Geometry notebook foldables ebook 341 pgs

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Geometry notebook foldables ebook 341 pgs

  1. 1. Contributing AuthorDinah ZikeConsultantDouglas Fisher, Ph.D.Director of Professional DevelopmentSan Diego, CA
  2. 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: 978-0-07-877340-2 Geometry Noteables™: InteractiveMHID: 0-07-877340-7 Study Notebook with Foldables™1 2 3 4 5 6 7 8 9 10 005 16 15 14 13 12 11 10 09 08 07
  3. 3. Glencoe Geometry iiiContentsFoldables. . . . . . . . . . . . . . . . . . . . . . . . . . . 1Vocabulary Builder . . . . . . . . . . . . . . . . . . 21-1 Points, Lines, and Planes . . . . . . . . . 41-2 Linear Measure and Precision . . . . . 81-3 Distance and Midpoints. . . . . . . . . 111-4 Angle Measure . . . . . . . . . . . . . . . . 131-5 Angle Relationships . . . . . . . . . . . . 161-6 Two-Dimensional Figures. . . . . . . . 191-7 Three-Dimensional Figures . . . . . . 23Study Guide . . . . . . . . . . . . . . . . . . . . . . . 26Foldables. . . . . . . . . . . . . . . . . . . . . . . . . . 31Vocabulary Builder . . . . . . . . . . . . . . . . . 322-1 Inductive Reasoning andConjecture. . . . . . . . . . . . . . . . . . . . 342-2 Logic . . . . . . . . . . . . . . . . . . . . . . . . 362-3 Conditional Statements . . . . . . . . . 392-4 Deductive Reasoning . . . . . . . . . . . 422-5 Postulates and ParagraphProofs . . . . . . . . . . . . . . . . . . . . . . . 452-6 Algebraic Proof . . . . . . . . . . . . . . . 482-7 Proving Segment Relationships. . . 512-8 Proving Angle Relationships . . . . . 53Study Guide . . . . . . . . . . . . . . . . . . . . . . . 57Foldables. . . . . . . . . . . . . . . . . . . . . . . . . . 61Vocabulary Builder . . . . . . . . . . . . . . . . . 623-1 Parallel Lines and Transversals. . . . 643-2 Angles and Parallel Lines. . . . . . . . 673-3 Slopes of Lines . . . . . . . . . . . . . . . . 693-4 Equations of Lines . . . . . . . . . . . . . 723-5 Proving Lines Parallel. . . . . . . . . . . . 753-6 Perpendiculars and Distance . . . . . . 79Study Guide . . . . . . . . . . . . . . . . . . . . . . . 82Foldables. . . . . . . . . . . . . . . . . . . . . . . . . . 85Vocabulary Builder . . . . . . . . . . . . . . . . . 864-1 Classifying Triangles . . . . . . . . . . . . . 884-2 Angles of Triangles . . . . . . . . . . . . . . 914-3 Congruent Triangles . . . . . . . . . . . . . 954-4 Proving Congruence—SSS, SAS . . . . 984-5 Proving Congruence—ASA, AAS. . 1024-6 Isosceles Triangles . . . . . . . . . . . . . 1054-7 Triangles and Coordinate Proof . . 108Study Guide . . . . . . . . . . . . . . . . . . . . . . 111Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 115Vocabulary Builder . . . . . . . . . . . . . . . . 1165-1 Bisectors, Medians, andAltitudes . . . . . . . . . . . . . . . . . . . . 1185-2 Inequalities and Triangles . . . . . . 1225-3 Indirect Proof . . . . . . . . . . . . . . . . 1255-4 The Triangle Inequality . . . . . . . . 1285-5 Inequalities Involving TwoTriangles . . . . . . . . . . . . . . . . . . . . 130Study Guide . . . . . . . . . . . . . . . . . . . . . . 133Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 137Vocabulary Builder . . . . . . . . . . . . . . . . 1386-1 Angles of Polygons. . . . . . . . . . . . 1396-2 Parallelograms . . . . . . . . . . . . . . . 1426-3 Tests for Parallelograms. . . . . . . . 1446-4 Rectangles. . . . . . . . . . . . . . . . . . . 1476-5 Rhombi and Squares . . . . . . . . . . 1526-6 Trapezoids. . . . . . . . . . . . . . . . . . . 1556-7 Coordinate Proof withQuadrilaterals. . . . . . . . . . . . . . . . 159Study Guide . . . . . . . . . . . . . . . . . . . . . . 162Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 167Vocabulary Builder . . . . . . . . . . . . . . . . 1687-1 Proportions . . . . . . . . . . . . . . . . . . 1707-2 Similar Polygons . . . . . . . . . . . . . . 1727-3 Similar Triangles . . . . . . . . . . . . . . 1757-4 Parallel Lines and ProportionalParts. . . . . . . . . . . . . . . . . . . . . . . . 1787-5 Parts of Similar Triangles . . . . . . . 181Study Guide . . . . . . . . . . . . . . . . . . . . . . 184
  4. 4. iv Glencoe GeometryContentsFoldables. . . . . . . . . . . . . . . . . . . . . . . . . 187Vocabulary Builder . . . . . . . . . . . . . . . . 1888-1 Geometric Mean. . . . . . . . . . . . . . 1908-2 The Pythagorean Theorem andIts Converse. . . . . . . . . . . . . . . . . . 1948-3 Special Right Triangles. . . . . . . . . 1988-4 Trigonometry . . . . . . . . . . . . . . . . 2018-5 Angles of Elevation andDepression . . . . . . . . . . . . . . . . . . 2048-6 The Law of Sines . . . . . . . . . . . . . 2078-7 The Law of Cosines . . . . . . . . . . . 211Study Guide . . . . . . . . . . . . . . . . . . . . . . 214Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 219Vocabulary Builder . . . . . . . . . . . . . . . . 2209-1 Reflections . . . . . . . . . . . . . . . . . . 2229-2 Translations. . . . . . . . . . . . . . . . . . 2269-3 Rotations. . . . . . . . . . . . . . . . . . . . 2299-4 Tessellations . . . . . . . . . . . . . . . . . 2319-5 Dilations . . . . . . . . . . . . . . . . . . . . 2339-6 Vectors . . . . . . . . . . . . . . . . . . . . . 236Study Guide . . . . . . . . . . . . . . . . . . . . . . 239Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 243Vocabulary Builder . . . . . . . . . . . . . . . . 24410-1 Circles and Circumference . . . . . . 24610-2 Measuring Angles and Arcs. . . . . 24910-3 Arcs and Chords . . . . . . . . . . . . . . 25210-4 Inscribed Angles . . . . . . . . . . . . . . 25610-5 Tangents . . . . . . . . . . . . . . . . . . . . 26010-6 Secants, Tangents, and AngleMeasures. . . . . . . . . . . . . . . . . . . . 26210-7 Special Segments in a Circle . . . . 26510-8 Equations of Circles . . . . . . . . . . . 267Study Guide . . . . . . . . . . . . . . . . . . . . . . 269Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 273Vocabulary Builder . . . . . . . . . . . . . . . . 27411-1 Areas of Parallelograms. . . . . . . . 27511-2 Areas of Triangles, Trapezoids,and Rhombi . . . . . . . . . . . . . . . . . 27711-3 Areas of Regular Polygons andCircles . . . . . . . . . . . . . . . . . . . . . . 28011-4 Areas of Composite Figures. . . . . 28311-5 Geometric Probability andAreas of Sectors . . . . . . . . . . . . . . 285Study Guide . . . . . . . . . . . . . . . . . . . . . . 288Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 291Vocabulary Builder . . . . . . . . . . . . . . . . 29212-1 Representations ofThree-Dimensional Figures . . . . . 29412-2 Surface Areas of Prisms. . . . . . . . . 29812-3 Surface Areas of Cylinders. . . . . . 30012-4 Surface Areas of Pyramids. . . . . . 30212-5 Surface Areas of Cones . . . . . . . . 30512-6 Surface Areas of Spheres. . . . . . . 307Study Guide . . . . . . . . . . . . . . . . . . . . . . 310Foldables. . . . . . . . . . . . . . . . . . . . . . . . . 315Vocabulary Builder . . . . . . . . . . . . . . . . 31613-1 Volumes of Prisms andCylinders . . . . . . . . . . . . . . . . . . . . 31713-2 Volumes of Pyramids andCones. . . . . . . . . . . . . . . . . . . . . . . 32013-3 Volumes of Spheres . . . . . . . . . . . 32313-4 Congruent and Similar Solids . . . 32513-5 Coordinates in Space . . . . . . . . . . 328Study Guide . . . . . . . . . . . . . . . . . . . . . . 330
  5. 5. Glencoe Geometry vOrganizing Your FoldablesCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Make this Foldable to help you organize andstore your chapter Foldables. Begin with onesheet of 11" × 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.Foldables OrganizerReading and Taking Notes As you read and study each chapter, recordnotes in your chapter Foldable. Then store your chapter Foldables insidethis Foldable organizer.
