Gch7 l4
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This document discusses applying properties of similar triangles to find unknown segment lengths. It begins with examples showing how to use the triangle proportionality theorem and triangle angle bisector theorem to find missing side lengths. An example application involves an artist's sketch and finding lengths based on similar right triangles. The lesson concludes with a quiz to practice applying the concepts. The key ideas covered are using proportions set up with the cross product property to find unknowns, and applying theorems about similar triangles to verify or determine parallelism or lengths of segments.
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This document discusses calculating the volumes of prisms and cylinders. It provides examples of using the formulas for volume of a prism (V=lwh) and cylinder (V=πr^2h) to find volumes. It also explores how changing the dimensions of prisms and cylinders affects their volumes. For prisms, doubling the length, width, and height increases the volume by 8 times. For cylinders, doubling the radius and height quadruples the volume.
Gch10 l5
This document provides instruction on calculating the surface areas of pyramids and cones. It begins with key vocabulary terms like vertex, slant height, and altitude. Examples are then given of using the surface area formulas to find the lateral and total surface areas of regular pyramids and right cones when given dimensions like base length and slant height. The effects of changing dimensions on surface area are explored, and a manufacturing application involving cones is presented.
Gch10 l4
This document provides instruction on calculating the surface areas of prisms and cylinders. It includes definitions of key terms like lateral face and altitude. Formulas are provided for calculating the surface area of a right rectangular prism and a right cylinder. Examples show how to use the formulas to find the surface area of various prisms and cylinders. Effects of changing dimensions on surface area are also explored through examples.
Gch10 l1
This document is from a Holt Geometry textbook and covers solid geometry. It defines three-dimensional figures like prisms, pyramids, cylinders and cones. It discusses how to classify solids based on their vertices, edges and bases. It also covers nets, which are diagrams that can be folded to form three-dimensional figures, and cross sections, which are shapes revealed when a solid is cut by a plane. Examples are provided to identify solids from nets and describe cross sections. Food applications involving cutting cheese and watermelon are also discussed.
Gch8 l4
This document is from a Holt Geometry textbook and covers angles of elevation and depression. It includes examples of classifying angles as elevation or depression, using tangent ratios to find distances given an elevation or depression angle, and multi-step word problems combining these concepts. Warm-up questions, vocabulary, lesson objectives, examples with step-by-step solutions, and a lesson quiz are provided.
Gch8 l3
This document discusses solving right triangles using trigonometric ratios. It includes examples of identifying angles from trigonometric ratios, calculating angle measures from ratios using inverse functions, solving right triangles when given side lengths or an angle measure, and solving triangles in the coordinate plane. One example problem solves for the angle of a road given its grade in percentage form.
Gch5 l7
This document is a lesson on the Pythagorean theorem from a Holt Geometry textbook. It begins with examples of using the Pythagorean theorem to solve for missing sides of right triangles. It then discusses how to identify if a set of three sides forms a Pythagorean triple. The lesson continues with examples of using the converse of the Pythagorean theorem and Pythagorean inequalities to classify triangles as right, acute, or obtuse based on their side lengths. It concludes with a two-part lesson quiz reviewing the material.
Gch5 l6
This document discusses using inequalities to compare angles and side lengths in two triangles. It begins with examples that use the hinge theorem and its converse to determine relationships between angles and sides. Application examples are provided, including comparing distances traveled from school. Proofs are presented to demonstrate triangle relationships using statements and reasons. The document concludes with a lesson quiz to assess understanding.
Gch5 l5
This document provides a lesson on indirect proofs and inequalities in one triangle from Holt Geometry. It includes examples of writing indirect proofs, ordering triangle side lengths and angle measures, applying the triangle inequality theorem, finding possible side lengths of a triangle, and using triangle inequalities in a travel application problem. Practice problems are provided to check understanding. The key concepts covered are indirect proof, triangle inequalities, and relating triangle side lengths to angle measures.
Gch04 l5
This document discusses different methods for proving triangles congruent:
- ASA (Angle-Side-Angle) congruence can be used if two angles and the included side are congruent.
- AAS (Angle-Angle-Side) congruence can be used if two angles of one triangle are congruent to two angles of another triangle and one side included between the angles is congruent.
- HL (Hypotenuse-Leg) congruence can be used if two right triangles have one leg and the hypotenuse congruent.
Examples are provided to demonstrate applications of each method of triangle congruence.
