solve practical problems,given similar geometric objects.Theorem 12-15If two solids are similar with a scale factor of a:b, then thesurface areas have a ratio of a2:b2and the volumes have aratio of a3:b3.181812121283 344652211
  • 254. ©Glencoe/McGraw-Hill12–7246 Geometry: Concepts and ApplicationsFor the similar prisms, find thescale factor of the prism on theleft to the prism on the right.Then find the ratios of thesurface areas and the volumes.The scale factor is ᎏ271ᎏ ϭ ᎏ3100ᎏ or .The ratio of the surface areas is ᎏ3122ᎏ or .The ratio of the volumes is ᎏ3133ᎏ or .Find the scale factor ofthe prism on the left to the prism on theright. Then find the ratios of the surfaceareas and the volumes.Sara made a scale model of the Great AmericanPyramid in Memphis, Tennessee, which has a base sidelength of 544 ft and a lateral area of 456,960 ft2. If thescale factor of the model to the original is 1:136, whatwill be the lateral area of the model?ϭ ᎏ113262ᎏϭ18,496L ϭ 456,960L Ϸ ft2A scale model of a house is made using a scalefactor of ᎏ1112ᎏ. What fraction of the actual house material wouldwould the dollhouse need to cover all of its floors?Your Turnsurface area of the modelᎏᎏᎏᎏsurface area of Great Amer. Pyr.Your TurnL 130 cm21 cm21 cm 10 cm7 cm7 cmPage(s):Exercises:HOMEWORKASSIGNMENT18 ft30 ft
  • 255. BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillBUILD YOURVOCABULARYUse your Chapter 12 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 12, go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder(pages 228–229) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R12Geometry: Concepts and Applications 247Complete each sentence.1. Two faces of a polyhedron intersect at a(n) .2. A triangular pyramid is called a .3. A is a figure that encloses a part of space.4. Three faces of a polyhedron intersect at a point called a(n).Find the lateral area and surface area ofeach solid to the nearest hundredth.5. a regular pentagonal prism with apothema ϭ 4, side length s ϭ 6, and height h ϭ 12a. L ϭb. S ϭ6. a cylinder with radius r ϭ 42 and height h ϭ 10a. L Ϸ b. S Ϸ12-1Solid Figures12-2Surface Areas of Prisms and Cylinders4612STUDY GUIDE
  • 256. Chapter BRINGING IT ALL TOGETHER©Glencoe/McGraw-Hill12248 Geometry: Concepts and ApplicationsFind the volume of each solid and round to the nearesthundredth.7. the regular pentagonal prism from Exercise #58. How much water will a 24 in. by 15 in. by 10 in. fishtank hold?Find the lateral and surface areas of each solid.9. a rectangular pyramid with base dimensions 2 ft by 3 ft andlateral height h = 1 fta. L Ϸb. S Ϸ10. a cone with diameter 3.6 cm and lateral height 2.4 cma. L Ϸb. S ϷFind the volume of each solid rounded to the nearesthundredth, if necessary.11. a cone with its height as three times the radius12. the cone in Exercise #1012-3Volumes of Prisms and Cylinders12-4Surface Areas of Pyramids and Cones12-5Volumes of Pyramids and Cones
  • 257. ©Glencoe/McGraw-HillChapter BRINGING IT ALL TOGETHER12Geometry: Concepts and Applications 249Complete the sentence.13. The set of all points a given distance from the center is a.A beach ball will have a diameter of 30 in.14. How much material will be used to make the beach ball?15. How much air will be needed to fill it?16. Solids having the same shape but not always the same sizeare .If the radius of a sphere is doubled:17. How does the surface area change?18. How does the volume change?The diameter of the moon is about 2160 miles. The diameterof the Earth is about 7900 miles.19. Assuming both are spheres, what is the scale factor of theEarth to the moon?20. Are they similar solid figures?12-7Similarity of Solid Figures12-6Spheres
  • 258. ©Glencoe/McGraw-HillVisit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 12.C H A P T E R12Checklist250 Geometry: Concepts and ApplicationsARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 12 Practice Test on page543 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 12 Study Guide and Reviewon pages 540–542 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 12 Practice Test onpage 543.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 12 Foldable.• Then complete the Chapter 12 Study Guide and Review onpages 540–542 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 12 Practice Test onpage 543.Student Signature Parent/Guardian SignatureTeacher Signature
  • 259. Right Triangles and Trigonometry©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R13Geometry: Concepts and Applications 251NOTE-TAKING TIP: When taking notes, it is oftenhelpful to remember what you’ve learned if youcan paraphrase or summarize key terms andconcepts in your own words.StackStack sheets of paper withedges ᎏ14ᎏ inch apart.FoldFold up bottom edges.All tabs should be thesame size.CreaseCrease and staple along fold.TurnTurn and label the tabswith the lesson names.Begin with three sheets of lined 8ᎏ12ᎏ" ϫ 11" paper.Chapter1313-1 Simplifying Square Roots13-2 45º -45º -90º Triangles13-3 30º -60º -90º Triangles13-4 Tangent Ratios13-5 Sine and Cosine RatiosRight Trianglesand Trigonometry
  • 260. ©Glencoe/McGraw-Hill252 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 13.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in thesecond column for reference when you study.C H A P T E R13BUILD YOUR VOCABULARYVocabulary TermFoundDefinitionDescription oron Page Example30°-60°-90° triangle45°-45°-90° triangleangle of depressionangle of elevationcosinehypsometerperfect squareradical expression[RAD-ik-ul]
  • 261. ©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY13Geometry: Concepts and Applications 253Vocabulary TermFoundDefinitionDescription oron Page Exampleradical signradicand[RAD-i-KAND]simplest formsinesquare roottangent[TAN-junt]trigonometric identity[TRIG-guh-no-MET-rik]trigonometric ratiotrigonometry
  • 262. Simplifying Square RootsSimplify each expression.͙36ෆ͙36ෆ ϭ , because 62ϭ 36.͙81ෆ͙81ෆ ϭ , because 92ϭ 81.͙24ෆ͙24ෆ ϭ ͙2 ؒ 2 ؒෆ 2 ؒ 3ෆ Prime factorizationϭ ͙2 ؒ 2ෆ ؒ ͙2 ؒ 3ෆ Product Property ofSquare Rootsϭ 2 ؒ ͙6ෆ ͙2 ؒ 2ෆ ϭ 2ϭ©Glencoe/McGraw-Hill254 Geometry: Concepts and Applications• Multiply, divide, andsimplify radicalexpressions.WHAT YOU’LL LEARNPerfect squares are products of two factors, orwhen a number multiplies itself.The square root, therefore, is one of equal factors.A number has both positive (ϩ) and negative (Ϫ)square roots, indicated by the radical sign ͙ෆ.A radical expression is an expression that contains a.The number under the radical sign ͙ෆ is the radicand.BUILD YOUR VOCABULARY (pages 252–253)What are the nextthree perfect squaresafter 16?WRITE ITProduct Property ofSquare Roots The squareroot of a product isequal to the product ofeach square root.KEY CONCEPT13–1
  • 263. ͙6ෆ ؒ ͙30ෆ͙6ෆ ؒ ͙30ෆ ϭ ͙6ෆ ؒ Prime factorizationϭ ͙6 ؒ 6 ؒෆ 5ෆ Product Property ofSquare Rootsϭ ؒ ͙5ෆ Product Property ofSquare Rootsϭ ͙6 ؒ 6ෆ ϭ 6Simplify each expression.a. ͙25ෆ b. ͙121ෆc. ͙18ෆ d. ͙3ෆ ؒ ͙12ෆSimplify each expression.ϭ Ίᎏ186ᎏ๶ Quotient PropertyϭΊᎏ14291ᎏ๶Ίᎏ14291ᎏ๶ ϭ Quotient PropertyϭSimplify each expression.a. b.͙144ෆᎏ͙25ෆ͙20ෆᎏ͙4ෆYour Turn͙121ෆᎏ͙49ෆ͙16ෆᎏ͙8ෆ͙16ෆᎏ͙8ෆYour Turn©Glencoe/McGraw-Hill13–1Geometry: Concepts and Applications 255REMEMBER ITSimplifying afraction with a radical inthe denominator iscalled rationalizing thedenominator.Quotient Property ofSquare Roots The squareroot of a quotient isequal to the quotient ofeach square root.On the tabfor Lesson 13-1, writethe names of the twoproperties introduced inthis lesson. Then writeyour own example ofeach property on theback of the tab.KEY CONCEPT
  • 264. Simplify .ϭ ؒ ϭ 1ϭ Product Property ofSquare Rootsϭ ͙7 ؒ 7ෆ ϭ 7We used the Identity Property and the Product Property ofSquare Roots to simplify the above radical expression. Thedenominator does not have a radical sign.Simplify .ϭ ؒ ϭ 1ϭ Product Property ofSquare Rootsϭ ͙6 ؒ 6ෆ ϭ 6ϭ orWe used the Identity Property and the Product Property ofSquare Roots to simplify the above expression and eliminatethe radical in the denominator.Simplify.a.b.4ᎏ͙5ෆ͙7ෆᎏ͙2ෆYour Turn16 ͙6ෆᎏ616 ؒ ͙6ෆᎏᎏ͙6ෆ ؒ ͙6ෆ16ᎏ͙6ෆ16ᎏ͙6ෆ16ᎏ͙6ෆ͙70ෆᎏ7͙10 ؒ 7ෆᎏᎏ͙7 ؒ 7ෆ͙7ෆᎏ͙7ෆ͙7ෆᎏ͙7ෆ͙10ෆᎏ͙7ෆ͙10ෆᎏ͙7ෆ͙10ෆᎏ͙7ෆ©Glencoe/McGraw-Hill13–1256 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTRules for SimplifyingRadical Expressions1. There are no perfectsquare factors otherthan 1 in the radicand.2. The radicand is not afraction.3. The denominator doesnot contain a radicalexpression.KEY CONCEPT16͙6ෆ
  • 265. In a scale model of a town, a baseball diamond has sides36 inches long. What is the distance from first base tothird base on the model? Round to the nearest tenth.h ϭ s͙2ෆ Theorem 13-1ϭ ͙2ෆ SubstitutionϷThe distance from first to third base on the scale model isabout inches.Find the length of the diagonal of a squarewhose side measures 22 inches.Your Turn©Glencoe/McGraw-Hill45°-45°-90° TrianglesGeometry: Concepts and Applications 257Describe two differentways to find the lengthof the hypotenuse of a45°-45°-90° triangle.WRITE ITThe of a square separates the square intotwo 45°-45°-90° triangles.BUILD YOUR VOCABULARY (page 252)Theorem 13-1 45°-45°-90° Triangle TheoremIn a 45°-45°-90° triangle, the hypotenuse is ͙2ෆ timesthe length of a leg.• Use the properties of45°-45°-90° triangles.WHAT YOU’LL LEARN13–2SOL G.7 The student will solve practical problems involving right triangles by using thePythagorean Theorem, properties of special right triangles, and right triangle trigonometry.Solutions will be expressed in radical form or as decimal approximations.