  6. 6. vi Glencoe GeometryThis note-taking guide is designed to help you succeed in Geometry.Each chapter includes:Glencoe Geometry 187Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter8Use 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.Right Triangles and TrigonometryNOTE-TAKING TIP: When you take notes, drawa visual (graph, diagram, picture, chart) thatpresents the information introduced in the lessonin a concise, easy-to-study format.Stack the sheets. Foldthe top right cornerto the bottom edgeto form a square.Fold the rectangularpart in half.Staple the sheetsalong the fold infour places.Label each sheet witha lesson number andthe rectangular partwith the chapter title.Begin with seven sheets of grid paper.C H A P T E R8188 Glencoe GeometryBUILD YOUR VOCABULARYCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.This 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 the secondcolumn for reference when you study.C H A P T E R8Vocabulary TermFoundDefinitionDescription oron Page Exampleangle of depressionangle of elevationcosinegeometric meanLaw of CosinesLaw of SinesThe 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.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
  7. 7. STUDY GUIDE214 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.BRINGING IT ALL TOGETHERBUILD YOURVOCABULARYVOCABULARYPUZZLEMAKERUse your Chapter 8 Foldable tohelp you study for your chaptertest.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 8, go to:glencoe.comYou can use your completedVocabulary Builder(pages 188–189) to help yousolve the puzzle.฀฀ ฀฀Find the geometric mean between each pair of numbers.1. 4 and 9 2. 20 and 303. Find x and y.4. For the figure shown, which statements are true?a. m2 + n2 = p2 b. n2 = m2 + p2c. m2 = n2 + p2 d. m2 = p2 - n2e. p2 = n2 - m2 f. n2 - p2 = m2g. n = √m2 + p2 h. p = √m2 - n2Which of the following are Pythagorean triples? Write yes or no.5. 10, 24, 26 6. √2, √2, 2 7. 10, 6, 88-1Geometric Mean8-2The Pythagorean Theorem and Its ConverseC H A P T E R8Glencoe Geometry 203Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENT8–4Use Trigonometric Ratios to Find a LengthEXERCISING A fitness trainer sets the incline on atreadmill to 7°. The walking surface is 5 feet long.Approximately how many inches did the trainer raisethe end of the treadmill from the floor?Let y be the height of the treadmill from the floor in inches.The length of the treadmill is feet, or inches.sin 7° = sin =leg opposite_hypotenusesin 7° = y Multiply each side by .The treadmill is about inches high.Check Your Progress The bottom of a handicap ramp is15 feet from the entrance of a building. If the angle of the rampis about 4.8°, how high does the ramp rise off the ground to thenearest inch?Glencoe Geometry viiExamples parallel theexamples in your textbook.Bringing It All TogetherStudy Guide reviews themain ideas and keyconcepts from each lesson.Check Your ProgressExercises allow you tosolve similar exerciseson your own.Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.฀ ฀฀฀฀฀฀฀฀฀฀฀ ฀ ฀ ฀ ฀ ฀ ฀฀ ฀ ฀ ฀ ฀฀฀฀฀ ฀ ฀ ฀ ฀฀฀฀ ฀ ฀ ฀฀ ฀ ฀฀฀฀ ฀฀฀฀ ฀ ฀฀ ฀ ฀ ฀฀ ฀ ฀ ฀ ฀ ฀ ฀฀ ฀ ฀ ฀ ฀ ฀ ฀฀Foldables featurereminds you to takenotes in your Foldable.48 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A two-column proof, or formal proof, containsand organized intwo columns.Algebraic Proof2–6• Use algebra to writetwo-column proofs.• Use properties ofequality in geometryproofs.MAIN IDEASVerify Algebraic RelationshipsSolve 2(5 - 3a) - 4(a + 7) = 92.Algebraic Steps Properties2(5 - 3a) - 4(a + 7) = 92 Original equation10 - 6a - 4a - 28 = 92 Property-18 - 10a = 92 Property= Addition Propertya = PropertyCheck Your Progress Solve -3(a + 3) + 5(3 - a) = -50.BUILD YOUR VOCABULARY (pages 32–33)Write a two-columnproof for the statementif 5(2x - 4) = 20, thenx = 4 under the tab forLesson 2-6.ORGANIZE ITFoldables featurereminds you to takenotes in your Foldable.Lessons cover the content ofthe lessons in your textbook.As your teacher discusses eachexample, follow along andcomplete the fill-in boxes.Take notes as appropriate.
  8. 8. viii Glencoe GeometryYour 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 (#) or an asterisk (*) to emphasize 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 equal ≠such as i.e. approximately ≈with w/ therefore ∴without w/o versus vsand + angle ∠Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.
  9. 9. Glencoe Geometry 1Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter1C H A P T E R1Use 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.Tools of GeometryNOTE-TAKING TIP: When you take notes, listen orread for main ideas. Then record what you knowand apply these concepts by drawing, measuring,and writing about the process.Begin with a sheet of 11" × 17" paper.Fold the short sidesto meet in the middle.Fold the top to thebottom.Open. Cut flaps alongsecond fold to makefour tabs.Label the tabs as shown.Points, Lines,PlanesLength andPerimeterAnglesAngleMeasure
  10. 10. 2 Glencoe GeometryBUILD YOUR VOCABULARYCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.This 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 Exampleadjacent angles[uh-JAY-suhnt]angleangle bisectorcollinear[koh-LIN-ee-uhr]complementary anglescongruent[kuhn-GROO-uhnt]coplanar[koh-PLAY-nuhr]degreelineline segmentlinear pairC H A P T E R1
  11. 11. Glencoe Geometry 3Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter BUILD YOUR VOCABULARY1Vocabulary TermFoundDefinitionDescription oron Page Examplemidpointperpendicularplanepointpolygon[PAHL-ee-gahn]polyhedronprecisionraysegment bisectorsidessupplementary anglesvertexvertical angles
  12. 12. 4 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–1Points on the same are collinear.Points that lie on the same are coplanar.Points, Lines, and PlanesName Lines and PlanesUse the figure to name each ofthe following.a. a line containing point KThe line can be named as line .There are three points on the line. Any two of the pointscan be used to name the line. The possible names are.b. a plane containing point LThe plane can be named as plane .You can also use the letters of any threepoints to name the plane.plane plane planeCheck Your Progress Use the figure to name each ofthe following.a. a line containing point Xb. a plane containing point Z• Identify and modelpoints, lines, andplanes.• Identify collinear andcoplanar points andintersecting lines andplanes in space.MAIN IDEASPoint A point hasneither shape nor size.Line There is exactlyone line through anytwo points.Plane There is exactlyone plane throughany three noncollinearpoints.KEY CONCEPTS(page 2)BUILD YOUR VOCABULARY
  13. 13. Glencoe Geometry 5Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–1Model Points, Lines, and PlanesName the geometric term modeled by each object.a. the long hand on a clockThe long hand on a clock models a .b. a 10 × 12 patioThe patio models a .c. a water glass on the tableThis models a .Check Your Progress Name the geometric shapemodeled by each object.a. a colored dot on a map usedto mark the location of a cityb. the ceiling of your classroomDraw Geometric FiguresDraw and label a figure for each situation.a. ALGEBRA Plane R contains lines AB and DE, whichintersect at point P. Add point C on plane R so that itis not collinear with AB or DE.Draw a surface to representand label it.Draw a line anywhere on the planeand draw dots on the line for pointsA and B. Label the points.Draw a line intersectingand draw dots on theline for points D and E.Label the points.Label the intersection of thetwo lines as .Draw and label a pointP, a line AB, and a planeXYZ under the Points,Lines, and Planes tab.Points, Lines,PlanesLength andPerimeterAnglesAngleMeasureORGANIZE IT
  14. 14. 6 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–1Draw a dot for point C in plane Rsuch that it will not lie on AB orDE. Label the point.b. QR on a coordinate plane contains Q(-2, 4) andR(4, -4). Add point T so that T is collinear withthese points.Graph each point and draw QR.There are an infinite number of points that are collinearwith Q and R. In the graph, one such point is .Check Your Progress Draw and label a figure foreach relationship.a. Plane D contains line a, line m, and line t, with all threelines intersecting at point Z. Add point F on plane D so thatit is not collinear with any of the three given lines.b. BA on a coordinate plane contains B(-3, -2) and A(3, 2).Add point M so that M is collinear with these points.REMEMBER ITThe prefix co-means together. So,collinear means lyingtogether on the sameline. Coplanar meanslying together in thesame plane.
  15. 15. Glencoe Geometry 7Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENT1–1Interpret DrawingsUse the figure for parts a–d.a. How many planes appear in thisfigure?b. Name three points that are collinear.Points , , and are collinear.c. Are points A, B, C, and D coplanar? Explain.Points A, B, C, and D all lie in , so theyare coplanar.d. At what point do DB and CA intersect?The two lines intersect at point .Check Your Progress Refer tothe figure.a. How many planes appear inthis figure?b. Name three points thatare collinear.c. Are points X, O, and R coplanar?Explain.d. At what point do BN and XO intersect?Explain the differentways of naming a plane.WRITE IT
  16. 16. 8 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–2A line segment can be measured because it hastwo .Linear Measure and PrecisionLength in Metric UnitsFind the length of−−−LM using the ruler.The long marks indicate , and theshorter marks indicate .L−−−LM is about millimeters long.Check Your Progressa. Find the length of−−AB.Length in Customary UnitsFind the length of each segment using each ruler.a.−−DEEach inch is divided into .−−−DE is aboutinches long.• Measure segments anddetermine accuracy ofmeasurement.• Compute with measures.MAIN IDEAS(page 2)BUILD YOUR VOCABULARY
  17. 17. Glencoe Geometry 9Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.b.−−FG−−−FG is about inches long.Check Your Progressa. Find the length of−−AZ.b. Find the length of−−IX.PrecisionFind the precision for each measurement. Explainits meaning.a. 323_4inchesThe measuring tool is divided into increments.Thus, the measurement is precise to within 1_2(1_4) or 1_8inch.The measurement could be to inches.b. 15 millimetersThe measuring tool is divided into millimeters. Thus themeasurement is precise to within 1_2of 1 millimeter. Themeasurement could be 14.5 to 15.5 millimeters.Check Your ProgressFind the precision for eachmeasurement.Explain how to findthe precision of ameasurement. Writethis under the Lengthand Perimeter tab.Points, Lines,PlanesLength andPerimeterAnglesAngleMeasureORGANIZE IT1–2
  18. 18. 10 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENT1–2Find MeasurementsFind the measurement of each segment.a.−−−LMLM + MN = LN ฀LM + = SubstitutionLM + - = - Subtract.LM = Simplify.−−−LM is centimeters long.b. x and ST if T is between S and U, ST = 7x, SU = 45,and TU = 5x - 3.ST + TU = SU7x + = Substitute known values.7x + 5x - 3 + 3 = 45 + 3 Add 3 to each side.12x = 48 Simplify.12x_12= 48_12Divide each side by 12.= Simplify.ST = 7x Given= 7(4) x = 4= Thus, x = 4, ST = .Check Your Progressa. Find SE.b. Find a and AB if AB = 4a + 10, BC = 3a - 5, and AC = 19.