Gch04 l4
This document discusses triangle congruence using the Side-Side-Side (SSS) and Side-Angle-Side (SAS) postulates. It provides examples of using SSS and SAS to prove that two triangles are congruent. Key concepts covered include triangle rigidity, included angles, and applying SSS and SAS to solve problems and construct congruent triangles. Worked examples demonstrate using SSS and SAS, along with step-by-step proofs of triangle congruence.
Gch04 l3
The document is from a geometry textbook and covers congruent triangles. It defines congruent triangles as those that are the same size and shape, with corresponding angles and sides being equal. Examples are provided of naming corresponding parts of congruent triangles and using properties of congruent triangles to prove other triangles are congruent or find missing angle measures. An engineering application example proves two triangles congruent based on bisecting lines and equal corresponding parts.
Gch04 l2
This document is from a geometry textbook and covers angle relationships in triangles. It begins with examples of finding missing angle measures using the Triangle Sum Theorem, Linear Pair Theorem, and exterior angle theorem. It then discusses properties of interior and exterior angles and applying the Third Angle Theorem. The document includes examples, practice problems, and key vocabulary terms like interior, exterior, and remote interior angles. It provides instruction and practice for finding missing angle measures in different types of triangles.
Gch6 l3
The document discusses conditions for identifying whether a quadrilateral is a parallelogram. It provides examples of showing quadrilateral figures are parallelograms using different criteria such as having two pairs of parallel opposite sides, two pairs of congruent opposite sides, or diagonals bisecting each other. Theorems are presented for determining parallelograms based on angles or side lengths. Students are given practice identifying parallelograms and justifying their reasoning.
Gch9 l1
This document discusses formulas for finding measurements of triangles, quadrilaterals, and other geometric shapes. It includes examples of using formulas to find heights, bases, areas, and perimeters given certain measurements. Formulas are provided for triangles, parallelograms, rectangles, squares, trapezoids, rhombuses, kites, and applying these formulas to problems involving geometric grids. Lesson objectives and examples with step-by-step solutions demonstrate how to use the formulas to calculate missing values.
Gch9 l2
The document discusses formulas for circles and regular polygons. It defines key terms like radius, diameter, circumference, area, apothem, and central angle. Examples are provided for calculating circle measurements like circumference and area using formulas like C=2πr and A=πr^2. A formula is developed for the area of a regular polygon by dividing it into congruent triangles: A=(1/2)aps, where a is the apothem, p is the perimeter, and s is the side length. Worked examples demonstrate calculating measurements for circles and finding areas of regular polygons.
Gch9 l3
This document discusses finding the areas of composite figures, which are made up of simple shapes. It provides examples of adding or subtracting the areas of individual shapes to find the total area of composite figures. Estimating the area of irregular shapes using composite figures is also demonstrated. The key steps are to divide composite figures into simple shapes, calculate the area of each, and then add or subtract areas as needed using the Area Addition Postulate.
Gch9 l4
This document contains a geometry lesson on calculating the perimeters and areas of polygons in the coordinate plane. It includes examples of estimating areas of irregular shapes using grids, finding perimeters and areas of parallelograms, and subtracting areas of additional shapes from a bounding rectangle. One example shows two composite figures that appear to have different areas but are proved to have the same area through calculation. The lesson concludes with a quiz to assess understanding of finding perimeters and areas of polygons in the coordinate plane.
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Chapter 2
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Chapter 3
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2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
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- 1. Holt Geometry 7-4 Applying Properties of Similar Triangles7-4 Applying Properties of Similar Triangles Holt Geometry Warm UpWarm Up Lesson PresentationLesson Presentation Lesson QuizLesson Quiz
- 2. Holt Geometry 7-4 Applying Properties of Similar Triangles Warm Up Solve each proportion. 1. 2. 3. 4. AB = 16 QR = 10.5 x = 21 y = 8
- 3. Holt Geometry 7-4 Applying Properties of Similar Triangles Use properties of similar triangles to find segment lengths. Apply proportionality and triangle angle bisector theorems. Objectives
- 4. Holt Geometry 7-4 Applying Properties of Similar Triangles Artists use mathematical techniques to make two- dimensional paintings appear three-dimensional. The invention of perspective was based on the observation that far away objects look smaller and closer objects look larger. Mathematical theorems like the Triangle Proportionality Theorem are important in making perspective drawings.
- 5. Holt Geometry 7-4 Applying Properties of Similar Triangles
- 6. Holt Geometry 7-4 Applying Properties of Similar Triangles Example 1: Finding the Length of a Segment Find US. Substitute 14 for RU, 4 for VT, and 10 for RV. Cross Products Prop.US(10) = 56 Divide both sides by 10. It is given that , so by the Triangle Proportionality Theorem.