  • 266. If ᭝DJT is an isosceles right triangle and the measureof the hypotenuse is ͙200ෆ, find the measure ofeither leg.h ϭ s͙2ෆ Theorem 13-1ϭ s͙2ෆ Substitutionϭ sThe length of each leg measures .If ᭝XYZ is an isoscelesright triangle and the measure of thehypotenuse is 25, find the measure ofeither leg.Your Turn25X YZss©Glencoe/McGraw-Hill13–2258 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTOn the tab forLesson 13-2, draw a45°-45°-90° triangleand label its parts.Then write your ownreal-world example thatcan be solved using thisinformation.13-1 Simplifying Square Roots13-2 45º -45º -90º Triangles13-3 30º -60º -90º Triangles13-4 Tangent Ratios13-5 Sine and Cosine RatiosRight Trianglesand TrigonometryORGANIZE ITREMEMBER ITA 45°-45°-90°triangle is isosceles, sothe legs are alwayscongruent.
  • 267. In ᭝ABC, a ϭ 12. Find b and c.a ϭ b͙3ෆ The longer leg is ͙3ෆ timesthe length of the shorter leg.Replace a with .ϭ b͙3ෆϭ bc ϭ 2b The hypotenuse is twicethe shorter leg.Replace b with .c ϭ 2΂ ΃c ϭRefer to Example 1.a. If b ϭ 3.5, find a and c. b. If b ϭ ᎏ13ᎏ, find a and c.Your Turn©Glencoe/McGraw-Hill30°-60°-90° TrianglesGeometry: Concepts and Applications 259• Use the properties of30°-60°-90° triangles.WHAT YOU’LL LEARNREMEMBER ITThe shorter leg isalways opposite the30° angle, and thelonger leg is alwaysopposite the 60° angle.The median of an equilateral triangle separates it intotwo 30°-60°-90° triangles.BUILD YOUR VOCABULARY (page 252)Theorem 13-2 30°-60°-90° Triangle TheoremIn a 30°-60°-90° triangle, the hypotenuse is twice thelength of the shorter leg, and the longer leg is ͙3ෆ timesthe length of the shorter leg.13–3SOL G.7 The student will solve practical problems involving right triangles by using thePythagorean Theorem, properties of special right triangles, and right triangle trigonometry.Solutions will be expressed in radical form or as decimal approximations.cbaA CB30˚60˚
  • 268. In ᭝DEF, DE ϭ 18. Find EF and DF.Use Theorem 13–2.DE ϭ EF͙3ෆ The longer leg is ͙3ෆ timesthe shorter leg.Replace DE with .ϭ EF͙3ෆ Divide each side by ͙3ෆ.ϭ EFDF ϭ 2(EF) The hypotenuse is twicethe shorter leg.Replace EF with .DF ϭ 2΂ ΃DF ϭ Associative PropertyRefer to Example 2. If DE ϭ 11, find EF and DF.Find the length, to the nearest tenth,of the median in the equilateral triangle.The median bisects one side into two 5-metersegments and is opposite the 60° angle.x ϭ 5͙3ෆ Theorem 13-2Ϸ metersFind the length, to the nearesttenth, of the median in the equilateral triangle.Your Turn10 m10 m10 mxYour TurnED F60˚30˚©Glencoe/McGraw-Hill13–3260 Geometry: Concepts and ApplicationsUnder the tab forLesson 13-3, draw a30°-60°-90° triangle andlabel its parts. Thenwrite a summary of howyou can find the lengthof the longer leg giventhe length of theshorter leg.13-1 Simplifying Square Roots13-2 45º -45º -90º Triangles13-3 30º -60º -90º Triangles13-4 Tangent Ratios13-5 Sine and Cosine RatiosRight Trianglesand TrigonometryORGANIZE ITPage(s):Exercises:HOMEWORKASSIGNMENT232323x
  • 269. Find tan K and tan M.tan K ϭ ᎏMKLLᎏϭ ᎏ2218ᎏ or Substitutiontan M ϭ ᎏMKLLᎏϭ ᎏ2281ᎏ or SubstitutionFind tan 30°, tan 45°, tan 60°.Your Turnoppositeᎏᎏadjacentoppositeᎏᎏadjacent282135K LM©Glencoe/McGraw-HillTangent Ratio13–4Geometry: Concepts and Applications 261Trigonometry is the study of the properties of.A trigonometric ratio is a ratio of the measures of two sidesof a triangle.The tangent is the ratio of one to the other.If A is an acute angle of a right triangle,tan A ϭ .measure of leg opposite ЄAᎏᎏᎏᎏmeasure of leg adjacent to ЄABUILD YOUR VOCABULARY (page 253)REMEMBER ITThe ratio of themeasures of the legs ofa right triangle can becompared to the ratioof rise to run in thedefinition of slopeof a line.• Use the tangent ratio tosolve problems.WHAT YOU’LL LEARNSOL G.7 The student will solve practical problems involving right triangles by using thePythagorean Theorem, properties of special right triangles, and right triangle trigonometry.Solutions will be expressed in radical form or as decimal approximations.
  • 270. Find QR to the nearesttenth of a meter.tan ϭ ᎏQPQRᎏtan ϭ Substitutionؒ ϭ QR Multiplication Propertyof EqualityϷ QRA ranger sights the top of a tree at a 40° angle ofelevation. Find the height of the tree if it is 80 feet fromwhere the ranger is standing.tan ϭtan ϭ height of tree Multiplication Propertyof EqualityϷ height of treeThe height of the tree is about feet.oppositeᎏᎏadjacentoppositeᎏᎏadjacent©Glencoe/McGraw-Hill13–4262 Geometry: Concepts and ApplicationsThe line of sight and a horizontal line when looking upform the angle of elevation.Angles of elevation can be measured usinga hypsometer.The line of sight and a horizontal line when lookingform the angle of depression.BUILD YOUR VOCABULARY (page 252)REMEMBER ITThe symbol tan B isread the tangent ofangle B.55Њ20 mQPRQRheight of treeWrite the definition oftangent on the tab forLesson 13-4. Under thetab, sketch and label atriangle. Then expressthe tangent of one ofthe angles.13-1 Simplifying Square Roots13-2 45º -45º -90º Triangles13-3 30º -60º -90º Triangles13-4 Tangent Ratios13-5 Sine and Cosine RatiosRight Trianglesand TrigonometryORGANIZE IT
  • 271. a. Find YZ to the nearest tenth of a foot.b. The ranger sights the top of another tree at a 52° angle ofelevation. Find the height of the tree if it is 20 feet fromwhere he stands.Find mЄ1 to the nearest tenth.tan(Є1) ϭtanϪ1΂ᎏ2427ᎏ΃Ϸ Definition of arctangentThe measure of Є1 is about .Find mЄ2 to the nearest tenth.Your Turnoppositeᎏᎏadjacent122 in.47 in.Your Turn©Glencoe/McGraw-Hill13–4Geometry: Concepts and Applications 263Page(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITThe inverse tangentis also called thearctangent.1771YXZK J1192R
  • 272. Sine and Cosine RatiosFind sin K, cos K, sin M,and cos M.sin K sin Mϭ ᎏKLMMᎏ ϭ ᎏKKMLᎏϭ Substitution ϭ SubstitutionϷ Ϸcos K cos Mϭ ᎏKKMLᎏ ϭ ᎏKLMMᎏϭ Substitution ϭ SubstitutionϷ ϷFind sin 30°, cos 30°, sin 45°, cos 45°, sin 60°,cos 60°.Your Turnadjacentᎏᎏhypotenuseadjacentᎏᎏhypotenuseoppositeᎏᎏhypotenuseoppositeᎏᎏhypotenuse©Glencoe/McGraw-Hill264 Geometry: Concepts and ApplicationsWrite the definitions ofsine and cosine on thetab for Lesson 13-5. Onthe back of the tab,describe a similarity anda difference betweensine and cosine.13-1 Simplifying Square Roots13-2 45º -45º -90º Triangles13-3 30º -60º -90º Triangles13-4 Tangent Ratios13-5 Sine and Cosine RatiosRight Trianglesand TrigonometryORGANIZE ITBoth the sine and the cosine ratios relate an angle measureto the ratio of the measures of a triangle’s to its.If A is an acute angle of a right triangle,sin A ϭ , andcos A ϭ .measure of leg adjacent ЄAᎏᎏᎏᎏmeasure of hypotenusemeasure of leg opposite ЄAᎏᎏᎏᎏmeasure of hypotenuseBUILD YOUR VOCABULARY (pages 252–253)• Use the sine and cosineratios to solve problems.WHAT YOU’LL LEARNSOL G.7 The student will solve practical problems involving right triangles by usingthe Pythagorean Theorem, properties of special right triangles, and right triangletrigonometry. Solutions will be expressed in radical form or as decimal approximations.LMK1517813–5
  • 273. Find the value of x to thenearest tenth.sin 26 ϭ ᎏ20x0ᎏ200 sin 26 ϭ x Multiplication Propertyof EqualityϷ xFind the measure of ЄK to thenearest degree.sin K ϭ ᎏKLMMᎏsin K ϭ ᎏ6882ᎏ SubstitutionmЄK ϭ sinϪ1΂ᎏ6882ᎏ΃ Inverse sinemЄK Ϸa. Find the value of x to the nearest tenth.b. Find the measure of ЄA to the nearest degree.Your Turnoppositeᎏᎏhypotenuseoppositeᎏᎏhypotenuse©Glencoe/McGraw-Hill13–5Geometry: Concepts and Applications 265REMEMBER ITSinϪ1and cosϪ1arealso known as arcsinand arccos.Page(s):Exercises:HOMEWORKASSIGNMENT26Њ200 mx m6882LKMTheorem 13-3If x is a measure of an acute angle of a right triangle,then ᎏcsoinsxxᎏ ϭ tan x.x5833C8632BA
  • 274. ©Glencoe/McGraw-Hill266 Geometry: Concepts and ApplicationsBRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 13 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 13, go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder(pages 252–253) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R13Simplify.1. ͙63ෆ 2. ᎏ͙13ෆᎏ 3. ͙10ෆ ؒ ͙8ෆ4. Find the value of x if ᎏ͙2xෆᎏ ϭ ᎏ2͙3xෆᎏ.A fabric square is cut on the diagonal for a quilt.The perimeter of the square is 116 in.5. What is the length of each leg/side?6. What is the length of the hypotenuse/diagonal?7. What is the measure of each leg of an isosceles right triangleif its hypotenuse measures 10?13-1Simplifying Square Roots13-245°-45°-90° TrianglesSTUDY GUIDE
  • 275. ©Glencoe/McGraw-HillChapter BRINGING IT ALL TOGETHER13Geometry: Concepts and Applications 2678. The Gothic arch, similar to the figure, is basedon an equilateral triangle. Find the width of thebase of the triangle if the median is 4 ft long.Find the missing measure. Simplify all radicals.9. 10.11. You spot a cat on the roof of a house 80 feet away from whereyou’re standing. Your eye level is 5 feet above ground level, andthe angle of elevation from eye level is 33Њ. How tall is the house?Find the missing measures.12. If y ϭ 20, find x and z.13. If z ϭ 2.3, find x and y.14. If x ϭ 9, find y and z.66 3c60Њ5.5b1160Њ413-4Tangent Ratio13-330°-60°-90° Triangles30YXZ60yxz13-5Sine and Cosine Ratios
  • 276. ©Glencoe/McGraw-HillCheck the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 13 Practice Test onpage 581 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 13 Study Guide and Reviewon pages 578–580 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 13 Practice Test onpage 581.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 13 Foldable.• Then complete the Chapter 13 Study Guide and Review onpages 578–580 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 13 Practice Test onpage 581.Visit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 13.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R13Checklist268 Geometry: Concepts and Applications
  • 277. Circle Relationships©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R14Geometry: Concepts and Applications 269NOTE-TAKING TIP: When taking notes, definenew terms and write about the new ideas andconcepts you are learning in your own words.Write your own examples that use the new termsand concepts.FoldFold in half alongthe width.OpenOpen and fold thebottom to form apocket. Glue edges.RepeatRepeat steps 1 and 2three times and glue allthree pieces together.LabelLabel each pocket withthe lesson names. Place anindex card in each pocket.Begin with three sheets of plain 8ᎏ12ᎏ" ϫ 11"paper.Chapter14CircleRelationships
  • 278. ©Glencoe/McGraw-Hill270 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 14.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R14BUILD YOUR VOCABULARYVocabulary TermFoundDefinitionDescription oron Page Exampleexternal secant segment[SEE-kant]externally tangent[TAN-junt]inscribed angleintercepted arcinternally tangent
  • 279. ©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY14Geometry: Concepts and Applications 271Vocabulary TermFoundDefinitionDescription oron Page Examplepoint of tangencysecant anglesecant-tangent anglesecant segmenttangenttangent-tangent angle
  • 280. Inscribed AnglesDetermine whether ЄABC is aninscribed angle. Name theintercepted arc for the angle.The vertex of ЄABC, point B, is oncircle Q. Therefore, ЄABC is anangle. The interceptedarc is AC២.Determine whether ЄJKL is an inscribedangle. Name the intercepted arc for the angle.Your TurnABCQ©Glencoe/McGraw-Hill272 Geometry: Concepts and Applications14–1• Identify and useproperties of inscribedangles.WHAT YOU’LL LEARNUnder the tab forInscribed Angles, writethe definition of aninscribed angle anddraw a picture toillustrate the concept.Record the theoremsand other importantinformation from thislesson.CircleRelationshipsORGANIZE ITAn inscribed angle is an angle whose lies on acircle and whose sides contain of the circle.An intercepted arc is an arc of a circle, formed by an angle,such that the of the arc lie on the sidesof the angle and all other points of the arc lie on theof the angle.BUILD YOUR VOCABULARY (page 270)Theorem 14-1The degree measure of an inscribed angle equals one-halfthe degree measure of its intercepted arc.KJMPLSOL G.10 The student will investigate and solve practical problems involving circles, usingproperties of angles, arcs, chords, tangents, and secants. Problems will include finding arclength and . . . may be drawn from applications of architecture, art, and construction.