  19. 19. Glencoe Geometry 11Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–3 Distance and MidpointsFind Distance on a Number LineUse the number line to find QR.The coordinates of Q and Rare and .QR = |-6 - (-3)| Distance Formula= |-3| or 3 Simplify.Check Your Progress Use the number line to find AX.Find Distance on a Coordinate PlaneFind the distance betweenE(-4, 1) and F(3, -1).Method 1 Pythagorean Theorem(EF)2 = (ED)2 + (DF)2(EF)2 = 2+ 2(EF)2 = 53EF = √53Method 2 Distance Formulad = √(x2 - x1)2 + (y2 - y1)2 Distance FormulaEF = √[3 - (-4)]2 + (-1 - 1)2 (x1, y1) = (-4, 1) and(x2, y2) = (3, -1)EF = ( )2+ ( )2Simplify.EF = √53 Simplify.• Find the distancebetween two points.• Find the midpoint of asegment.MAIN IDEASDistance FormulasNumber Line If P and Qhave coordinates a and b,respectively, then thedistance betweenP and Q is givenby PQ = |b - a|or |a - b|.Coordinate Plane Thedistance d between twopoints with coordinates(x1, y1) and (x2, y2) isgiven by d =√(x2 - x1)2 + (y2 - y1)2.KEY CONCEPT
  20. 20. 12 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENT1–3Check Your Progress Find thedistance between A(-3, 4) and M(1, 2).Find Coordinate of EndpointFind the coordinates of D if E(-6, 4) is the midpointof−−DF and F has coordinates (-5, -3).Let F be (x2, y2) in the Midpoint Formula.E(-6, 4) = E(x1 + (-5)_2,y1 + (-3)_2 ) (x2, y2) = (-5, -3)Write and solve two equations to find the coordinates of D.-6 =x1 + (-5)_24 =y1 + ( )__2= 8 = y1 - 3= x1 = y1The coordinates of D are .Check Your Progress Find the coordinates of R if N(8, -3)is the midpoint of−−RS and S has coordinates (-1, 5).The midpoint of a segment is the point halfwaybetween the of the segment.Any segment, line, or plane that intersects a segment atits is called a segment bisector.Draw a coordinatesystem and show howto find the distancebetween two pointsunder the Length andPerimeter tab.Points, Lines,PlanesLength andPerimeterAnglesAngleMeasureORGANIZE ITMidpoint The midpointM of−−PQ is the pointbetween P and Q suchthat PM = MQ.KEY CONCEPT(page 3)BUILD YOUR VOCABULARY
  21. 21. Glencoe Geometry 13Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–4 Angle MeasureAngles and Their PartsRefer to the figure.a. Name all angles that have B as a vertex.b. Name the sides of ∠5.and or are the sides of ∠5.c. Write another name for ∠6., , , and areother names for ∠6.Check Your Progressa. Name all angles that have X as a vertex.b. Name the sides of ∠3.c. Write another name for ∠3.• Measure and classifyangles.• Identify and usecongruent anglesand the bisector ofan angle.MAIN IDEASAngle An angle isformed by twononcollinear rays thathave a commonendpoint.KEY CONCEPTWhen can you use asingle letter to namean angle?WRITE ITA ray is a part of a . A ray starts at aon the line and extends endlessly in .An angle is formed by two rays thathave a common endpoint.BUILD YOUR VOCABULARY (pages 2–3)
  22. 22. 14 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–4Measure and Classify AnglesMeasure each angle named and classify it as right,acute, or obtuse.a. ∠XYV∠XYV is marked with a rightangle symbol, so measuring isnot necessary. m∠XYV = ,so ∠XYV is a(n) .b. ∠WYTUse a protractorto find thatm∠WYT = 130.180 > m∠WYT > 90,so ∠WYT is a(n).c. ∠TYUUse a protractorto find thatm∠TYU = 45.45 < 90, so ∠TYU isa(n) .Check Your Progress Measure each angle named andclassify it as right, acute, or obtuse.a. ∠CZDb. ∠CZEc. ∠DZXClassify AnglesA right angle has adegree measure of 90.An acute angle has adegree measure lessthan 90. An obtuseangle has a degreemeasure greater than 90and less than 180.Congruent AnglesAngles that have thesame measure arecongruent angles.KEY CONCEPTSDraw and label ∠RSTthat measures 70° underthe Angle Measure tab.Classify ∠RST as acute,right, or obtuse.Points, Lines,PlanesLength andPerimeterAnglesAngleMeasureORGANIZE IT
  23. 23. Glencoe Geometry 15Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTA that divides an angle into congruentis called an angle bisector.1–4Use Algebra to Find Angle MeasuresINTERIOR DESIGN Wall stickers of standardshapes are often used to provide a stimulatingenvironment for a young child’s room.A five-pointed star sticker is shown withvertices labeled. Find m∠GBH and m∠HCIif ∠GBH ∠HCI, m∠GBH = 2x + 5, andm∠HCI = 3x - 10.∠GBH ∠HCI Givenm∠GBH = m∠HCI2x + 5 = 3x - 10 Substitution= Add 10 to each side.15 = x Subtract 2x from each side.Use the value of x to find the measure of one angle.m∠GBH = 2x + 5 Given= 2(15) + 5 x == + or 35 Simplify.Since m∠GBH = m∠HCI, m∠HCI = .Check Your Progress A railroadcrossing sign forms congruent angles.In the figure, m∠WVX m∠ZVY.If m∠WVX = 7a + 13 andm∠ZVY = 10a - 20, find the actualmeasurements of m∠WVX and m∠ZVY.REMEMBER ITIf angles arecongruent, then theirmeasures are equal.BUILD YOUR VOCABULARY (page 2)
  24. 24. 16 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–5Identify Angle PairsRefer to the figure. Name two acute vertical angles.There are four acute anglesshown. There is one pair ofvertical angles. The acutevertical angles are.Check Your Progress Name an angle pair that satisfieseach condition.a. two angles that form a linear pairb. two adjacent angles whose measureshave a sum that is less than 90Angle MeasureALGEBRA Find the measures of two supplementaryangles if the measure of one angle is 6 less thanfive times the other angle.The sum of the measures of supplementary angles is .Draw two figures to represent the angles.If m∠A = x, then m∠B = .• Identify and use specialpairs of angles.• Identify perpendicularlines.MAIN IDEASAngle RelationshipsAngle PairsAdjacent angles aretwo angles that lie inthe same plane, havea common vertex, anda common side, but nocommon interior points.Vertical angles aretwo nonadjacentangles formed bytwo intersecting lines.A linear pair is a pair ofadjacent angles whosenoncommon sides areopposite rays.KEY CONCEPTSDraw andlabel examples under theAngles tab.
  25. 25. Glencoe Geometry 17Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Lines that form are perpendicular.1–5m∠A + m∠B = 180 Given+ ( )= 180m∠A = x andm∠B = 5x - 66x - 6 = 180 Simplify.6x = 186 Add to each side.x = 31 Divide each side by 6.m∠A = x m∠B =m∠A = 31 m∠B = 5(31) - 6 or 149Check Your Progress Find the measures of twocomplementary angles if one angle measures six degrees lessthan five times the measure of the other.Perpendicular LinesALGEBRA Find x so that KO ⊥ HM.If KO ⊥ HM, then m∠KJH = .m∠KJH = m∠KJI + m∠IJH90 = (3x + 6) + 9x Substitution90 = Add.84 = 12x Subtract 6 from each side.= x Divide each side by 12.Angle RelationshipsComplementary anglesare two angles whosemeasures have asum of 90°.Supplementary anglesare two angles whosemeasures have asum of 180°.KEY CONCEPTPerpendicular linesintersect to formfour right angles.Perpendicular linesintersect to formcongruent adjacentangles. Segments andrays can be perpendicularto lines or to other linesegments and rays.The right angle symbolis used in figures toindicate that lines,rays, or segmentsare perpendicular.KEY CONCEPTBUILD YOUR VOCABULARY (page 3)
  26. 26. 18 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENT1–5Check Your Progress Find xand y so that AD and CE areperpendicular.Interpret FiguresDetermine whether each statement can be assumedfrom the figure below.a. m∠VYT = 90The diagram is marked to show VY ⊥ XT.From the definition of perpendicular,perpendicular lines intersect to formcongruent adjacent angles.; VY and TX are perpendicular.b. ∠TYW and ∠TYU are supplementary.Yes; they form a of angles.c. ∠VYW and ∠TYS are adjacent angles.No; they do not share a .Check Your Progress Determine whether eachstatement can be assumed from the figure below.Explain.a. m∠XAY = 90b. ∠TAU and ∠UAY arecomplementary.c. ∠UAX and ∠UXA are adjacent.
  27. 27. Glencoe Geometry 19Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–6 Two-Dimensional FiguresIdentify PolygonsName each polygon by the number of sides. Thenclassify it as convex or concave, regular or irregular.a. There are 4 sides, so this isa .No line containing any of the sides will pass through theinterior of the quadrilateral, so it is . The sidesare not congruent, so it is .b. There are 9 sides, so this is a.A line containing a side will pass through the interior of thenonagon, so it is .The sides are not congruent, so it is .Check Your Progress Name each polygon by thenumber of sides. Then classify it as convex or concave,regular or irregular.a. b.• Identify and namepolygons.• Find perimeter orcircumference and areaof two-dimensionalfigures.MAIN IDEASA polygon is a figure whose areall segments.BUILD YOUR VOCABULARY (page 3)Polygon A polygon is aclosed figure formed bya finite number ofcoplanar segments suchthat (1) the sides thathave a commonendpoint arenoncollinear, and(2) each side intersectsexactly two other sides,but only at theirendpoints.KEY CONCEPT
  28. 28. 20 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–6Find Perimeter and AreaFind the perimeter or circumference and area ofthe figure.C = 2πr Circumference of a circleC = 2π( ) r =C = Multiply.C = Use a calculator.A = πr2 Area of a circleA = π( 2) r =A = Simplify.A = Use a calculator.The circumference of the circle is about inches and thearea is about square inches.Check Your Progress Find the perimeter or circumferenceand area of the figure.Perimeter The perimeterP of a polygon is the sumof the lengths of thesides of a polygon.Area The area is thenumber of square unitsneeded to cover asurface.KEY CONCEPTDraw a polygon andexplain how to find theperimeter. Place thisunder the Length andPerimeter tab.Points, Lines,PlanesLength andPerimeterAnglesAngleMeasureORGANIZE IT
  29. 29. Glencoe Geometry 21Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–6Largest AreaTEST EXAMPLE Terri has 19 feet of tape to make an areain the classroom where the students can read. Which ofthese shapes has a perimeter or circumference thatwould use most or all of the tape?A square with side length of 5 feetB circle with the radius of 3 feetC right triangle with each leg length of 6 feetD rectangle with a length of 8 feet and a width of 3 feetRead the Test ItemYou are asked to compare the perimeters of four differentshapes.Solve the Test ItemFind the perimeter of each shape.Square Circle RectangleP = 4s C = 2πr P = 2ℓ + 2wP = 4(5) C = 2π(3) P = 2(8) + 2(3)P = ft C = 6π or about ft P = ftRight TriangleUse the Pythagorean Theorem to find the length of thehypotenuse.c2 = a2 + b2c2 = 62 + 62c2 = 72c = 6√2P = 6 + 6 + 6√2P = or about ftThe shape that uses the most of the tape is the circle. Theanswer is .Check Your Progress Jason has 20 feet of fencing tomake a pen for his dog. Which of these shapes enclosesthe largest area?A square with a side length of 5 feetB circle with radius of 3 feetC right triangle with each leg about 6 feetD rectangle with length of 4 feet and width of 6 feet
  30. 30. 22 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENT1–6Perimeter and Area on the Coordinate PlaneFind the perimeter of pentagonABCDE with A(0, 4), B(4, 0),C(3, -4), D(-3, -4), and E(-3, 1).Since−−−DE is a vertical line segment,we can count the squares onthe grid. The length of−−−DE isunits. Likewise, since−−−CD is ahorizontal line segment, count the squares to find that thelength is units.To find AE, AB, and BC, use the distance formula.AE = √(0 - (-4))2 + (4 - 1)2 SubstitutionAE =2+2Subtract.AE = or 5 Simplify.AB = √(0 - 4)2 + (4 - 0)2 SubstitutionAB = ( )2+2Subtract.AB = √32 or about Simplify.BC = √(4 - 3)2 + (0 - (-4))2 SubstitutionBC =2+2Subtract.BC = or about 4.1 Simplify.To find the perimeter, add the lengths of each side.P = AB + BC + CD + DE + AEP ≈ 5.7 + 4.1 + 6 + 5 + 5P ≈The perimeter is approximately units.Check Your Progress Find the perimeter of quadrilateralWXYZ with W(2, 4), X(-3, 3), Y(-1, 0) and Z(3, -1).