- 7. Holt Geometry 7-4 Applying Properties of Similar Triangles Check It Out! Example 1 Find PN. Substitute in the given values. Cross Products Prop.2PN = 15 PN = 7.5 Divide both sides by 2. Use the Triangle Proportionality Theorem.
- 8. Holt Geometry 7-4 Applying Properties of Similar Triangles
- 9. Holt Geometry 7-4 Applying Properties of Similar Triangles Example 2: Verifying Segments are Parallel Verify that . Since , by the Converse of the Triangle Proportionality Theorem.
- 10. Holt Geometry 7-4 Applying Properties of Similar Triangles Check It Out! Example 2 AC = 36 cm, and BC = 27 cm. Verify that . Since , by the Converse of the Triangle Proportionality Theorem.
- 11. Holt Geometry 7-4 Applying Properties of Similar Triangles
- 12. Holt Geometry 7-4 Applying Properties of Similar Triangles Example 3: Art Application Suppose that an artist decided to make a larger sketch of the trees. In the figure, if AB = 4.5 in., BC = 2.6 in., CD = 4.1 in., and KL = 4.9 in., find LM and MN to the nearest tenth of an inch.
- 13. Holt Geometry 7-4 Applying Properties of Similar Triangles Example 3 Continued Given 2-Trans. Proportionality Corollary Substitute 4.9 for KL, 4.5 for AB, and 2.6 for BC. Cross Products Prop.4.5(LM) = 4.9(2.6) Divide both sides by 4.5.LM ≈ 2.8 in.
- 14. Holt Geometry 7-4 Applying Properties of Similar Triangles Example 3 Continued 2-Trans. Proportionality Corollary Substitute 4.9 for KL, 4.5 for AB, and 4.1 for CD. Cross Products Prop.4.5(MN) = 4.9(4.1) Divide both sides by 4.5.MN ≈ 4.5 in.
- 15. Holt Geometry 7-4 Applying Properties of Similar Triangles Check It Out! Example 3 Use the diagram to find LM and MN to the nearest tenth.
- 16. Holt Geometry 7-4 Applying Properties of Similar Triangles Given Check It Out! Example 3 Continued 2-Trans. Proportionality Corollary Substitute 2.6 for KL, 2.4 for AB, and 1.4 for BC. Cross Products Prop.2.4(LM) = 1.4(2.6) Divide both sides by 2.4.LM ≈ 1.5 cm
- 17. Holt Geometry 7-4 Applying Properties of Similar Triangles Check It Out! Example 3 Continued 2-Trans. Proportionality Corollary Substitute 2.6 for KL, 2.4 for AB, and 2.2 for CD. Cross Products Prop.2.4(MN) = 2.2(2.6) Divide both sides by 2.4.MN ≈ 2.4 cm
- 18. Holt Geometry 7-4 Applying Properties of Similar Triangles The previous theorems and corollary lead to the following conclusion.
- 19. Holt Geometry 7-4 Applying Properties of Similar Triangles Example 4: Using the Triangle Angle Bisector Theorem Find PS and SR. Substitute the given values. Cross Products Property Distributive Property by the ∆ ∠ Bisector Theorem. 40(x – 2) = 32(x + 5) 40x – 80 = 32x + 160
- 20. Holt Geometry 7-4 Applying Properties of Similar Triangles Example 4 Continued Simplify. Divide both sides by 8. Substitute 30 for x. 40x – 80 = 32x + 160 8x = 240 x = 30 PS = x – 2 SR = x + 5 = 30 – 2 = 28 = 30 + 5 = 35
- 21. Holt Geometry 7-4 Applying Properties of Similar Triangles Check It Out! Example 4 Find AC and DC. Substitute in given values. Cross Products Theorem So DC = 9 and AC = 16. Simplify. by the ∆ ∠ Bisector Theorem. 4y = 4.5y – 9 –0.5y = –9 Divide both sides by –0.5.y = 18
- 22. Holt Geometry 7-4 Applying Properties of Similar Triangles Find the length of each segment. 1. 2. Lesson Quiz: Part I SR = 25, ST = 15
- 23. Holt Geometry 7-4 Applying Properties of Similar Triangles Lesson Quiz: Part II 3. Verify that BE and CD are parallel. Since , by the Converse of the ∆ Proportionality Thm.