  • 281. Refer to the figure.If mAB២ϭ 76, find mЄADB.mЄADB ϭ ᎏ12ᎏ(mAB២) Theorem 14-1mЄADB ϭ ᎏ12ᎏ΂ ΃ Replace mAB២.mЄADB ϭIf mЄBDC ϭ 40, find mBC២.mЄBDC ϭ ᎏ12ᎏ(mAB២) Theorem 14-140 ϭ ᎏ12ᎏ(mBC២) Replace mЄBDC.2 ؒ 40 ϭ 2 ؒ ᎏ12ᎏ(mBC២) Multiply each side by 2.ϭ mBC២Refer to the figure.a. If mZW២ϭ 124,find mЄWXZ.b. If mЄYXZ ϭ 49,find mYZ២.In circle A, suppose mЄTLN ϭ 6y ϩ 7and mЄTWN ϭ 7y. Find the value of y.ЄTLN and ЄTWN both intercept TN២.ЄTLN Х ЄTWN Theorem 14-2mЄTLN ϭ mЄTWN Definition of congruent anglesϭ Replace mЄTLN and mЄTWN.ϭ y Subtract 6y from each side.AT NL W1 2XYZWYour TurnADBCK©Glencoe/McGraw-Hill14–1Geometry: Concepts and Applications 273Theorem 14-2If inscribed angles intercept the same arc or congruent arcs,then the angles are congruent.What is the differencebetween a central angleand an inscribed angle?WRITE ITREMEMBER ITThere are 360Њ in acircle and 180Њ in a semi-circle.
  • 282. In the circle, if mЄAHM ϭ 10x andmЄATM ϭ 20x Ϫ 30, find the value of x.In circle G, mЄ1 ϭ 6x Ϫ 5 andmЄ2 ϭ 3x Ϫ 4. Find the value of x.Inscribed angle DEF intercepts semicircleDF២. ЄDEF is a right angle by Theorem 14-3.Therefore, Є1 and Є2 are complementary.mЄ1 ϩ mЄ2 ϭ 90 Complementary angles(6x Ϫ 5) ϩ (3x Ϫ 4) ϭ 90 Substitutionϭ 90 Combine like terms.9x Ϫ 9 ϩ ϭ 90 ϩ Add to each side.ϭᎏ99xᎏ ϭ ᎏ999ᎏ Divide each side by 9.x ϭIn circle W, mЄRKP ϭ ΂ᎏ12ᎏ΃x andmЄKRP ϭ ΂ᎏ13ᎏ΃x ϩ 5. Find the value of x.KW RPYour TurnFEDG12HM ATYour Turn©Glencoe/McGraw-Hill14–1274 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTTheorem 14-3If an inscribed angle of a circle intercepts a semicircle, thenthe angle is a right angle.How does the measureof an inscribed anglerelate to the measure ofits intercepted arc?WRITE IT
  • 283. AB៮៬ is tangent to circle C at B. Find BC.AB៮៬ Ќ CB៮៮ by Theorem 14-4, making ЄCBAa right angle by definition. Therefore, ᭝ABCis a right triangle.(BC)2ϩ (AB)2ϭ (AC)2Pythagorean Theorem(BC)2ϩ2ϭ2Replace AB and AC.(BC)2ϩ ϭ Square and .(BC)2ϩ 784 Ϫ 784 ϭ 1225 Ϫ 784 Subtract 784 from each side.(BC)2ϭ͙(BC)2ෆ ϭ ͙441ෆ Take the square root ofeach side.BC ϭAB3528C©Glencoe/McGraw-HillTangents to a Circle14–2Geometry: Concepts and Applications 275Under the tab forTangents to a Circle,write the definition oftangent and draw apicture to illustrate theconcept. Record thetheorems and otherimportant informationfrom this lesson.CircleRelationshipsORGANIZE ITTheorem 14-4In a plane, if a line is tangent to a circle, then it isperpendicular to the radius drawn to the point of tangency.Theorem 14-5In a plane, if a line is perpendicular to a radius of a circle atits endpoint on the circle, then the line is a tangent.• Identify and applyproperties of tangentsto circles.WHAT YOU’LL LEARNSOL G.10 The student will investigate and solve practical problems involving circles, usingproperties of angles, arcs, chords, tangents, and secants. Problems will include finding arclength and . . . may be drawn from applications of architecture, art, and construction.In a plane, a line is a tangent if and only if it intersects acircle in exactly point.The point of intersection is the point of tangency.BUILD YOUR VOCABULARY (page 271)
  • 284. EF៮៮ and EG៮៮ are tangent to circle H.Find the value of x.EෆFෆ Х EෆGෆ Theorem 14-6ϭ Replace EF៮ and EG៮៮.3x ϩ 10 Ϫ ϭ 43 Ϫ Subtract 10 from each side.3x ϭ 33ᎏ33xᎏ ϭ ᎏ333ᎏ Divide each side by .x ϭAD៮៮, AC៮៮, and AB៮៮ aretangents to circles Q and R, respectively.Find the value of x.DQRABC6x ϩ 5 Ϫ2x ϩ 37Your Turn©Glencoe/McGraw-Hill14–2276 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTTheorem 14-6If two segments from the same exterior point are tangentto a circle, then they are congruent.If two circles are tangent and one circle is theother, the circles are internally tangent.If two circles are tangent and circle is insidethe other, the circles are externally tangent.BUILD YOUR VOCABULARY (pages 270–271)AE៮៮ is tangent to circle Cat E. Find AE.C AE16 in.34 in.Your TurnFEG43 m(3x + 10) mH
  • 285. Find mЄ1.The vertex of Є1 is inside circle P.mЄ1 ϭ ᎏ12ᎏ(mAB២ϩ mCD២) Theorem 14-8mЄ1 ϭ ᎏ12ᎏ΂ ϩ ΃ Replace mAB២and mCD២.mЄ1 ϭ ᎏ12ᎏ΂ ΃orIf mMA២ϭ 40 and mHT២ϭ 50,find mЄ1.HSMAT1Your Turn©Glencoe/McGraw-HillSecant Angles14–3Geometry: Concepts and Applications 277• Find measures of arcsand angles formed bysecants.WHAT YOU’LL LEARNA secant segment is a segment that contains aof a circle.A secant angle is the angle formed when twosegments intersect.BUILD YOUR VOCABULARYTheorem 14-7A line or line segment is a secant to a circle if and only if itintersects the circle in two points.Theorem 14-8If a secant angle has its vertex inside a circle, then itsdegree measure is one-half the sum of the degree measuresof the arcs intercepted by the angle and its vertical angle.Theorem 14-9If a secant angle has its vertex outside a circle, then itsdegree measure is one-half the difference of the degreemeasures of the intercepted arcs.(page 271)ACDP40Њ76ЊB1Under the tab forSecant Angles, write thedefinition of a secantsegment. Draw a pictureof secant angles toillustrate the concept.Record the theoremsand other importantinformation from thislesson.CircleRelationshipsORGANIZE ITSOL G.10 The student will investigate and solve practical problems involving circles, usingproperties of angles, arcs, chords, tangents, and secants. Problems will include finding arclength and . . . may be drawn from applications of architecture, art, and construction.