  31. 31. Glencoe Geometry 23Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–7A solid with all that enclose a singleregion of space is called a polyhedron.A prism is a polyhedron with congruentfaces called bases.A regular prism is a prism with that are regularpolygons.A polyhedron with all faces (except for one) intersecting atis a pyramid.A polyhedron is a regular polyhedron if all of its faces areand all of theare congruent.A cylinder is a solid with congruentin a pair of parallel planes.A cone has a base and a .A sphere is a set of in space that are a givendistance from a given point.Three-Dimensional Figures• Identify three-dimensional figures.• Find surface area andvolume.MAIN IDEASBUILD YOUR VOCABULARY (page 3)
  32. 32. 24 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.1–7Identify SolidsIdentify each solid. Name the bases, faces, edges, andvertices.a.The bases and faces are rectangles. This is a rectangularprism.Bases: rectangles andFaces:Edges:Vertices:b.This figure has two faces that are hexagons. Therefore, it isa hexagonal prism.Bases: hexagons andFaces: rectangles EFLK, FGML, GHMN, HNOI, IOPJ, andJPKEEdges:−−FL,−−−GM,−−−HN,−−IO,−−JP,−−−EK,−−EF,−−−FG,−−−GH,−−HI,−−IJ,−−JE,−−KL,−−−LM,−−−MN,−−−NO,−−−OP, and−−PKVertices: E, F, G, H, I, J, K, L, M, N, O, Pc.The base of the solid is a circle and the figure comes to apoint. Therefore, it is a cone.Base: Vertex:There are faces or edges.
  33. 33. Glencoe Geometry 25Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENT1–7Check Your Progress Identify the solid.Name the bases, faces, edges, and vertices.Surface Area and VolumeRefer to the cone with r = 3 cm, h = 4 cm, and ℓ = 5 cm.a. Find the surface area of the cone.T = πrℓ + πr2 Surface area of a coneT = -π( )( )+ π( 2) SubstitutionT = 15π + 9π Simplify.T ≈ cm2 Use a calculator.b. Find the volume of the cone.V = 1_3πr2h Volume of a coneV = 1_3π( 2)( ) SubstitutionV = 12π Simplify.V ≈ cm3 Use a calculator.Check Your Progress Find the surfacearea and volume of the triangular prism.
  34. 34. 26 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.STUDY GUIDEBRINGING IT ALL TOGETHERBUILD YOURVOCABULARYVOCABULARYPUZZLEMAKERC H A P T E R1Refer to the figure.1. Name a point contained in line n.2. Name the plane containing lines n and p.3. Draw a model for therelationship AK and CGintersect at point Min plane T.Find the measure of each segment.4.−−−AD 5.−−−WX1-1Points, Lines, and PlanesUse your Chapter 1 Foldableto help you study for yourchapter test.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 1, go to:glencoe.comYou can use your completedVocabulary Builder(pages 2–3) to help you solvethe puzzle.1-2Linear Measure and Precision
  35. 35. Glencoe Geometry 27Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter BRINGING IT ALL TOGETHER16. CARPENTRY Jorge used the figure at the right to make apattern for a mosaic he plans to inlay on a tabletop. Name allof the congruent segments in the figure.Use the number line to find each measure.7. LN 8. JLFind the distance between each pair of points.9. F(-3, -2), G(1, 1) 10. Y(-6, 0), P(2, 6)Find the coordinates of the midpoint of a segment havingthe given endpoints.11. A(3, 1), B(5, 3) 12. T(-4, 9), U(7, 5)For Exercises 13–16, refer to the figure at the right.13. Name a right angle.14. Name an obtuse angle.15. Name a point in the interior of ∠EBC.16. What is the angle bisector of ∠EBC?1-3Distance and Midpoints1-4Angle Measure
  36. 36. 28 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter BRINGING IT ALL TOGETHER1In the figure, CB and CD are opposite rays, CE bisects ∠DCF,and CG bisects ∠FCB.17. If m∠DCE = 4x + 15 and m∠ECF = 6x - 5, find m∠DCE.18. If m∠FCG = 9x + 3 and m∠GCB = 13x - 9, find m∠GCB.19. TRAFFIC SIGNS The diagram shows a sign usedto warn drivers of a school zone or crossing. Measure andclassify each numbered angle.For Exercises 20–23, use the figure at the right.20. Name two acute vertical angles.21. Name a linear pair.22. Name two acute adjacent angles.23. Name an angle supplementary to ∠FKG.24. Find x so that−−TR ⊥−−TSif m∠RTS = 8x + 18.25. Find m∠PTQ if−−TR ⊥−−TSand m∠PTQ = m∠RTQ - 18.1-5Angle Relationships
  37. 37. Glencoe Geometry 29Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter BRINGING IT ALL TOGETHER1Determine whether each statement can be assumed from the figure.Explain.26. ∠YZU and ∠UZV are supplementary.27. ∠VZU is adjacent to ∠YZX.Name each polygon by its number of sides and then classify it as convexor concave and regular or irregular.28. 29.30. The length of a rectangle is 8 inches less than six times itswidth. The perimeter is 26 inches. Find the length of each side.Find the surface area and volume of each solid.31. 32. 33.1-6Two-Dimensional Figures1-7Three-Dimensional Figures
  38. 38. 30 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Student Signature Parent/Guardian SignatureTeacher SignatureChecklistC H A P T E R1Visit glencoe.com toaccess your textbook,more examples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 1.ARE 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 1 Practice Test on page 73of 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 68–72 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 73 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 68–72 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 73 of your textbook.
  39. 39. Glencoe Geometry 31Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Chapter2Reasoning and ProofC H A P T E R2NOTE-TAKING TIP: When you take notes, listen orread for main ideas. Then record concepts, defineterms, write statement in if-then form, and writeparagraph proofs.Stack the sheets ofpaper with edges3_4-inch apart. Foldthe bottom edges upto create equal tabs.Staple along the fold.Label the top tab withthe chapter title. Labelthe next 8 tabs withlesson numbers. Thelast tab is forKey Vocabulary.Begin with five sheets of 81_2" × 11" plain paper.Use 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.
  40. 40. 32 Glencoe GeometryBUILD YOUR VOCABULARYCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.C H A P T E R2Vocabulary TermFoundDefinitionDescription oron Page ExampleThis 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.conclusionconditional statementconjecture[kuhn-JEK-chur]conjunctioncontrapositiveconversecounterexampledeductive argumentdeductive reasoningdisjunctionhypothesisif-then statement
  41. 41. Chapter BUILD YOUR VOCABULARY2Glencoe Geometry 33Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundDefinitionDescription oron Page Exampleinductive reasoninginversenegationparagraph proofpostulaterelated proofconditionalsstatementtheoremtruth tabletruth valuetwo-column proof
  42. 42. 34 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A conjecture is an educated based on knowninformation.Inductive reasoning is reasoning that uses a numberof specific examples to arrive at a plausible generalizationor .Inductive Reasoning and ConjecturePatterns and ConjectureMake a conjecture about the next number based on thepattern 2, 4, 12, 48, 240.Find a Pattern: 2 4 12 48 240Conjecture: The next number will be multiplied by 6.So, it will be · 240 or .Check Your Progress Make a conjecture about the nextnumber based on the pattern 1, 1_4, 1_9, 1_16, 1_25.Geometric ConjectureFor points L, M, and N, LM = 20, MN = 6, and LN = 14.Make a conjecture and draw a figure to illustrate yourconjecture.Given: Points L, M, and N; LM = 20, MN = 6, and LN = 14.Since LN + MN = LM, the points can be collinearwith point N between points L and M.Conjecture: L, M and N are .REMEMBER ITWhen looking forpatterns in a sequence ofnumbers, test allfundamental operations.Sometimes twooperations can be used.(pages 32–33)BUILD YOUR VOCABULARY• Make conjectures basedon inductive reasoning.• Find counterexamples.MAIN IDEAS2–1
  43. 43. Glencoe Geometry 35Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTA counterexample is one false example showing that aconjecture is not true.BUILD YOUR VOCABULARY (pages 32–33)Check Your Progress ACE is a right triangle withAC = CE. Make a conjecture and draw a figure to illustrateyour conjecture.Find a CounterexampleUNEMPLOYMENT Refer to the table. Find acounterexample for the following statement.The unemployment rate is highest in the cities withthe most people.City Population RateArmstrong 2163 3.7%Cameron 371,825 7.2%El Paso 713,126 7.0%Hopkins 33,201 4.3%Maverick 50,436 11.3%Mitchell 9402 6.1%Maverick has a population of people, and it hasa higher rate of unemployment than El Paso, which has apopulation of people.Check Your Progress Refer to the table. Find acounterexample for the following statement.The unemployment rate is lowest in the cities with theleast people.2–1Write definitions for aconjecture and inductivereasoning under the tabfor Lesson 2-1. Design apattern and state aconjecture about thepattern.ORGANIZE IT
  44. 44. 36 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.2–2 Logic• Determine truth valuesof conjunctions anddisjunctions.• Construct truth tables.MAIN IDEAS A statement is any sentence that is either true or false,but not . The truth or falsity of a statement iscalled its truth value.(pages 32–33)BUILD YOUR VOCABULARYTruth Values of ConjunctionsUse the following statements to write a compoundstatement for each conjunction. Then find its truth value.p: One foot is 14 inches.q: September has 30 days.r: A plane is defined by three noncollinear points.a. p and qOne foot is 14 inches, and September has 30 days. p and qis because is false and q is .b. ∼p ∧ rA foot is not 14 inches, and a plane is defined by threenoncollinear points. ∼p ∧ r is , because ∼p isand r is .Check Your Progress Use the following statements towrite a compound statement for each conjunction. Thenfind its truth value.p: June is the sixth month of the year.q: A square has five sides.r: A turtle is a bird.a. p and rb. ∼q ∧ ∼rNegation If a statementis represented by p, thennot p is the negationof the statement.Conjunction Aconjunction is acompound statementformed by joining twoor more statements withthe word and.KEY CONCEPTS
  45. 45. Glencoe Geometry 37Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A convenient method for organizing the truth values ofstatements is to use a truth table.Disjunction A disjunctionis a compound statementformed by joining twoor more statements withthe word or.KEY CONCEPTTruth Values of DisjunctionsUse the following statements to write a compoundstatement for each disjunction. Then find its truth value.p:−−AB is proper notation for “segment AB.”q: Centimeters are metric units.r: 9 is a prime number.a. p or q−−AB is proper notation for “segment AB,” or centimeters aremetric units. p or q is because q is .It does not matter that is false.b. q ∨ rCentimeters are metric units, or 9 is a prime number. q ∨ ris because q is . It does not matterthat is false.Check Your Progress Use the following statements towrite a compound statement for each disjunction. Thenfind its truth value.p: 6 is an even number.q: A cow has 12 legs.r: A triangle has three sides.a. p or rb. ∼q ∨ ∼rBUILD YOUR VOCABULARY (pages 32–33)2–2
  46. 46. 38 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTConstruct a truth tablefor the compoundstatement ∼q ∨ p andwrite it under the tabfor Lesson 2-2.ORGANIZE ITUse Venn DiagramsDANCING The Venn diagram shows the number ofstudents enrolled in Monique’s Dance School for tap,jazz, and ballet classes.Tap2817 2513 Jazz4329Ballet9a. How many students are enrolled in all three classes?The number of students that are enrolled in allclasses is represented by the of the threecircles. There are students enrolled in all three classes.b. How many students are enrolled in tap or ballet?The number of students enrolled in tap or ballet isrepresented by the union of the two sets. There areor studentsenrolled in tap or ballet.c. How many students are enrolled in jazz and ballet,but not tap?The number of students enrolled in and ballet butnot tap is represented by the intersection of the jazz andsets. There are students enrolled inonly.Check Your Progress How many students are enrolled inballet and tap, but not jazz?2–2
  47. 47. Glencoe Geometry 39Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.2–3 Conditional StatementsWrite a Conditional in If-Then FormIdentify the hypothesis and conclusion of eachstatement. Then write each statement inif-then form.a. Distance is positive.Hypothesis:Conclusion:If a distance is measured, then it is positive.b. A five-sided polygon is a pentagon.Hypothesis:Conclusion:If a polygon has , then it is a .Check Your Progress Identify the hypothesis and conclusionof the statement.To find the distance between two points, you can use theDistance Formula.• Analyze statements inif-then form.• Write the converse,inverse, andcontrapositive ofif-then statements.MAIN IDEASIf-Then Statement Anif-then statement iswritten in the form if p,then q. The phraseimmediately followingthe word if is called thehypothesis, and thephrase immediatelyfollowing the word thenis called the conclusion.KEY CONCEPT
  48. 48. 40 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Truth Values of ConditionalsDetermine the truth value of the following statementfor each set of conditions. If Yukon rests for 10 days, hisankle will heal.a. Yukon rests for 10 days, and he still has a hurt ankle.The hypothesis is , since he rested for 10 days. Theconclusion is since his ankle will heal. Thus,the conditional statement is .b. Yukon rests for 10 days, and he does not have a hurtankle anymore.The hypothesis is since Yukon rested for 10 days,and the conclusion is because he does not have ahurt ankle. Since what was stated is true, the conditionalstatement is .c. Yukon rests for 7 days, and he does not have a hurtankle anymore.The hypothesis is , and the conclusion is .The statement does not say what happens if Yukon onlyrests for 7 days. In this case, we cannot say that thestatement is false. Thus, the statement is .Check Your Progress Determine the truth value ofthe following statement for each set of conditions. If itrains today, then Michael will not go skiing.a. It rains today; Michael does not go skiing.b. It rains today; Michael goes skiing.Under the tab forLesson 2-3, explain howto determine the truthvalue of a conditional.Be sure to include anexample.ORGANIZE IT2–3
  49. 49. Glencoe Geometry 41Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTRelated ConditionalsWrite the converse, inverse, and contrapositive of thestatement All squares are rectangles. Determinewhether each statement is true or false. If a statementis false, give a counterexample.Conditional: If a shape is a square, then it is a rectangle.The conditional statement is true.Write the converse by switching the andconclusion of the conditional.Converse: If a shape is a rectangle, then it is a square.The converse is . A rectangle with ℓ = 2and w = 4 is not a square.Inverse: If a shape is not a square, then it is not arectangle. The inverse is . A rectanglewith side lengths 2, 2, 4, and 4 is not a square.The contrapositive is the of the hypothesis andconclusion of the converse.Contrapositive: If a shape is not a rectangle, then it is nota square. The contrapositive is .Check Your Progress Write the converse, inverse, andcontrapositive of the statement The sum of the measures oftwo complementary angles is 90. Determine whether eachstatement is true or false. If a statement is false, give acounterexample.Related ConditionalsConditional Formed bygiven hypothesis andconclusionConverse Formed byexchanging thehypothesis andconclusion of theconditionalInverse Formed bynegating both thehypothesis andconclusion of theconditionalContrapositive Formedby negating both thehypothesis andconclusion of theconverse statementKEY CONCEPTS2–3
  50. 50. 42 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Deductive reasoning uses facts, rules, , orto reach logical conclusions.Deductive Reasoning• Use the Law ofDetachment.• Use the Law ofSyllogism.MAIN IDEASDetermine Valid ConclusionsThe following is a true conditional. Determinewhether the conclusion is valid based on the giveninformation. Explain your reasoning.If two segments are congruent and the second segmentis congruent to a third segment, then the first segmentis also congruent to the third segment.a. Given:−−−WX−−UV;−−UV−−RTConclusion:−−−WX−−RTThe hypothesis states that W andU . Since the conditional is true and thehypothesis is true, the conclusion is .b. Given:−−UV,−−−WX−−RTConclusion:−−−WX−−UV and−−UV−−RTThe hypothesis states that W and .Not enough information is provided to reach the conclusion.The conclusion is .Check Your Progress The following is a true conditional.Determine whether the conclusion is valid based on the giveninformation. Explain your reasoning.If a polygon is a convex quadrilateral, then the sum of themeasures of the interior angles is 360°.BUILD YOUR VOCABULARY (pages 32–33)Law of Detachment Ifp → q is true and p istrue, then q is also true.KEY CONCEPT2–4
  51. 51. Glencoe Geometry 43Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Given: m∠X + m∠N + m∠O = 360°Conclusion: If you connect X, N, and O with segments, thefigure will be a convex quadrilateral.Determine Valid Conclusions From Two ConditionalsPROM Use the Law of Syllogism to determine whethera valid conclusion can be reached from each set ofstatements.a. (1) If Salline attends the prom, she will go with Mark.(2) If Salline goes with Mark, Donna will go with Albert.The conclusion of the first statement is theof the second statement. Thus, if Salline attends the prom,Donna will go with .b. (1) If Mel and his date eat at the Peddler Steakhouse beforegoing to the prom, they will miss the senior march.(2) The Peddler Steakhouse stays open until 10 P.M.There is . While both statementsmay be true, the conclusion of one statement is not used asthe of the other statement.Check Your Progress Use the Law of Syllogism todetermine whether a valid conclusion can be reachedfrom each set of statements.a. (1) If you ride a bus, then you attend school.(2) If you ride a bus, then you go to work.b. (1) If your alarm clock goes off in the morning, then youwill get out of bed.(2) You will eat breakfast, if you get out of bed.Law of Syllogism Ifp → q and q → r aretrue, then p → r isalso true.KEY CONCEPTUse symbols and wordsto write the Law ofDetachment and theLaw of Syllogism underthe tab for Lesson 2-4.ORGANIZE IT2–4
  52. 52. 44 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTAnalyze ConclusionsDetermine whether statement (3) follows from statements(1) and (2) by the Law of Detachment or the Law ofSyllogism. If it does, state which law was used. If it doesnot, write invalid.(1) If the sum of the squares of two sides of a triangleis equal to the square of the third side, then thetriangle is a right triangle.(2) For △XYZ, (XY)2 + (YZ)2 = (ZX)2.(3) △XYZ is a right triangle.p: The of the squares of the lengths of the two sidesof a is equal to the square of the length ofthe side.q: the triangle is a triangle.By the Law of , if p → q is true andp is true, then q is also true. Statement (3) is aconclusion by the Law of Detachment.Check Your Progress Determine whether statement(3) follows from statements (1) and (2) by the Law ofDetachment or the Law of Syllogism. If it does, state whichlaw was used. If it does not, write invalid.(1) If a children’s movie is playing on Saturday, Janine willtake her little sister Jill to the movie.(2) Janine always buys Jill popcorn at the movies.(3) If a children’s movie is playing on Saturday, Jill willget popcorn.2–4
  53. 53. Glencoe Geometry 45Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.2–5A postulate is a statement that describes a fundamentalrelationship between the basic terms of .Postulates are accepted as .Postulates and Paragraph Proofs• Identify and use basicpostulates about points,lines, and planes.• Write paragraph proofs.MAIN IDEASPoints and LinesSNOW CRYSTALS Some snow crystals are shaped likeregular hexagons. How many lines must be drawn tointerconnect all vertices of a hexagonal snow crystal?Draw a diagram of a hexagon toillustrate the solution. Connect eachpoint with every other point. Then,count the number of segments.Between every two points there isexactly segment.Be sure to include the sides of thehexagon. For the six points,segments can be drawn.Check Your Progress Jodi is making a string art design.She has positioned ten nails, similar to the vertices of adecagon, onto a board. How many strings will she need tointerconnect all vertices of the design?BUILD YOUR VOCABULARY (pages 32–33)Postulate 2.1Through any two points, there is exactly one line.Postulate 2.2Through any three points not on the same line, there isexactly one plane.