  • 286. Page(s):Exercises:HOMEWORKASSIGNMENTFind mЄJ.The vertex of ЄJ is outside circle Q.mЄJ ϭ ᎏ12ᎏ(mMN២Ϫ mKL២) Theorem 14-9mЄJ ϭ ᎏ12ᎏ΂ Ϫ ΃mЄJ ϭ ᎏ12ᎏ΂ ΃orFind the value of x. Then find mCD២.The vertex lies inside circle P.57 ϭ ᎏ12ᎏ(mAB២ϩ mCD២)57 ϭ ᎏ12ᎏ΄΂ ΃ϩ ΂ ΃΅57 ϭ ᎏ12ᎏ(9x ϩ 6) Combine like terms.2 ؒ 57 ϭ 2 ؒ ᎏ12ᎏ(9x ϩ 6) Multiply each side by 2.ϭ114 Ϫ 6 ϭ 9x ϩ 6 Ϫ 6 Subtract 6 from each side.ϭ Subtraction Propertyϭ x Division PropertymCD២ϭ 6x ϩ 7 ϭ 6΂ ΃ϩ 7 ϭ ϩ 7 ϭa. If mCE២ϭ 85 and mBD២ϭ 40,find mЄA.b. Find the value of x. Then find mTH២.ETCBADYour Turn©Glencoe/McGraw-Hill14–3278 Geometry: Concepts and ApplicationsREMEMBER ITThe diameter of acircle is also a secant.MQKJLN50Њ100ЊACDP57Њ(6x + 7)Њ(3x – 1)ЊBMAKHTE30x ϩ 10Њ40x Ϫ 2Њ 39Њ
  • 287. In the figure, AෆDෆ is tangentto circle K at A.Find mЄ1.Vertex D of the secant-tangent angle is outside circle K.Apply Theorem 14-10.The degree measure of the whole circle is 360°. So, themeasure of AC២is 360° Ϫ 160° Ϫ 110° ϭ 90°.mЄ1 ϭ ᎏ12ᎏ(mAB២Ϫ mAC២) Theorem 14-10mЄ1 ϭ ᎏ12ᎏ΂ Ϫ ΃ SubstitutionmЄ1 ϭ ᎏ12ᎏ΂ ΃orFind mЄ2.Vertex A of the secant-tangent angle is on circle K.mЄ2 ϭ ᎏ12ᎏ(mABC២) Theorem 14-11mЄ2 ϭ ᎏ12ᎏ΂ ϩ ΃ SubstitutionmЄ2 ϭ ᎏ12ᎏ΂ ΃or©Glencoe/McGraw-HillSecant-Tangent Angles14–4Geometry: Concepts and Applications 279• Find measures of arcsand angles formed bysecants and tangents.WHAT YOU’LL LEARNADBCK160Њ110Њ12Secant – Tangent AnglesVertex Outside the CircleSecant – tangent anglePQR intercepts PR២andPS២.Vertext on the CircleSecant - tangent angleABC intercepts AB២.BDACRSPQUnder thetab for Secant-TangentAngles, write thedefinitions of secant-tangent angles andtangent-tangent angles.KEY CONCEPTSOL G.10 The student will investigate and solve practical problems involving circles, usingproperties of angles, arcs, chords, tangents, and secants. Problems will include finding arclength and . . . may be drawn from applications of architecture, art, and construction.Theorem 14-10If a secant-tangent angle has its vertex outside the circle,then its degree measure is one-half the difference of thedegree measures of the intercepted arcs.Theorem 14-11If a secant-tangent angle has its vertex on the circle, thenits degree measure is one-half the degree measure of theintercepted arc.
  • 288. a. AෆZෆ is tangent to circle D at A.If mAB២ϭ 150, find mЄZ.b. EF៮៬ is tangent to circle D at E.If mEGC២ϭ 230, find mЄFEC.DCEFGZDYAB80ЊYour Turn©Glencoe/McGraw-Hill14–4280 Geometry: Concepts and ApplicationsA tangent-tangent angle is formed by two .Its vertex is always outside the circle.BUILD YOUR VOCABULARY (page 271)Explain the differencebetween a minor arcand a major arc of acircle. (Lesson 11-2)Theorem 14-12The degree measure of a tangent-tangent angle is one-halfthe difference of the degree measures of the intercepted arcs.REVIEW ITREMEMBER ITThe vertex of asecant-tangent anglecannot be located insidethe circle.Page(s):Exercises:HOMEWORKASSIGNMENTFind mЄG.ЄG is a tangent-tangent angle. ApplyTheorem 14-12.By definition of a right angle, mЄFOH ϭ 90. So,mFH២ϭ 90, because a minor arc is congruent to its central angle.Since the sum of the measures of a minor arc and its major arcis 360°, major arc FH២is 360° Ϫ 90° ϭ 270°.mЄG ϭ ᎏ12ᎏ(major arc FH២Ϫ minor arc FH២)mЄG ϭ ᎏ12ᎏ(270 Ϫ 90)mЄG ϭ ᎏ12ᎏ΂ ΃orFind mЄB.Your TurnFGHOBXQZE152Њ
  • 289. In circle A, find the value of x.PT ؒ TR ϭ QT ؒ TS Theorem 14-136 ؒ ϭ ؒ 12 Substitution48 ϭ 12xᎏ448ᎏ ϭ ᎏ142xᎏ Divide each side by .ϭ x Division PropertyFind the value of x in the circle.8123x2xYour TurnASRQP Tx8612©Glencoe/McGraw-HillSegment Measures14–5Geometry: Concepts and Applications 281• Find measures ofchords, secants, andtangents.WHAT YOU’LL LEARNUnder the tab forSegment Measures,write the definition ofan external secantsegment. Record thetheorems and othermain ideas from thislesson.CircleRelationshipsORGANIZE ITTheorem 14-13If two chords of a circle intersect, then the product of themeasures of the segments of one chord equals the productof the measures of the segments of the other chord.Theorem 14-14If two secant segments are drawn to a circle from anexterior point, then the product of the measures of onesecant segment and its external secant segment equals theproduct of the measures of the other secant segment andits external secant segment.Theorem 14-15If a tangent segment and a secant segment are drawn to acircle from an exterior point, then the square of the measureof the tangent segment equals the product of the measuresof the secant segment and its external secant segment.SOL G.10 The student will investigate and solve practical problems involving circles, usingproperties of angles, arcs, chords, tangents, and secants. Problems will include finding arclength and . . . may be drawn from applications of architecture, art, and construction.A segment is an external secant segment if and only if itis the part of a secant segment that is a circle.BUILD YOUR VOCABULARY (page 270)
  • 290. Explain the differencebetween Theorem 14-13and Theorem 14-14 inyour own words.WRITE ITFind the value of x to thenearest tenth.(x ϩ 6) ؒ 6 ϭ (5 ϩ 7) ؒ 5 Theorem 14-14ϭ 60 Distributive Property6x ϩ 36 Ϫ 36 ϭ 60 Ϫ 36 Subtract from each side.6x ϭᎏ66xᎏ ϭ ᎏ264ᎏ Divide each side by .x ϭUse the value of x to find the value of y.y2 ϭ (x ϩ 5) ؒ 5 Theorem 14-15y2 ϭ (4 ϩ 5) ؒ 5 Substitutiony2 ϭ͙y2ෆ ϭ ͙45ෆ Take the square root.y ϭ Ϸa. Find the value of x.b. Find the value of x.1848x468xYour Turnxy6557©Glencoe/McGraw-Hill14–5282 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENT
  • 291. Write the equation of a circle with center at (Ϫ4, 0) anda radius of 5 units.(x Ϫ h)2ϩ (y Ϫ k)2ϭ r2Equation of a Circle΄x Ϫ ΂ ΃΅2ϩ ΂y Ϫ ΃2ϭ (h, k) ϭ (Ϫ4, 0), r ϭ 5ϩ ϭThe equation for the circle is .Find the coordinates of the center and the measure ofthe radius of a circle whose equation is΂x ϩ ᎏ32ᎏ΃2ϩ ΂y Ϫ ᎏ12ᎏ΃2ϭ ᎏ14ᎏ.Rewrite the equation.(x Ϫ h)2ϩ (y Ϫ k)2ϭ r2΄x Ϫ ΂ ΃΅2ϩ ΂y Ϫ ΃2ϭ ΂ ΃2Since h ϭ , k ϭ , and r ϭ , the centerof the circle is at . Its radius is .©Glencoe/McGraw-HillEquations of Circles14–6Geometry: Concepts and Applications 283• Write equations ofcircles using the centerand the radius.WHAT YOU’LL LEARNUnder the tab forEquations of Circles,write the GeneralEquation of a Circle, anddraw a picture, labelingthe center and radius.Record several examplesto help you rememberthe main idea.CircleRelationshipsORGANIZE ITTheorem 14-16 General Equation of a CircleThe equation of a circle with center at (h, k) and a radius ofr units is (x Ϫ h)2ϩ (y Ϫ k)2ϭ r2.
  • 292. a. Write the equation of a circle with center C(5, Ϫ3) anda radius of 6 units.b. Find the coordinates of the center and the measure of theradius of a circle whose equation is (x ϩ 2)2ϩ (y ϩ 7)2ϭ 81.Your Turn©Glencoe/McGraw-Hill14–6284 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENT
  • 293. BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillIn circle P, AC៮៮ is a diameter; mCD២ϭ 68 andmBE២ϭ 96. Find each of the following.1. mЄABC2. mЄCED3. mAD២In circle A, HE៮៮ is a diameter.4. If mЄHTC ϭ 52, find mCH២.5. Find mHCE២.6. If mЄHTC ϭ 52, find mCEH២.CE HATUAB CPE DBUILD YOURVOCABULARYUse your Chapter 14 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumble puzzleof the vocabulary words inChapter 14, go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder(pages 270–271) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R14STUDY GUIDEGeometry: Concepts and Applications 28514-1Inscribed Angles
  • 294. Chapter BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillUnderline the best term to complete the statement.7. If a line is tangent to a circle, then it is perpendicular to theradius drawn to the [point of tangency/vertex].8. AෆBෆ is tangent to circle C. Find the value of x.9. Circle P is inscribed in right ᭝CTA.Find the perimeter of ᭝CTA if theradius of circle P is 5, CT ϭ 18, andJT ϭ 11.Underline the best term to complete the statement.10. A [radius/secant segment] is a line segment that intersects acircle in exactly two points.Find the value of x.11. 12.Sx1228Sx8452JACTPAxBC12161614286 Geometry: Concepts and Applications14-3Secant Angles14-2Tangents to a Circle
  • 295. Geometry: Concepts and Applications 287©Glencoe/McGraw-HillUnderline the best term to complete the statement.13. The measure of a(n) [tangent-tangent/inscribed] angle is alwaysone-half the difference of the measures of the intercepted arcs.Find the value of x. Assume that segments that appear to betangent are tangent.14. 15.Find the value of x.16. 17.18. Write the equation of the circle with center (Ϫ5, 9) and radius2͙5ෆ.19. What are the coordinates of the center and length of theradius for the circle (x ϩ 4)2ϩ y2ϭ 121.x1868x 92xxCDBA1481284xS20Chapter BRINGING IT ALL TOGETHER1414-5Segment Measures14-6Equations of Circles14-4Secant–Tangent Angles
  • 296. ©Glencoe/McGraw-HillVisit geomconcepts.com toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 14.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R14Checklist288 Geometry: Concepts and ApplicationsARE YOU READY FORTHE CHAPTER TEST?Check the one that applies. Suggestions to help you study aregiven with each item.I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 14 Practice Test onpage 627 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 14 Study Guide and Reviewon pages 624–626 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 14 Practice Test onpage 627.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 14 Foldable.• Then complete the Chapter 14 Study Guide and Review onpages 624–626 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 14 Practice Test onpage 627.
  • 297. Formalizing Proof©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R15Geometry: Concepts and Applications 289NOTE-TAKING TIP: To help you organize data,create a study guide or study cards when takingnotes, solving equations, defining vocabularywords and explaining concepts.FoldFold each sheet of paperin half along the width.Then cut along the crease.StapleStaple the eighthalf-sheets togetherto form a booklet.CutCut seven lines from thebottom of the top sheet,six lines from the secondsheet, and so on.LabelLabel each tab witha lesson number. The lasttab is for vocabulary.Begin with four sheets of 8ᎏ12ᎏ" ϫ 11" grid paper.Chapter1515-115-215-315-415-515-6VocabularyFormalizingProof
  • 298. ©Glencoe/McGraw-Hill290 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 15.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term’s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R15BUILD YOUR VOCABULARYVocabulary TermFoundDefinitionDescription oron Page Examplecompound statementconjunctioncontrapositivecoordinate proofdeductive reasoning[dee-DUK-tiv]disjunctionindirect proofindirect reasoninginverseLaw of Detachment
  • 299. ©Glencoe/McGraw-HillChapter BUILD YOUR VOCABULARY15Geometry: Concepts and Applications 291Vocabulary TermFoundDefinitionDescription oron Page ExampleLaw of Syllogism[SIL-oh-jiz-um]logically equivalentnegationparagraph proofproofproof by contradictionstatementtruth tabletruth valuetwo-column proof
  • 300. Logic and Truth Tables©Glencoe/McGraw-Hill292 Geometry: Concepts and Applications15–1Let p represent “An octagon has eight sides” and qrepresent “Water does not boil at 90°C.”Write the negation of statement p.ϳ p: An octagon have eight sides.Write the negation of statement q.ϳ q: Water boil at 90°C.• Find the truth values ofsimple and compoundstatements.WHAT YOU’LL LEARN A statement is any sentence that is either true or false, butnot both.Every has a truth value, true (T) or false (F).If a statement is represented by p, then p is thenegation of the statement.The relationship between the of astatement are organized on a truth table.When two statements are , they form acompound statement.A conjunction is a statement formed byjoining two statements with the word .A disjunction is a statement formed byjoining two statements with the word .BUILD YOUR VOCABULARY (pages 290–291)Under the tab forLesson 15-1, list anddefine the followingsymbols used in thelesson: ϳ , ٙ, ٚ, and →.Under the last tab, listthe vocabulary wordsand their definitionsfrom Lesson 15-1.15-115-215-315-415-515-6VocabularyFormalizingProofORGANIZE ITSOL G.1.b The student will construct and judge the validity of a logical argumentconsisting of a set of premises and a conclusion. This will include translating a shortverbal argument into symbolic form.
  • 301. ©Glencoe/McGraw-Hill15–1Geometry: Concepts and Applications 293Let p represent “Tofu is a protein source”and q represent “␲ is not a rational number.”a. Write the negation b. Write the negationof statement p. of statement q.Your TurnLet p represent “92ϭ 99”, q represent “An equilateraltriangle is equiangular”, and r represent “A rectangularprism has six faces.” Write the statement for eachconjunction or disjunction. Then find the truth value.ϳ p ٙ q9299 and an equilateral triangle is equiangular. Because pis , ϳ p is . Therefore, ϳ p ٙ q isbecause both ϳ p and q are .p ٚ ϳ r92ϭ 99 or a rectangular prism does not have six faces.Because r is , ϳ r is . Therefore, p ٚ ϳ r isbecause both p and ϳ r are .ϳ q ٙ ϳ rAn equilateral triangle is not equiangular and a rectangularprism does not have six faces. Because q is , ϳ q is; and because r is , ϳ r is .Therefore, ϳ q ٙ ϳ r is because both ϳ q and ϳ rare .REMEMBER ITIn the Negationtruth table, p doesnot have to be a truestatement and ϳp isnot necessarily a falsestatement.Write in if-then form:All natural numbersare whole numbers.(Lesson 1-4)REVIEW IT
  • 302. Let p represent “0.5 is an integer”, qrepresent “A rhombus has four congruent sides”, and rrepresent “A parallelogram has congruent diagonals.”Write the statement for each conjunction or disjunction.Then find the truth value.a. ϳ p ٙ q b. ϳ p ٚ r c. ϳ q ٙ ϳ rConstruct a truth tablefor the conjunctionϳ (p ٙ q).Make columns with theheadings p, q, ,and ϳ (p ٙ q). Then, listall possible combinationsof truth values for p and q.Use these truth values tocomplete the last two columns of the andits .Constructa truth table for thedisjunction ϳ (p ٚ q).Your TurnYour Turn©Glencoe/McGraw-Hill15–1294 Geometry: Concepts and ApplicationsREMEMBER ITA disjunction isfalse only when bothstatements are false.The converse of aconditional is falsewhen p is false and q istrue. A conditional isfalse only when p is trueand q is false.Page(s):Exercises:HOMEWORKASSIGNMENTThe inverse of a conditional is formed byboth p and q.The contrapositive of a conditional statement is formed bynegating the of the statement.Two statements are logically equivalent if their truth tablesare the .BUILD YOUR VOCABULARY (pages 290–291)p q p ٙ q ϳ (p ٙ q)
  • 303. Geometry: Concepts and Applications 295©Glencoe/McGraw-HillDeductive Reasoning15–2Use the Law of Detachment to determine a conclusionthat follows from statements (1) and (2). If a validconclusion does not follow, then write no valid conclusion.(1) In a plane, if a line is perpendicular to one of twoparallel lines, then it is perpendicular to the other line.(2) AෆBෆ ʈ CෆDෆ and EෆFෆ Ќ AෆBෆ.p: AෆBෆ ʈ CCෆDෆ and .q: EෆFෆ ЌStatement (1) indicates that p → q is , and statement(2) indicates that p is . So, is true. Therefore,EෆFෆ Ќ CCෆDෆ.(1) Two nonvertical lines have the same slope if andonly if they are parallel.(2) AෆBෆ is a vertical line.p: Two lines are nonvertical and .q: Two lines have the same .Statement (2) indicates that p is . Therefore,there is no valid conclusion.• Use the Law ofDetachment and theLaw of Syllogism indeductive reasoning.WHAT YOU’LL LEARN Deductive reasoning is the process of using facts, rules,definitions, and properties in a logical order.The Law of Detachment allows us to reach logicalfrom statements.The Law of Syllogism is similar to the Transitive Propertyof Equality.BUILD YOUR VOCABULARY (pages 290–291)Law of DetachmentIf p → q is a trueconditional and p istrue, then q is true.Under the tabfor Lesson 15-2,summarize the Law ofDetachment and theLaw of Syllogism in yourown words.KEY CONCEPTSOL G.1.d The student will construct and judge the validity of a logical argumentconsisting of a set of premises and a conclusion. This will include using deductivereasoning, including the law of syllogism.
  • 304. ©Glencoe/McGraw-Hill15–2296 Geometry: Concepts and ApplicationsUse the Law of Detachment to determinea conclusion that follows from statements (1) and (2).If a valid conclusion does not follow, then write novalid conclusion.a. (1) If a figure is an isosceles triangle, then it has twocongruent angles.(2) A figure is an isosceles triangle.b. (1) If a hexagon is regular, each interior anglemeasures 120°.(2) The hexagon is regular.Use the Law of Syllogism to determine a conclusion thatfollows from statements (1) and (2).(1) If mЄK ϭ 90, then ЄK is a right angle.(2) If ЄK is a right angle, then ᭝JKL is a right triangle.p: mЄK ϭq: ЄK is a right angle.r: ᭝JKL is a triangle.Use the Law of Syllogism to conclude p → r.Therefore, if , then ᭝JKL is atriangle.Use the Law of Syllogism to determine aconclusion that follows from statements (1) and (2).(1) If it is rainy tomorrow, then Alan cannot play golf.(2) If Alan cannot play golf, then he will watch television.Your TurnYour TurnPage(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITIn the Law ofSyllogism, bothconditionals must betrue for the conclusionto be true.
  • 305. ©Glencoe/McGraw-HillGeometry: Concepts and Applications 297Paragraph Proofs15–3Write a paragraph proof for the conjecture.In ᭝RST, if TX៮៮ Ќ RS៮៮ and TX៮៮ bisects ЄRTS, thenRX៮៮ Х XS៮៮.Given: TෆXෆ Ќ RෆSෆ; TෆXෆ bisects ЄRTS.Prove: RෆXෆ Х XෆSෆProof: If TෆXෆ Ќ RෆSෆ, then ЄRXT and ЄTXS areangles and ᭝RXT and are right triangles.If TෆXෆ bisects ЄRTS, then ЄRTX Х ЄSTX by thedefinition of angle . Also, TෆXෆ Хsince congruence is . So, ᭝RTX Х ᭝STXby the Theorem. Therefore, RෆXෆ Х XෆSෆ becauseparts of congruent triangles arecongruent (CPCTC).SX TR• Use paragraph proofsto prove theorems.WHAT YOU’LL LEARNUnder the tab forLesson 15-3, summarizewhat information islisted as “Given” and“Prove” in a paragraphproof.15-115-215-315-415-515-6VocabularyFormalizingProofORGANIZE ITA proof is a logical argument in which each statement isbacked up by a that is accepted as .Statements and reasons are written inform in a paragraph proof.BUILD YOUR VOCABULARY (page 291)SOL G.1.b The student will construct and judge the validity of a logical argumentconsisting of a set of premises and a conclusion. This will include translating a shortverbal argument into symbolic form.
  • 306. ©Glencoe/McGraw-Hill15–3298 Geometry: Concepts and ApplicationsIf Є1 and Є2 are congruent, then ᐉis parallel to m.Given: Є1 Х Є2Prove: ᐉ ʈProof: Vertical angles are congruent so Є2 Х Є3. SinceЄ1 Х Є2, Є1 Х by substitution. If two linesin a are cut by a so thatcorresponding angles are , then thelines are . Therefore, .Write a paragraph proof for eachconjecture.a. If A is the midpoint of DෆCෆand EෆBෆ,then ᭝DAE Х ᭝CAB.Given: A is the midpoint of DෆCෆand EෆBෆProve: ᭝DAE Х ᭝CABb. If Є3 Х Є4, then Є5 Х Є6. JXY K516423EDACBYour Turnᐉ13 mt2Page(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITThere is more thanone way to plan aproof.What can you say aboutcorresponding anglesformed when parallellines are cut by atransversal? (Lesson 4-3)REVIEW IT
  • 307. ©Glencoe/McGraw-HillGeometry: Concepts and Applications 299Preparing for Two-Column Proofs15–4Justify the steps for the proof of theconditional. If ЄXWY Х ЄXYW, thenЄAWX Х ЄBYX.XWA BY• Use properties ofequality in algebraicand geometric proofs.WHAT YOU’LL LEARN A two-column proof is a deductive argument withand organized in twocolumns.BUILD YOUR VOCABULARY (page 291)Statements Reasons1. Х 1. Given2. mЄXWY ϭ mЄXYW 2.3. mЄAWX ϩ mЄXWY ϭ 180; 3.mЄBYX ϩ mЄXYW ϭ 1804. mЄAWX ϩ mЄXWY ϭ 4. Substitutionϩ5. mЄAWX ϩ mЄXWY ϭ 5. Substitutionϩ6. mЄAWX ϭ 6. Subtraction property7. Х 7. Definition of congruentanglesUnder the tab forLesson 15-4, summarizethe Properties ofEquality fromLesson 2-2.15-115-215-315-415-515-6VocabularyFormalizingProofORGANIZE ITSOL G.1.b The student will construct and judge the validity of a logical argumentconsisting of a set of premises and a conclusion. This will include translating a shortverbal argument into symbolic form.