  54. 54. 46 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Use PostulatesDetermine whether the statement is always, sometimes,or never true. Explain.a. If plane T contains EF and EF contains point G, thenplane T contains point G.; Postulate 2.5 states that if twolie in a plane, then the entire containing thosepoints lies in the plane.b. For XY, if X lies in plane Q and Y lies in plane R, thenplane Q intersects plane R.; planes Q and R can both be ,and can intersect both planes.c. GH contains three noncollinear points.; noncollinear points do not lie on the sameby definition.Check Your Progress Determine whether the statementis always, sometimes, or never true. Explain.Plane A and plane B intersect in one point.Postulate 2.3A line contains at least two points.Postulate 2.4A plane contains at least three points not on the same line.Postulate 2.5If two points lie on a plane, then the entire line containingthose points lies in that plane.Postulate 2.6If two lines intersect, then their intersection is exactlyone point.Postulate 2.7If two planes intersect, then their intersection is a line.Collinear means lyingon the same line, sononcollinear means notlying on the same line.(Lesson 1-1)REVIEW IT2–5
  55. 55. Glencoe Geometry 47Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENT2–5Write a Paragraph ProofGiven AC intersects CD, write a paragraph proof toshow that A, C, and D determine a plane.Given: AC intersects CD.Prove: A, C, and D determine a plane.AC and CD must intersect at C because if linesintersect, then their intersection is exactly point.Point A is on AC and point D is on CD. Therefore, pointsA and D are . Therefore, A, C, and Ddetermine a plane.Check Your Progress Given−−RT−−TY, S is the midpoint of−−RT,and X is the midpoint of−−TY, writea paragraph proof to show that−−ST−−TX.Theorem 2.1 Midpoint TheoremIf M is the midpoint of−−AB, then−−−AM−−−MB.Proofs Five essentialparts of a good proof:• State the theorem orconjecture to beproven.• List the giveninformation.• If possible, draw adiagram to illustratethe given information.• State what is to beproved.• Develop a system ofdeductive reasoning.CopyExample 3 under the tabfor Lesson 2-5 as anexample of a paragraphproof.KEY CONCEPT
  56. 56. 48 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.A two-column proof, or formal proof, containsand organized intwo columns.Algebraic Proof2–6• Use algebra to writetwo-column proofs.• Use properties ofequality in geometryproofs.MAIN IDEASVerify Algebraic RelationshipsSolve 2(5 - 3a) - 4(a + 7) = 92.Algebraic Steps Properties2(5 - 3a) - 4(a + 7) = 92 Original equation10 - 6a - 4a - 28 = 92 Property-18 - 10a = 92 Property= Addition Propertya = PropertyCheck Your Progress Solve -3(a + 3) + 5(3 - a) = -50.BUILD YOUR VOCABULARY (pages 32–33)Write a two-columnproof for the statementif 5(2x - 4) = 20, thenx = 4 under the tab forLesson 2-6.ORGANIZE IT
  57. 57. Glencoe Geometry 49Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.2–6Write a Two-Column ProofWrite a two-column proof to show that if7d + 3_4= 6,then d = 3.Statements Reasons1. 7d + 3_4= 6 1. Given2. 4(7d + 3_4 ) = 4(6) 2.3. 7d + 3 = 24 3. Substitution4. 7d = 21 4.5. 5.Check Your Progress Write a two-column proof for thefollowing.Given: 10 - 8n_3= -2Proof: n = 2What are the fiveessential parts of a goodproof? (Lesson 2-5)WRITE IT
  58. 58. 50 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENT2–6Geometric ProofSEA LIFE A starfish has five arms. If the length of arm 1is 22 centimeters, and arm 1 is congruent to arm 2, andarm 2 is congruent to arm 3, prove that arm 3 has alength of 22 centimeters.Given: arm 1 arm 2arm 2 arm 3m arm 1 = 22 cmProve: m arm 3 = 22 cmProof:Statements Reasons1. arm 1 arm 2; 1. Givenarm 2 arm 32. arm 1 arm 3 2.3. m arm 1 = m arm 3 3. Definition of congruence4. 4. Given5. m arm 3 = 22 cm 5. Transitive PropertyCheck Your Progress A stopsign as shown at right is a regularoctagon. If the measure of angle Ais 135 and angle A is congruent toangle G, prove that the measureof angle G is 135.
  59. 59. Glencoe Geometry 51Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.2–7• Write proofs involvingsegment addition.• Write proofs involvingsegment congruence.MAIN IDEASProving Segment RelationshipsProof with Segment AdditionProve the following.Given: PR = QSProve: PQ = RSStatements Reasons1. PR = QS 1. Given2. PR - QR = QS - QR 2. Subtraction Property3. PR - QR = PQ; 3. Segment AdditionQS - QR = RS Postulate4. PQ = RS 4. SubstitutionCheck Your Progress Prove the following.Given: AC = ABAB = BXCY = XDProve: AY = BDPostulate 2.8 Ruler PostulateThe points on any line or line segment can be paired withreal numbers so that, given any two points A and B on aline, A corresponds to zero, and B corresponds to a positivereal number.Postulate 2.9 Segment Addition PostulateIf B is between A and C, then AB + BC = AC.If AB + BC = AC, then B is between A and C.Under the tab forLesson 2-7, write theSegment AdditionPostulate, draw anexample, and writean equation for yourexample.ORGANIZE IT
  60. 60. 52 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENT2–7Transitive Property of CongruenceProve the following.Given: WY = YZ,−−YZ−−XZ−−XZ−−−WYProve:−−−WX−−−WYStatements Reasons1. WY = YZ 1. Given2.−−−WY−−YZ 2. Definition ofsegments3.−−YZ−−XZ;−−XZ−−−WX 3. Given4. 4. Transitive Property5.−−−WX−−−WY 5.Check Your Progress Prove the following.Given:−−−GD−−−BC,−−−BC−−−FH,−−−FH−−AEProve:−−AE−−−GDTheorem 2.2Congruence of segments is reflexive, symmetric, andtransitive.
  61. 61. Glencoe Geometry 53Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.2–8 Proving Angle RelationshipsAngle AdditionTIME At 4 o’clock, the angle betweenthe hour and minute hands of a clockis 120°. If the second hand stopswhere it bisects the angle betweenthe hour and minute hands, what arethe measures of the angles betweenthe minute and second hands andbetween the second and hour hands?If the second hand stops where the angle is bisected, then theangle between the minute and second hands isthe measure of the angle formed by the hour and minute hands,or (120) = .By the Angle Addition Postulate, the sum of the two angles is, so the angle between the second and hour hands isalso .Check Your Progress The diagramshows one square for a particular quiltpattern. If m∠BAC = m∠DAE = 20°,and ∠BAE is a right angle, find m∠CAD.Postulate 2.10 Protractor PostulateGiven AB and a number r between 0 and 180, there isexactly one ray with endpoint A, extending on either sideof AB, such that the measure of the angle formed is r.Postulate 2.11 Angle Addition PostulateIf R is in the interior of ∠PQS, then m∠PQR + m∠RQS =m∠PQS. If m∠PQR + m∠RQS = m∠PQS, then R is in theinterior of ∠PQS.• Write proofs involvingsupplementary andcomplementary angles.• Write proofs involvingcongruent and rightangles.MAIN IDEAS
  62. 62. 54 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.2–8The angles of a linearpair are alwayssupplementary, butsupplementary anglesneed not form a linearpair. (Lesson 1-5)REVIEW ITSupplementary AnglesIf ∠1 and ∠2 form a linear pair and m∠2 = 166, find m∠1.m∠1 + m∠2 = 180m∠1 + = 180 m∠2 =m∠1 = 14 Subtraction PropertyCheck Your Progress If ∠1 and ∠2 are complementaryangles and m∠1 = 62, find m∠2.Theorem 2.5Congruence of angles is reflexive, symmetric, and transitive.Theorem 2.6Angles supplementary to the same angle or to congruentangles are congruent.Theorem 2.7Angles complementary to the same angle or to congruentangles are congruent.Theorem 2.3 Supplement TheoremIf two angles form a linear pair, then they are supplementaryangles.Theorem 2.4 Complement TheoremIf the noncommon sides of two adjacent angles form aright angle, then the angles are complementary angles.
  63. 63. Glencoe Geometry 55Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.2–8Use Supplementary AnglesIn the figure, ∠1 and ∠4 form a linear pair, andm∠3 + m∠1 = 180. Prove that ∠3 and ∠4 are congruent.Given: ∠1 and ∠4 form a linear pair.m∠3 + m∠1 = 180Prove: ∠3 ∠4Proof:StatementsReasons1. m∠3 + m∠1 = 180; ∠1 1. Givenand ∠4 form a linear pair.2. ∠1 and ∠4 are 2.supplementary.3. ∠3 and ∠1 are 3. Definition of supplementarysupplementary. angles.4. ∠3 ∠4 4.Check Your ProgressIn the figure, ∠NYR and∠RYA form a linear pair,∠AXY and ∠AXZ form alinear pair, and ∠RYAand ∠AXZ are congruent.Prove that ∠RYN and∠AXY are congruent.Under the tab forLesson 2-8, copyTheorem 2.12: If twoangles are congruentand supplementary, theneach angle is a rightangle. Illustrate thistheorem with a diagram.ORGANIZE IT
  64. 64. 56 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENT2–8Vertical AnglesIf ∠1 and ∠2 are vertical angles and m∠1 = d - 32 andm∠2 = 175 - 2d, find m∠1 and m∠2.∠1 ∠2 Theoremm∠1 = m∠2 Definition of congruent angles= Substitution3d = 207 Addition Propertyd = 69 Divide each side by 3.m∠1 = m∠2 = 175 - 2d= - 32 = 175 - 2( )= =Check Your Progress If ∠A and ∠Z are vertical angles andm∠A = 3b - 23 and m∠Z = 152 - 4b, find m∠A and m∠Z.Theorem 2.8 Vertical Angle TheoremIf two angles are vertical angles, then they are congruent.Theorem 2.9Perpendicular lines intersect to form four right angles.Theorem 2.10All right angles are congruent.Theorem 2.11Perpendicular lines form congruent adjacent angles.Theorem 2.12If two angles are congruent and supplementary, then eachangle is a right angle.Theorem 2.13If two congruent angles form a linear pair, then they areright angles.REMEMBER ITBe sure to readproblems carefully inorder to provide theinformation requested.Often the value of thevariable is used to findthe answer.