  • 308. Justify the stepsfor the proof of the conditional.If mЄAOC ϭ mЄBOD, thenmЄAOB ϭ mЄCOD.Given: mЄAOC ϭ mЄBODProve: mЄAOB ϭ mЄCODProof:Show that if A ϭ ᎏ12ᎏbh, then b ϭ ᎏ2hAᎏ.Given: A ϭ ᎏ12ᎏbhProve: b ϭ ᎏ2hAᎏProof:Your Turn©Glencoe/McGraw-Hill15–4300 Geometry: Concepts and ApplicationsStatements Reasons1. 1.2.3.4.5.2.3.4.5.Statements Reasons1. A ϭ ᎏ12ᎏbh 1. Given2. ϭ bh 2. Multiplication property3. ᎏ2hAᎏ ϭ b 3.4. b ϭ 4. Symmetric propertyREMEMBER ITYou cannot write astatement unlessyou give a reason tojustify it.What information isalways in the firststatement of a proof?What information canalways be found in thelast satement?WRITE ITAODCB
  • 309. Show that if PV ϭ nRT, then R ϭ ᎏPnTVᎏ.Given:Prove:Proof:Your Turn©Glencoe/McGraw-Hill15–4Geometry: Concepts and Applications 301Page(s):Exercises:HOMEWORKASSIGNMENTStatements Reasons1. 1.2.3.2.3.
  • 310. Two-Column ProofsWrite a two-column proof for the conjecture.If Є1 ϭ Є2, then quadrilateral ABCD is a trapezoid.Given: Є1 ϭ Є2Prove: ᐉ ʈ mProof:Write a two-columnproof. If ᭝XYZ is isosceles withXෆZෆ Х XෆYෆ and OෆZෆ Х NෆYෆ, thenOෆYෆ Х NෆZෆ.Given: ᭝XYZ is isosceles withXZ៮ Х XY៮ and OZ៮ Х NY៮Prove: OY៮ Х NZ៮XO NZ YYour TurnAB CD ᐉmtn132©Glencoe/McGraw-Hill302 Geometry: Concepts and Applications15–5• Use two-column proofsto prove theorems.WHAT YOU’LL LEARNUnder the tab forLesson 15-5, summarizethe process to write atwo-column proof.15-115-215-315-415-515-6VocabularyFormalizingProofORGANIZE ITStatements Reasons1. Х 1. Given2. Є2 Х 2.3. Х 3. Substitution4. 4. If two lines in a planeare cut by a transversalso that correspondingangles are congruent,then the lines areparallel.5. Quadrilateral ABCD is 5.a trapezoid.SOL G.1.b The student will construct and judge the validity of a logical argumentconsisting of a set of premises and a conclusion. This will include translating a shortverbal argument into symbolic form.
  • 311. Write a two-column proof.Given: X is the midpoint of both BෆDෆ and AෆCෆ.Prove: ᭝DXC Х ᭝BXAProof:ADXBC©Glencoe/McGraw-Hill15–5Geometry: Concepts and Applications 303Statements Reasons1. 1.2.3.4.5.6.2.3.4.5.6.Statements Reasons1. X is the midpoint of both 1. GivenBෆDෆ and AෆCෆ.2. DෆXෆ Х BෆXෆ; Х 2.3. Х 3. Vertical angles arecongruent.4. Х 4.Proof:
  • 312. ©Glencoe/McGraw-Hill15–5304 Geometry: Concepts and ApplicationsPage(s):Exercises:HOMEWORKASSIGNMENTWrite a two-column proof.Given: AD and CE bisect each other.Prove: AE ʈ CDYour TurnStatements Reasons1. 1.2.3.4.5.6.2.3.4.5.6.1EBADC243Proof:
  • 313. ©Glencoe/McGraw-HillGeometry: Concepts and Applications 305Coordinate Proofs15–6Position and label a rectanglewith length b and height d ona coordinate plane.• Use the origin as a .• Place one side on the x-axis and one side on the .• Label the A, B, C and D.• Label the coordinates D΂ ΃, C΂ , 0΃,B΂ ΃, and A΂0, ΃.Position and label an isosceles triangle withbase m units long and height n units on a coordinate plane.Write a coordinate proof to prove that the oppositesides of a parallelogram are congruent.Given: parallelogram ABDCProve: AෆBෆ Х CෆDෆ and AෆCෆ Х BෆDෆYour TurnyO xA(0, d) B(b, d)D(0, 0) C(b, 0)• Use coordinate proofsto prove theorems.WHAT YOU’LL LEARNA proof that uses on a coordinate plane is acoordinate proof.BUILD YOUR VOCABULARY (page 290)Guidelines for PlacingFigures on a CoordinatePlane1. Use the origin as avertex or center.2. Place at least oneside of a polygon onan axis.3. Keep the figurewithin the firstquadrant, if possible.4. Use coordinates thatmake computations assimple as possible.Under thetab for Lesson 15-6,summarize the Guidelinesfor Placing Figures on aCoordinate Plane.KEY CONCEPTSOL G.1.b The student will construct and judge the validity of a logical argumentconsisting of a set of premises and a conclusion. This will include translating a shortverbal argument into symbolic form.
  • 314. ©Glencoe/McGraw-Hill15–6306 Geometry: Concepts and ApplicationsProof:Label the vertices A(0, 0), B(a, 0),D(a ϩ b, c), and C(b, c). Use theDistance Formula to find AB, CD,AC, and BD.AB ϭ ͙(a Ϫ 0ෆ)2ϩ (0ෆ Ϫ 0)2ෆϭ ͙a2ෆ or aCD ϭ ͙[(a ϩෆb) Ϫ bෆ]2ϩ (cෆ Ϫ c)2ෆ ϭ or aAC ϭ ϭ͙b2ϩ cෆ2ෆBD ϭϭSo, AB ϭ CD and AC ϭ BD.Therefore, and ; oppositesides of a parallelogram are .Write a coordinateproof to prove that a WXYZ is arectangle by proving the diagonalsare congruent.Your TurnyO xC(b, c) D(a + b, c)A(0, 0) B(a, 0)(0, 0) (a, 0)ZWXY(0, b)(a, b)What is slope and howwould you determinethe slope of a line?(Lesson 4-6)REVIEW ITWhat is the DistanceFormula? (Lesson 6-7)REVIEW IT(b Ϫ )2ϩ (c Ϫ )2[(a ϩ b) Ϫ]2ϩ (c Ϫ )2
  • 315. ©Glencoe/McGraw-Hill15–6Geometry: Concepts and Applications 307Write a coordinate proof to provethat the length of the segmentjoining the midpoints of two sidesof a triangle is one-half the lengthof the third side.Given: ᭝EFG with midpoints J and K, of EෆFෆ and FෆGෆProve: JK ϭ ᎏ12ᎏEGLabel the vertices E , F , and G(2c, 0).Use the Midpoint Formula to find the coordinates of J and K,and the Distance Formula to find JK and EG.Coordinates of J: ΂ᎏ0 ϩ22aᎏ, ᎏ0 ϩ22bᎏ΃ ϭCoordinates of K: ΂ᎏ2a ϩ22cᎏ, ᎏ2b2ϩ 0ᎏ΃ϭ (a ϩ c, b)JK ϭ ͙[(a ϩෆc) Ϫ aෆ]2ϩ (bෆ Ϫ b)2ෆ ϭ ͙c2ෆ or cEG ϭ ͙(2c Ϫෆ0)2ϩෆ(0 Ϫ 0ෆ)2ෆ ϭ ͙(2c)2ෆ orᎏ12ᎏEG ϭ ᎏ12ᎏ ϭTherefore, .Write acoordinate proof to provethat the length of a mediansegment joining themidpoints of two legs ofa trapezoid is one-halfthe sum of the length ofthe bases.Your TurnWhat is the MidpointFormula? (Lesson 2-5)REVIEW IT(0, 0)UMR SNT(2c, o)(a + c, b)(2a, 2b)(0, b)(0, 2b)Page(s):Exercises:HOMEWORKASSIGNMENTyO xF(2a, 2b)J KE(0, 0) G(2c, 0)
  • 316. Indicate whether the statement is true or false.1. A table that lists all truth values of a statement is a truthtable.2. p → q is an example of a disjunction.3. ϳ p → ϳ q is the inverse of a conditional statement.4. p ٚ q is an example of a conjunction.5. Complete the truth table.15-1Logic and Truth Tablesp q ϳ p ϳ q p ٙ q p ٚ q ϳ p → ϳ qTTFFTFTF©Glencoe/McGraw-Hill308 Geometry: Concepts and ApplicationsBRINGING IT ALL TOGETHERBUILD YOURVOCABULARYUse your Chapter 15 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumble puzzleof the vocabulary Chapter 15,go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder(pages 290–291) to help yousolve the puzzle.VOCABULARYPUZZLEMAKERC H A P T E R15STUDY GUIDE
  • 317. ©Glencoe/McGraw-HillChapter BRINGING IT ALL TOGETHER15Geometry: Concepts and Applications 309Draw a conclusion from statements (1) and (2).6. (1) All functions are relations.(2) x ϭ y2is a relation.7. (1) Integers are rational numbers.(2) (Ϫ6) is an integer.8. (1) If it is Saturday, I see my friends.(2) If I see my friends, we laugh.Indicate whether the statement is true or false.9. A proof is a logical argument where each statement is backedup by a reason accepted as true.Write a paragraph proof.10. Given: mЄ1 ϭ mЄ2; mЄ3 ϭ mЄ4Prove: mЄ1 ϩ mЄ4 ϭ 9015-3Paragraph Proofs1 43215-2Deductive Reasoning
  • 318. Chapter BRINGING IT ALL TOGETHER©Glencoe/McGraw-Hill15310 Geometry: Concepts and ApplicationsComplete the statement.11. A proof containing statements and reasons and is organized bysteps is a proof.Complete the proof.12. Given: AෆBෆ and AෆRෆ aretangent to circle K.Prove: ЄBAK Х ЄRAK13. Write a two-column proof.Given: AෆBෆ is tangent to circle X at B.AෆCෆ is tangent to circle X at C.Prove: AB៮៮ Х AC៮៮BRK AStatements Reasons1. AෆBෆ and AෆRෆ are tangent 1.to circle K2. Х 2. If 2 segments from the sameexterior point are tangent toa circle, then they are Х.3. BෆKෆ Х RෆKෆ 3.4. Х 4.5. Х ᭝RAK 5.6. Х ЄRAK 6.15-4Preparing for Two-Column Proofs15-5Two Column ProofsBCXAProof:
  • 319. ©Glencoe/McGraw-HillComplete the statement.14. The vertex or center of the figure should beplaced on the .15. Position and label a rhombus on a coordinate plane with base r and height t.(a, t)(0, 0) (r, 0)(r + a, t)Chapter BRINGING IT ALL TOGETHERGeometry: Concepts and Applications 3111515-6Coordinate ProofsStatements Reasons1. AෆBෆ is tangent to circle X 1.at B. AෆCෆ is tangent tocircle X at C.2. Draw BෆXෆ, CX៮៮, and AෆXෆ. 2. Through any 2there is 1 .3. ЄABX and ЄACX are 3. If a line is tangent to acircle, then it is Ќ to theradius drawn to the point oftangency.4. BෆXෆ Х CෆXෆ 4.5. Х 5. Reflexive Property6. ᭝AXB Х 6. HL7. 7. CPCTC.Givenpointslineright anglesAll radii of acircle are Х.AX៮៮ AX៮៮᭝AXCAB៮៮ Х AC៮៮originProof:
  • 320. ©Glencoe/McGraw-HillCheck the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 15 Practice Test onpage 671 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 15 Study Guide and Reviewon pages 668–670 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 15 Practice Test onpage 671.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 15 Foldable.• Then complete the Chapter 15 Study Guide and Review onpages 668–670 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 15 Practice Test onpage 671.Visit geomconcepts.com toaccess your textbook,more examples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 15.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R15Checklist312 Geometry: Concepts and Applications
  • 321. More Coordinate Graphing and Transformations©Glencoe/McGraw-HillUse the instructions below to make a Foldable to help youorganize your notes as you study the chapter. You will seeFoldable reminders in the margin of this Interactive StudyNotebook to help you in taking notes.C H A P T E R16Geometry: Concepts and Applications 313NOTE-TAKING TIP: When taking notes, markanything you do not understand with a questionmark. Be sure to ask your instructor to explain theconcepts or sections before your next quiz or exam.StapleStaple the six sheets ofgraph paper onto theposter board.LabelLabel the six pages withthe lesson titles.Begin with six sheets of graph paper and an8ᎏ12ᎏ" ϫ 11" poster board.Chapter16
  • 322. ©Glencoe/McGraw-Hill314 Geometry: Concepts and ApplicationsThis is an alphabetical list of new vocabulary terms you will learn in Chapter 16.As you complete the study notes for the chapter, you will see Build YourVocabulary reminders to complete each term´s definition or description onthese pages. Remember to add the textbook page number in the secondcolumn for reference when you study.C H A P T E R16BUILD YOUR VOCABULARYVocabulary TermFoundDefinitionDescription oron Page Examplecenter of rotationcomposition oftransformationsdilation[dye-LAY-shun]elimination[ee-LIM-in-AY-shun]reflectionrotationsubstitution[SUB-sti-TOO-shun]system of equationstranslationturn
  • 323. ©Glencoe/McGraw-HillSolving Systems of Equations by Graphing16–1Geometry: Concepts and Applications 315Solve each system of equations by graphing.y ϭ x Ϫ 1y ϭ Ϫx ϩ 3Find ordered pairs by choosing values for x and finding thecorresponding y-values.Graph the ordered pairs and draw thegraphs of the equations. The graphsintersect at the point whose coordinatesare . Therefore, the solutionof the system of equations is .y ϭ Ϫ2xy ϭ Ϫ2x ϩ 3Use the slope and y-intercept to grapheach equation.The slope of each line is so the graphs areand do not intersect. Therefore, there is .xyOy ϭ Ϫ2x ϩ 3y ϭ Ϫ2xxyO(2, 1)y ϭ Ϫx ϩ 3y ϭ x Ϫ 1• Solve systems ofequations by graphing.WHAT YOU’LL LEARNOn the page labeledSolving Systems ofEquations by Graphing,sketch graphs of systemsof equations. Explainwhy each graph producesthe result that it does.ORGANIZE ITA set of two or more equations is called a system ofequations.BUILD YOUR VOCABULARY (page 314)y ϭ x Ϫ 1x x Ϫ 1 y (x, y)3 2 2 (3, 2)2 1 1 (2, 1)1 0 0 (1, 0)y ϭ Ϫx ϩ 3x Ϫx ϩ 3 y (x, y)3 0 0 (3, 0)2 1 1 (2, 1)1 2 2 (1, 2)Equation Slope y-intercepty ϭ Ϫ2x Ϫ2 0y ϭ Ϫ2x ϩ 3 Ϫ2 3
  • 324. ©Glencoe/McGraw-Hill16–1316 Geometry: Concepts and ApplicationsSolve each system of equations by graphing.a. x Ϫ 2y ϭ 23x ϩ y ϭ 6b. 3x ϩ 2y ϭ 123x ϩ 2y ϭ 6Toshiro wants a wildflower garden. He wants the lengthto be 1.5 times the width and he has 100 meters offencing to put around the garden. If w represents thewidth of the garden and ᐉ represents the length, solvethe system of equations below to find the dimensions ofthe wildflower garden.ᐍ ϭ 1.5w2w ϩ 2ᐍ ϭ 100Solve the second equation for ᐍ.2w ϩ 2ᐍ ϭ 100 The perimeter is meters.2w ϩ 2ᐍ Ϫ 2w ϭ 100Ϫ 2w Subtract from eachside.ϭ 100 Ϫ 2wᎏ22ᐍᎏ ϭ ᎏ1002Ϫ 2wᎏ Divide.ᐍ ϭYour TurnExplain how to graph6x Ϫ 2y ϭ 8 using theslope-intercept method.(Lesson 4-6)REVIEW ITExplain how to solve asystem of equations bygraphing.WRITE ITOxy3x ϩ 3y ϭ 63x ϩ 2y ϭ 12O xyx Ϫ2y ϭ 23x ϩ y ϭ 6(2, 0)
  • 325. ©Glencoe/McGraw-Hill16–1Geometry: Concepts and Applications 317Use a graphing calculator to graph the equationsand to find the coordinates of theintersection point. Note that these equations can be written asy ϭ 1.5x and y ϭ 50 Ϫ x and then graphed.Enter: [y ϭ ] 1.5 50Next, use the intersection tool on to find the coordinatesof the point of intersection.The solution is . Since w ϭ and ᐍ ϭ ,the width of the garden is meters and the length ismeters.Check your answer by examining the original problem.Is the length of the garden 1.5 times the width? ✔Does the garden have a perimeter of 100 meters? ✔The solution checks.Ruth wants to enclose an area of her yard forher children to play. She has 72 meters of fence. The length ofthe play area is 4 meters greater than 3 times the width. Whatare the dimensions of the play area?Your TurnF5GRAPHϫ—ENTERϫPage(s):Exercises:HOMEWORKASSIGNMENTREMEMBER ITCheck the solutionto a system of equationsby substituting it intoeach equation.
  • 326. Solving Systems of Equations by Using Algebra©Glencoe/McGraw-Hill318 Geometry: Concepts and Applications16–2Use substitution to solve the system of equations.y ϭ x ϩ 42x ϩ y ϭ 1Substitute x ϩ 4 for y in the second equation.2x ϩ y ϭ 12x ϩ ϭ 13x ϩ 4 ϭ 1 Combine like terms.3x ϩ 4 Ϫ ϭ 1 Ϫ Subtract from eachside.3x ϭ Ϫ3ᎏ33xᎏ ϭ ᎏϪ33ᎏ Divide each side by .x ϭ Division PropertySubstitute Ϫ1 for x in the first equation and solve for y.y ϭ (Ϫ1) ϩ 4 ϭThe solution to this system of equations is .Use substitution to solve 2x Ϫ y ϭ 4 and x ϭ y ϩ 5.Your TurnOne algebraic method for solving a system of equations iscalled substitution.Another algebraic method for solving systems of equationsis called elimination.BUILD YOUR VOCABULARY (page 314)• Solve systems ofequations by using thesubstitution orelimination method.WHAT YOU’LL LEARNOn the page labeledSolving Systems ofEquations by UsingAlgebra, write a systemof equations and solveit using substitution andelimination. Explain theprocess you used witheach method.ORGANIZE IT
  • 327. ©Glencoe/McGraw-Hill16–2Geometry: Concepts and Applications 319Use elimination to solve the system of equations.3x Ϫ 2y ϭ 44x ϩ 2y ϭ 103x Ϫ 2y ϭ 4(ϩ) 4x ϩ 2y ϭ 10 Add the equations to eliminate the y terms.7x ϩ 0 ϭ 14ᎏ77xᎏ ϭ ᎏ174ᎏ Divide each side by 7.x ϭThe value of x in the solution is .Now substitute in either equation to find the value of y.3x Ϫ 2y ϭ 43΂ ΃Ϫ 2y ϭ 4Ϫ 2y ϭ 46 Ϫ 2y Ϫ 6 ϭ 4 Ϫ 6 Subtract from each side.ϭ Subtraction PropertyᎏϪϪ22yᎏ ϭ ᎏϪϪ22ᎏ Divide each side by .y ϭThe value of y in the solution is .The solution to the system is .Use elimination to solve x ϩ y ϭ 7 and2x Ϫ y ϭ Ϫ1.Your Turn
  • 328. ©Glencoe/McGraw-Hill16–2320 Geometry: Concepts and ApplicationsUse elimination to solve the system of equations.3x ϩ y ϭ 6x Ϫ 2y ϭ 93x ϩ y ϭ 6 6x ϩ 2y ϭ 12x Ϫ 2y ϭ 9 ϩ x Ϫ 2y ϭ 97x ϩ 0 ϭ 21 Combine like terms.ᎏ77xᎏ ϭ ᎏ271ᎏ Divide.x ϭSubstitute 3 into either equation to solve for y.3x ϩ y ϭ 63΂ ΃ϩ y ϭ 6 Replace x with .9 ϩ y ϭ 69 ϩ y Ϫ 9 ϭ 6 Ϫ 9 Subtract from each side.y ϭ Subtraction PropertyThe solution of this system is .Use elimination to solve 7x ϩ 3y ϭ Ϫ1 and4x ϩ y ϭ 3.Your TurnPage(s):Exercises:HOMEWORKASSIGNMENTExplain the differencebetween solving asystem of equations bysubstitution or by theelimination method.WRITE IT(ϫ 2)
  • 329. ©Glencoe/McGraw-HillTranslations16–3Geometry: Concepts and Applications 321Graph ᭝LMN with vertices L(0, 3), M(4, 2), and N(Ϫ3, Ϫ1).Then find the coordinates of its vertices if it is translatedby (5, 0). Graph the translation image.To find the coordinates of the vertices of ᭝LЈMЈNЈ, add 5 toeach x-coordinate and add 0 to each y-coordinate of ᭝LMN:(x ϩ 5, y ϩ 0).L(0, 3) ϩ (5, 0) LЈ(0 ϩ 5, 3 ϩ 0) ϭ LЈM(4, 2) ϩ (5, 0) MЈ(4 ϩ 5, 2 ϩ 0) ϭ MЈN(Ϫ3, Ϫ1) ϩ (5, 0) NЈ(Ϫ3 ϩ 5, Ϫ1 ϩ 0) ϭ NЈGraph ᭝ABC with vertices A(1, 2), B(Ϫ3, Ϫ1),and C(2, 1). Then find the coordinates of its vertices if it istranslated by (3, Ϫ2). Graph the translation image.Your TurnxyOL LЈMЈNЈNM• Investigate and drawtranslations on acoordinate plane.WHAT YOU’LL LEARNOn the page labeledTranslations, sketchgraphs of severaldifferent translations.Explain why eachtranslation producesthe result it does.ORGANIZE ITA translation is a slide of a figure from one position toanother.BUILD YOUR VOCABULARY (page 314)SOL G.2.c The student will use . . . coordinate methods, to solve problems involving symmetryand transformation. This will include determining whether a figure has been translated, reflected,or rotated.xyOCBACBAPage(s):Exercises:HOMEWORKASSIGNMENT
  • 330. ReflectionsGraph ᭝ABC with vertices A(0, 0), B(4, 1), and C(1, 5).Then find the coordinates of its vertices if it is reflectedover the x-axis and graph its reflection image.To find the coordinates of the vertices of᭝AЈBЈCЈ, use the definition of reflectionover the x-axis: (x, y) (x, Ϫy).A(0, 0) AЈB(4, 1) BЈC(1, 5) CЈThe vertices of ᭝AЈBЈCЈ are , ,and .In the same ᭝ABC, find the coordinates of the verticesof ᭝ABC after a reflection over the y-axis. Graph thereflected image.To find the coordinates of AЉ, BЉ, and CЉ,use the definition of reflection over they-axis: (x, y) (Ϫx, y).A(0, 0) AЉB(4, 1) BЉC(1, 5) CЉThe vertices of ᭝AЉBЉCЉ are , , and.©Glencoe/McGraw-Hill322 Geometry: Concepts and Applications16–4• Investigate and drawreflections on acoordinate plane.WHAT YOU’LL LEARNOn the pages labeledReflections, sketch graphsof several differentreflections. Explain whyeach reflection producesthe result it does.ORGANIZE ITA reflection is the flip of a figure over a line to produce amirror image.BUILD YOUR VOCABULARY (page 314)xyBCCBOxyBCBCOSOL G.2.c The student will use . . . coordinate methods, to solve problems involvingsymmetry and transformation. This will include determining whether a figure has beentranslated, reflected, or rotated.