  65. 65. STUDY GUIDEGlencoe Geometry 57Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.BUILD YOURVOCABULARYVOCABULARYPUZZLEMAKERUse your Chapter 2 Foldableto help you study for yourchapter test.To make a crossword puzzle,word search, or jumblepuzzle of the vocabulary wordsin Chapter 2, go to:glencoe.comYou can use your completedVocabulary Builder(pages 32–33) to help you solvethe puzzle.Make a conjecture about the next number in the pattern.1. -6, -3, 0, 3, 6 2. 4, -2, 1, -1_2, 1_43. Make a conjecture based on the given information.Points A, B, and C are collinear. D is between A and B.Use the statements p: -1 + 4 = 3, q: A pentagon has 5 sides,and r: 5 + 3 > 8 to find the truth value of each statement.4. p ∨ r 5. q ∧ r 6. (p ∨ q) ∧ r7. Construct a truth table for the compound statement p ∨ (∽p ∧ q).2-1Inductive Reasoning and Conjecture2-2LogicC H A P T E R2 BRINGING IT ALL TOGETHER
  66. 66. Chapter BRINGING IT ALL TOGETHER258 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.8. Identify the hypothesis and conclusion of the statement.If x - 7 = 4, then x = 11.9. Write the statement in if-then form.Supplementary angles have measures with a sum of 180°.10. Use the Law of Detachment to tell whether the followingreasoning is valid. Write valid or not valid.If a car has air bags it is safe. Maria’s carhas air bags. Therefore, Maria’s car is safe.11. Use the Law of Syllogism to write a valid conclusion fromthis set of statements.(1) All squares are rectangles.(2) All rectangles are parallelograms.12. Determine the number of line segments that canbe drawn connecting each pair of points.Name the definition, property, postulate, or theorem thatjustifies each statement.13.−−−CD−−−CD14. If A is the midpoint of−−−CD, then−−CA−−−AD.2-3Conditional Statements2-4Deductive Reasoning2-5Postulates and Paragraph Proofs
  67. 67. Chapter BRINGING IT ALL TOGETHER2Glencoe Geometry 59Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Name the definition, property, postulate, or theorem thatjustifies each statement.15. If x - 7 = 9, then x = 1616. 5(x + 8) = 5x + 4017. If−−−CD−−EF and−−EF−−−GH, then−−−CD−−−GH.Name the definition, property, postulate, or theorem thatjustifies each statement.18. If−−AB−−−CD, then−−−CD−−AB.19. If AB + BD = AD, then B is between A and D.Name the definition, property, postulate, or theorem thatjustifies each statement.20. If m∠1 + m∠2 = 90, and m∠2 + m∠3 = 90, then m∠1 = m∠3.21. If ∠A and ∠B are vertical angles, then ∠A ∠B.2-7Proving Segment Relationships2-6Algebraic Proof2-8Proving Angle Relationships
  68. 68. 60 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Student Signature Parent/Guardian SignatureTeacher SignatureC H A P T E R2ChecklistCheck 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 2 Practice Test onpage 137 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 2 Study Guide and Reviewon pages 132–136 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 2 Practice Test onpage 137 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 132–136 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 2 Practice Test onpage 137.ARE YOU READY FORTHE CHAPTER TEST?Visit glencoe.com to accessyour textbook, moreexamples, self-checkquizzes, and practice teststo help you study theconcepts in Chapter 2.
  69. 69. Chapter3Glencoe Geometry 61Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Use 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 R3 Parallel and Perpendicular LinesNOTE-TAKING TIP: When taking notes, writing aparagraph that describes the concepts, thecomputational skills, and the graphics will helpyou to understand the math in the lesson.Fold in half matchingthe short sides.Unfold and fold thelong side up 2 inchesto form a pocket.Staple or glue theouter edges tocomplete the pocket.Label each side asshown. Use indexcards to recordexamples.Begin with one sheet of 81_2" × 11" paper.Parallel Perpendicular
  70. 70. 62 Glencoe GeometryBUILD YOUR VOCABULARYCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.C H A P T E R3This 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.Vocabulary TermFoundDefinitionDescription oron Page Examplealternate exterioranglesalternate interioranglesconsecutive interioranglescorresponding anglesequidistant[ee-kwuh-DIS-tuhnt]non-Euclidean geometry[yoo-KLID-ee-yuhn]parallel lines
  71. 71. Chapter BUILD YOUR VOCABULARY3Glencoe Geometry 63Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Vocabulary TermFoundDefinitionDescription oron Page Exampleparallel planespoint-slope formrate of changeskew linesslopeslope-intercept formtransversal
  72. 72. 64 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3–1 Parallel Lines and TransversalsIdentify RelationshipsUse the figure shown at the right.a. Name all planes that areparallel to plane AEF.b. Name all segments thatintersect−−AF.c. Name all segments that are skew to−−AD.Check Your Progress Use the figure shown.a. Name all planes that are parallelto plane RST.Coplanar lines that do not are calledparallel lines.Two that do not are calledparallel planes.Lines that do not intersect and are not arecalled skew lines.A line that intersects or more lines in a planeat different points is called a transversal.BUILD YOUR VOCABULARY (pages 62–63)• Identify therelationships betweentwo lines or two planes.• Name angles formed bya pair of lines and atransversal.MAIN IDEAS
  73. 73. Glencoe Geometry 65Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3–1Identify TransversalsBUS STATION Some of a bus station’s driveways areshown. Identify the sets of lines to which each givenline is a transversal.a. line v If the lines are extended, line v intersects lines.b. line yc. line uCheck Your Progress A groupof nature trails is shown. Identifythe set of lines to which eachgiven line is a transversal.a. line ab. line bc. line cd. line db. Name all segments that intersect−−YZ.c. Name all segments that are skew to−−TZ.
  74. 74. 66 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3–1Page(s):Exercises:HOMEWORKASSIGNMENTconsecutive interior angles:∠4 and ∠5, ∠3 and ∠6alternate interior angles:∠4 and ∠6, ∠3 and ∠5alternate exterior angles:∠1 and ∠7, ∠2 and ∠8corresponding angles: ∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7,∠4 and ∠8BUILD YOUR VOCABULARY (page 62)1 23 4658710911 12nᐉ mpIdentify Angle RelationshipsIdentify each pair of angles as alternate interior, alternateexterior, corresponding, or consecutive interior angles.a. ∠7 and ∠3b. ∠8 and ∠2c. ∠4 and ∠11 d. ∠7 and ∠1e. ∠3 and ∠9 f. ∠7 and ∠10Check Your Progress Identify each pair of angles asalternate interior, alternate exterior, corresponding,or consecutive interior angles.a. ∠4 and ∠5b. ∠7 and ∠9c. ∠4 and ∠7 d. ∠2 and ∠11Using a separate cardfor each type of anglepair, draw two parallellines cut by a transversaland identify consecutiveinterior angles,alternate exteriorangles, alternateinterior angles, andcorresponding angles.Place your cards in theParallel Lines pocket.Parallel PerpendicularORGANIZE IT
  75. 75. Glencoe Geometry 67Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3–2 Angles and Parallel LinesSketch two parallel linesand a transversal on acard. Below the sketch,write a brief note toexplain how knowingone of the anglemeasures allows you tofind the measures of allthe other angles. Placethe card in the ParallelLines pocket.Parallel PerpendicularORGANIZE ITDetermine Angle MeasuresIn the figure x y and m∠11 = 51. Find m∠16.∠11 ∠15 Corresponding Angles PostulateVertical Angles Theorem∠11 ∠16 Transitive Propertym∠11 = m∠16 Definition of congruent angles= m∠16 SubstitutionCheck Your Progress In the figure, m∠18 = 42. Find m∠25.Postulate 3-1 Corresponding Angles PostulateIf two parallel lines are cut by a transversal, then each pairof corresponding angles is congruent.Theorem 3.1 Alternate Interior Angles TheoremIf two parallel lines are cut by a transversal, then each pairof alternate interior angles is congruent.Theorem 3.2 Consecutive Interior Angles TheoremIf two parallel lines are cut by a transversal, then each pairof consecutive interior angles is supplementary.• Use the propertiesof parallel lines todetermine congruentangles.• Use algebra to findangle measures.MAIN IDEAS
  76. 76. 68 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTWhen two lines intersect,what types of anglepairs are congruent?(Lesson 2-8)REVIEW ITFind Values of VariablesALGEBRA If m∠5 = 2x - 10,m∠6 = 4(y - 25), andm∠7 = x + 15, find x and y.Find x. Since p q, m∠5 m∠7by the Corresponding AnglesPostulate.m∠5 = m∠7 Definition of congruent angles= Substitutionx = Subtract x from each side andadd 10 to each side.Find y. Since m n, m∠5 m∠6 by the Alternate ExteriorAngles Theorem.m∠5 = m∠6Definition of congruentangles= Substitution= x = 25= y Add 100 to each side anddivide each side by 4.Check Your ProgressIf m∠1 = 9x + 6,m∠2 = 2(5x - 3), andm∠3 = 5y + 14, find x and y.Theorem 3.3 Alternate Exterior Angles TheoremIf two parallel lines are cut by a transversal, then each pairof alternate exterior angles is congruent.Theorem 3.4 Perpendicular Transversal TheoremIn a plane, if a line is perpendicular to one of two parallellines, then it is perpendicular to the other.3–2
  77. 77. Glencoe Geometry 69Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3–3 Slope of LinesFind the Slope of a Linea. Find the slope of the line.฀฀ ฀Use the rise_run method. From(-3, 7) to (-1, -1), go downunits and 2 units.rise_run = orb. Find the slope of the line.Use the slope formula.Let (0, 4) be (x1, y1) and(0, -3) be (x2, y2).m =y2 - y1_x2 - x1=--0 0or-7_0, which is .c. Find the slope of the line.฀m =y2 - y1_x2 - x1==The slope of a line is the ratio of the vertical rise to itshorizontal run.BUILD YOUR VOCABULARY (page 63)• Find slopes of lines.• Use slope to identifyparallel andperpendicular lines.MAIN IDEASSlope The slope m ofa line containing twopoints with coordinates(x1, y1) and (x2, y2) isgiven by the formulam =y2 - y1_x2 - x1, wherex1 ≠ x2.KEY CONCEPT-6 - (-2)
  78. 78. 70 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.The rate of change describes how a quantity isover .Postulate 3.2Two nonvertical lines have the same slope if and only ifthey are parallel.Postulate 3.3Two nonvertical lines are perpendicular if and only if theproduct of their slopes is -1.3–3d. Find the slope of the line.m =y2 - y1_x2 - x1== orCheck Your Progress Find the slope of each line.a. b.c. d.REMEMBER ITLines with positiveslopes rise as you movefrom left to right, whilelines with negativeslopes fall as you movefrom left to right.BUILD YOUR (page 63)
  79. 79. Glencoe Geometry 71Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTDetermine Line RelationshipsDetermine whether FG and HJ are parallel,perpendicular, or neither.a. F(1, -3), G(-2, -1), H(5, 0), J(6, 3)Find the slopes of FG and HJ.slope of FG =--= or -2_3slope of HJ =--= 3_1orThe slopes are not the same, so FG and HJ are. The product of the slopes is . SoFG and HJ are neither nor .b. F(4, 2), G(6, -3), H(-1, 5), J(-3, 10)slope of FG =--= or -5_2slope of HJ =--= or -5_2The slopes are the same, so FG and HJ are .Check Your Progress Determine whether AB and CDare parallel, perpendicular, or neither.a. A(-2, -1), B(4, 5), C(6, 1), D(9, -2)b. A(7, -3), B(1, -2), C(4, 0), D(-3, 1)On a study card, writethe equation of twolines that are paralleland explain what is trueabout their slopes. Placethis card in the ParallelLines pocket.Parallel PerpendicularORGANIZE IT3–3
  80. 80. 72 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3–4The slope-intercept form of a linear equation is, where m is the of theline and b is the y-intercept.The point-slope form is , where(x1, y1) are the coordinates of any point on the line and mis the slope of the line.Equations of LinesSlope and y-interceptWrite an equation in slope-intercept form of the linewith slope of 6 and y-intercept of -3.y = mx + b Slope-intercept formy = x + m = 6, b = -3The slope-intercept form of the equation is .Check Your Progress Write an equation in slope-interceptform of the line with slope of -1 and y-intercept of 4.Slope and a PointWrite an equation in point-slope form of the line whoseslope is -3_5and contains (-10, 8).y - y1 = m(x - x1) Point-slope formy - = -3_5[x - ] m = -3_5, (x1, y1) = (-10, 8)y - 8 = -3_5(x + 10) Simplify.• Write an equation of aline given informationabout its graph.• Solve problems bywriting equations.MAIN IDEASBUILD YOUR VOCABULARY (page 63)REMEMBER ITNote that the point-slope form of anequation is different foreach point used.However, the slope-intercept form of anequation is unique.
  81. 81. Glencoe Geometry 73Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Check Your Progress Write an equation in point-slopeform of the line whose slope is 1_3and contains (6, -3).Two PointsWrite an equation in slope-intercept form for a linecontaining (4, 9) and (-2, 0).Find the slope of the line.m =y2 - y1_x2 - x1Slope formula= 0 - 9_-2 - 4x1 = 4, x2 = -2,y1 = 9, y2 = 0= or Simplify.Use the point-slope form to write an equation.Using (4, 9):y - y1 = m(x - x1) Point-slope formy - = (x - 4) m = , (x1, y1) = (4, 9)y = 3_2x + 3 Distributive Property andadd 9 to both sides.Using (-2, 0):y - y1 = m(x - x1) Point-slope formy - = [x - (-2)] m = , (x1, y1) = (-2, 0)y = Distributive PropertyCheck Your Progress Write an equation in slope-interceptform for a line containing (3, 2) and (6, 8).3–4
  82. 82. 74 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTOne Point and an EquationWrite an equation in slope-intercept form for a linecontaining (1, 7) that is perpendicular to the liney = -1_2x + 1.Since the slope of the line is , the slopeof a line perpendicular to it is .y - y1 = m(x - x1) Point-slope formy - = (x - ) m = 2, (x1, y1) = (1, 7)y = Distributive Property andadd 7 to each side.Check Your Progress Write an equation in slope-interceptform for a line containing (-3, 4) that is perpendicular to theline y = 3_5x - 4.On a study card, writethe slope-interceptequations of two linesthat are perpendicular,and explain what istrue about their slopes.Place this card in thePerpendicular Linespocket.Parallel PerpendicularORGANIZE IT3–4
  83. 83. Glencoe Geometry 75Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3–5 Proving Lines ParallelIdentify Parallel LinesDetermine which lines,if any, are parallel.Since ∠RQT and ∠SQP areangles,m∠SQP = .Since m∠UPQ + m∠SQP = + or ,consecutive interior angles are supplementary. So, a b.Since m∠TQR + m∠VRQ = 77 + 100 or 177, consecutiveinterior angles are . So, c is notparallel to a or b.• Recognize angleconditions that occurwith parallel lines.• Prove that two lines areparallel based on givenangle relationships.MAIN IDEASPostulate 3.4If two lines in a plane are cut by a transversal so thatcorresponding angles are congruent, then the lines areparallel.Postulate 3.5 Parallel PostulateIf there is a line and a point not on the line, then thereexists exactly one line through the point that is parallel tothe given line.Theorem 3.5If two lines in a plane are cut by a transversal so that a pairof alternate exterior angles is congruent, then the two linesare parallel.Theorem 3.6If two lines in a plane are cut by a transversal so that a pairof consecutive interior angles is supplementary, then thelines are parallel.Theorem 3.7If two lines in a plane are cut by a transversal so that a pairof alternate interior angles is congruent, then the lines areparallel.Theorem 3.8In a plane, if two lines are perpendicular to the same line,then they are parallel.
  84. 84. 76 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3–5Check Your Progress Determine which lines, if any,are parallel.Solve Problems with Parallel LinesALGEBRA Find x and m∠ZYN so that PQ MN.m∠WXP = m∠ZYN Alternate exterior angles= Substitution4x Ϫ 25 = 35 Subract 7x from each side.4x = 60 Add 25 to each side.x = 15 Divide each side by 4.Now use the value of x to find m∠ZYN.m∠ZYN = 7x + 35 Original equation= 7΂ ΃+ 35 x = 15= 140 Simplify.So, x = 15 and m∠ZYN = 140.Check Your ProgressFind x and m∠GBAso that GH RS.On a card, write thebiconditional statementthat you obtain bycombining Postulates 3.1and 3.4. Place the cardin the Parallel Linespocket.Parallel PerpendicularORGANIZE IT
  85. 85. Glencoe Geometry 77Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3–5Prove Lines ParallelGiven: ℓ m∠4 ∠7Prove: r sWrite how you can useangles formed by twolines and a transversalto decide whether thetwo lines are parallel ornot parallel.WRITE ITStatements Reasons1. ℓ m, ∠4 ∠7 1. Given2. ∠4 and ∠6 are suppl. 2. Consecutive Interior ∠Thm.3. ∠4 + ∠6 = 180 3.4. = m∠7 4. Def. of congruent ∠s5. m∠7 + = 180 5.6. ∠7 and ∠6 are suppl. 6. Def. of suppl. Єs7. r s 7. If cons. int. Єs are suppl.,then lines are parallel.Slope and Parallel LinesDetermine whether p q.slope of p: m ==slope of q: m ϭϭSince the slopes are , p q.--ϪϪ΂ ΃
  86. 86. 78 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTCheck Your Progressa. Prove the following lines parallel.Given: x y∠1 ∠4Prove: a bb. Determine whether r s.3–5
  87. 87. Glencoe Geometry 79Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3–6Equidistant means that the between twolines measured along a to thelines is always the same.Perpendiculars and DistanceDistance from a Point to a LineDraw the segment that represents the distance fromA to BP.Since the distance from a line to a point not on the line is theof the segment to the linefrom the point, extend and draw so that.Check Your Progress Draw the segment that representsthe distance from R to−−XY.• Find the distancebetween a point anda line.• Find the distancebetween parallel lines.MAIN IDEASTheorem 3.9In a plane, if two lines are each equidistant from a thirdline, then the two lines are parallel to each other.BUILD YOUR VOCABULARY (page 62)Distance Between aPoint and a Line Thedistance from a line toa point not on the lineis the length of thesegment perpendicularto the line from thepoint.On a card,describe how to find thedistance from a pointin the coordinate planeto a line that does notpass through the point.Place the card in thePerpendicular Linespocket.KEY CONCEPT
  88. 88. 80 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.3–6Distance Between LinesFind the distance between parallel lines a and b whoseequations are y = 2x + 3 and y = 2x - 3, respectively.• First, write an equation of a line p perpendicular toa and b. The slope of p is the oppositeof 2, or . Use the y-intercept of line a, (0, 3), asone of the endpoints of the perpendicular segment.y - y1 = m(x - x1) Point-slope formy - = (x - ) x1 = , y1 = ,m =y = -1_2x + 3 Simplify and add 3 toeach side.• Next, use a system of equations to determine the point ofintersection of line b and p.= Substitutefor y in the secondequation.2x + 1_2x = 3 + 3 Group like terms on eachside.x = 2.4 Divide each side 5_2.y = ϭ Substitute 2.4 for x in theequation for p.The point of intersection is .Distance BetweenParallel Lines Thedistance between twoparallel lines is thedistance between one ofthe lines and any pointon the other line.KEY CONCEPT
  89. 89. Glencoe Geometry 81Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.Page(s):Exercises:HOMEWORKASSIGNMENTThen, use the Distance Formula to determine the distancebetween (0, 3) and (2.4, 1.8).d = Distance Formula=x2 = 2.4, x1 = 0,y2 = 1.8, y1 = 3=The distance between the lines is or aboutunits.Check Your Progress Find the distance between theparallel lines a and b whose equations are y = 1_3x + 1 andy = 1_3x - 2, respectively.What is the point-slopeform of a linearequation?WRITE IT3–6
  90. 90. 82 Glencoe GeometryCopyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.STUDY GUIDEBRINGING IT ALL TOGETHERBUILD YOURVOCABULARYVOCABULARYPUZZLEMAKERC H A P T E R3Use 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:glencoe.comYou can use your completedVocabulary Builder (pages 62–63)to help you solve the puzzle.Refer to the figure at the right.1. Name all planes that are parallel to plane ABC.2. Name all segments that are parallel to FෆGෆ.In the figure, m∠5 = 100. Find the measure of each angle.3. ∠1 14. ∠35. ∠4 16. ∠77. ∠6 18. ∠2Determine the slope of the line that contains the given points.9. F(-6, 2), M(7, 9) 10. Z(1, 10), L(5, -3)3-1Parallel Lines and Transversals3-2Angles and Parallel Lines3-3Slope of Lines
  91. 91. Chapter BRINGING IT ALL TOGETHER3Glencoe Geometry 83Copyright©Glencoe/McGraw-Hill,adivisionofTheMcGraw-HillCompanies,Inc.11. Determine whether−−EF and−−−PQ are parallel, perpendicular, orneither. E(0, 4), F(2, 3), P(-3, 5), Q(1, 3)12. Write an equation in slope-intercept form of the line with slope-2 that contains (2, 5).13. Write an equation in slope-intercept form of the line thatcontains (-4, -2) and (-1, 7).Given the following information, determine which lines,if any, are parallel. State the postulate or theorem thatjustifies your answer.14. ∠1 ∠1515. ∠9 ∠1116. ∠2 ∠617. Draw the segment that representsthe distance from m to NP.Find the distance between each pair of parallel lines.18. x = 3 19. y = x + 5x = -5 y = x - 53-4Equations of lines3-5Proving Lines Parallel3-6Perpendiculars and Distance

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