  • 331. a. Graph quadrilateral QUAD with vertices Q(Ϫ3, 3), U(3, 2),A(4, Ϫ4), and D(Ϫ4, Ϫ1). Then find the coordinates of itsvertices if it is reflected over the y-axis. Graph its reflectionimage.b. Graph ᭝STU with vertices S(1, 2), T(4, 4), and U(3, Ϫ3).Then find the coordinates of its vertices if it is reflectedover the y-axis and graph its reflection image.Your Turn©Glencoe/McGraw-Hill16–4Geometry: Concepts and Applications 323Page(s):Exercises:HOMEWORKASSIGNMENTReflect a figure over thex-axis and then reflectits image over the y-axis.Is this double reflectionthe same as atranslation? Explain.WRITE ITxyODQ QDUUA AxySTSUTUO
  • 332. RotationsRotate ᭝ABC 270° clockwise aboutpoint A.• The center of rotation is A. Use a protractor to draw anangle of clockwise about point A, using AෆBෆ as abaseline for your protractor.• Draw segment AAෆЈෆBෆЈෆ to AෆBෆ.• Trace the figure on a piece of paper and rotate the top paperclockwise, until the figure is rotated clockwise.• Draw ᭝AЈBЈCЈ congruent to ᭝ABC.Rotate ᭝XYZ 60° counterclockwise about point Y.Your TurnAAЈB CBЈCЈ©Glencoe/McGraw-Hill324 Geometry: Concepts and Applications16–5• Investigate and drawrotations on acoordinate plane.WHAT YOU’LL LEARNOn the page labeledRotations, sketch graphsof several differentrotations. Explain whyeach rotation producesthe result it does.ORGANIZE ITA rotation, also called a turn, is a movement of a figurearound a point. The fixed point may be in theof the object or a point theobject and is called the center of rotation.BUILD YOUR VOCABULARY (page 314)60ЊZXZXYYSOL G.2.c The student will use . . . coordinate methods, to solve problems involvingsymmetry and transformation. This will include determining whether a figure has beentranslated, reflected, or rotated.
  • 333. Graph ᭝XYZ with vertices X(Ϫ2, 1), Y(2, Ϫ3), and Z(3, 5).Then find the coordinates of the vertices after thetriangle is rotated 180° clockwise about the origin. Graphthe rotation image.• Draw a segment from the origin topoint X.• Use a protractor to reproduceOෆXෆ at a 180° angle so thatOX ϭ OXЈ.• Repeat this procedure with pointsY and Z.The rotation image ᭝XЈYЈZЈ has vertices XЈ ,YЈ , and ZЈ .Rotate ᭝ABC 90° counterclockwise around theorigin. The vertices are A(0, 4), B(3, 1), and C(4, 3).Your TurnXЈXYЈYZЈZxyxyOCBABCA©Glencoe/McGraw-Hill16–5Geometry: Concepts and Applications 325Page(s):Exercises:HOMEWORKASSIGNMENT
  • 334. DilationsGraph AෆBෆ with vertices A(0, 2) and B(2, 1). Then findthe coordinates of the dilation image of AෆBෆ with a scalefactor of 3, and graph its dilation image.Since k Ͼ 1, this is an enlargement. To find the dilation image,multiply each coordinate in the ordered pairs by 3.preimage imageA(0, 2) AЈB(2, 1) BЈThe coordinates of the endpoints of the dilation image areAЈ and BЈ .Graph ᭝JKL with vertices J(1, Ϫ2), K(4, Ϫ3),and L(6, Ϫ1). Then find the coordinates of the dilation imageof ᭝JKL with a scale factor of 2, and graph its dilation.Your TurnBЈAAЈBxyO©Glencoe/McGraw-Hill326 Geometry: Concepts and Applications16–6• Investigate and drawdilations on acoordinate plane.WHAT YOU’LL LEARNOn the page labeledDilations, sketch graphsof several differentdilations. Explain whyeach dilation producesthe result it does.ORGANIZE ITA dilation is a transformation that alters the size of a figure,but not its shape. It enlarges or reduces a figure by ak.BUILD YOUR VOCABULARY (page 314)xyOKJL4 8Ϫ4Ϫ12Ϫ812JLK(ϫ 3)(ϫ 3)
  • 335. Graph ᭝DEF with vertices D(3, 3), E(0, Ϫ3), and F(Ϫ6, 3).Then find the coordinates of the dilation image with ascale factor of ᎏ13ᎏ and graph its dilation image.Since k Ͻ 1, this is a reduction.preimage imageD(3, 3) DЈE(0, Ϫ3) EЈF(Ϫ6, 3) FЈThe coordinates of the vertices of the dilation image areDЈ , EЈ , and FЈ .Graph quadrilateral MNOP with verticesM(1, 2), N(3, 3), O(3, 5), and P(1, 4). Then find the coordinatesof the dilation image with a scale factor of ᎏ23ᎏ and graph itsdilation image.Your TurnyxFFЈEEЈDDЈ©Glencoe/McGraw-Hill16–6Geometry: Concepts and Applications 327How can you determinewhether a dilation is areduction or anenlargement?WRITE ITϫ ᎏ13ᎏϫ ᎏ13ᎏϫ ᎏ13ᎏxyOOPMNMOPN1 2 312345Page(s):Exercises:HOMEWORKASSIGNMENT
  • 336. ©Glencoe/McGraw-HillBRINGING IT ALL TOGETHERC H A P T E R16STUDY GUIDE328 Geometry: Concepts and ApplicationsSolve each system of equations by graphing.1. x Ϫ y ϭ 6 2. x ϩ y ϭ 27 3. y ϭ 4x ϩ 2y ϭ 9 3x Ϫ y ϭ 41 12x Ϫ 3y ϭ 916-2Complete each statement.4. Substitution and elimination are methods for solving.5. A linear system of equations can have at most solution.Solve the system of equations using substitution orelimination.6. 3x Ϫ y ϭ 4 7. y ϭ 3x Ϫ 82x Ϫ 3y ϭ Ϫ9 y ϭ 4 Ϫ x8. 2x ϩ 7y ϭ 3 9. 3x Ϫ 5y ϭ 11x ϭ 1 Ϫ 4y x Ϫ 3y ϭ 1BUILD YOURVOCABULARYUse your Chapter 16 Foldableto help you study for yourchapter test.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 16, go to:www.glencoe.com/sec/math/t_resources/free/index.phpYou can use your completedVocabulary Builder(page 314) to help you solvethe puzzle.VOCABULARYPUZZLEMAKER16-1Solving Systems of Equations by Graphing16-2Solving Systems of Equations by Using Algebra
  • 337. ©Glencoe/McGraw-HillComplete the statement.10. When a figure is moved from one position to another withoutturning, it is called a .Find the coordinates of the vertices after the translation.Graph each preimage and image.11. rectangle WXYZ with vertices W(Ϫ2, Ϫ2), X(Ϫ2, Ϫ10),Y(Ϫ7, Ϫ2), and Z(Ϫ7, Ϫ10) translated (6, 9)12. ᭝ABC with vertices A(4, 0), B(2, Ϫ1), and C(0, 1) translated(0, Ϫ4)13. ᭝JKL with vertices J(Ϫ5, Ϫ2), K(Ϫ2, 7), and L(1, Ϫ6)translated (6, 2)Chapter BRINGING IT ALL TOGETHER16Geometry: Concepts and Applications 32916-3TranslationsxyOY WZ X4 848Ϫ8 Ϫ4Ϫ8Ϫ4Y WZ XxyOCBACBAxyOLKJ4 848Ϫ8 Ϫ4Ϫ8Ϫ4LKJ
  • 338. Chapter BRINGING IT ALL TOGETHER©Glencoe/McGraw-HillComplete the statement.14. A is a flip of a figure over a line.Find the coordinates of the vertices after the reflection.Graph each preimage and image.15. quadrilateral ABCD with vertices A(1, 1), B(1, 4), C(6, 4), andD(6, 1) flipped over the x-axis16. quadrilateral JKLM with vertices J(3, 5), K(4, 0), L(0, Ϫ3),and M(Ϫ1, 2) flipped over the y-axis17. ᭝XYZ with vertices X(1, 1), Y(4, 1), and Z(1, 3) flipped overthe x-axis16330 Geometry: Concepts and ApplicationsxyOABDCABDCxyOM MLKJKJLxyOX YZXYZ16-4Reflections
  • 339. ©Glencoe/McGraw-HillFind the coordinates of the vertices after a rotation aboutthe origin. Graph the preimage and image.18. ᭝RST with vertices R(Ϫ4, 1), S(Ϫ1, 5), and T(Ϫ6, 9) rotated90° counterclockwiseUnderline the best term to complete the statement.19. A [dilation/rotation] alters the size of a figure but does notchange its shape.20. A figure is [reduced/enlarged] in a dilation if the scale factor isbetween 0 and 1.Find the coordinates of the dilation image for the givenscale factor. Graph the preimage and image.21. quadrilateral STUV with vertices S(2, 1), T(0, 2), U(Ϫ2, 0),and V(0, 0) and scale factor 3Chapter BRINGING IT ALL TOGETHER16Geometry: Concepts and Applications 33116-5Rotations16-6DilationsxyOTRS2 626Ϫ6 Ϫ2Ϫ6Ϫ2RSTxyOSSTTUVU V
  • 340. ©Glencoe/McGraw-HillCheck the one that applies. Suggestions to help you study aregiven with each item.ARE YOU READY FORTHE CHAPTER TEST?I completed the review of all or most lessons without usingmy notes or asking for help.• You are probably ready for the Chapter Test.• You may want to take the Chapter 16 Practice Test onpage 713 of your textbook as a final check.I used my Foldable or Study Notebook to complete the reviewof all or most lessons.• You should complete the Chapter 16 Study Guide and Reviewon pages 710–712 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 16 Practice Test onpage 713 of your textbook.I asked for help from someone else to complete the review ofall or most lessons.• You should review the examples and concepts in your StudyNotebook and Chapter 16 Foldable.• Then complete the Chapter 16 Study Guide and Review onpages 710–712 of your textbook.• If you are unsure of any concepts or skills, refer back to thespecific lesson(s).• You may also want to take the Chapter 16 Practice Test onpage 713 of your textbook.Visit geomconcepts.net toaccess your textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 16.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R16Checklist332 Geometry: Concepts and Applications

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