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4 Congruent Triangles CHAPTER FOCUS Learn about some of the Common Core State Standards that you will explore in this chapter. Answer the preview questions. As you complete each lesson, return to these pages to check your work. What You Will Learn Preview Question Lesson 4.1: Modeling: Two-Dimensional Figures G.MG.1 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). SMP 4 The end of a shaft in a machine is a square with side length 1 inch. The end of the shaft fits into a hole in a circular disc. The disc has a radius of 3 inches. Draw a model of the disc and then determine the area of the circular face of the disc. Lesson 4.2: Proving Theorems About Triangles G.CO.10 Prove theorems about triangles. G.MG.1 Use geometric shapes, their measures, SMP 2 Explain why the acute angles of a right triangle are supplementary. and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). Lesson 4.3: Proving Triangles Congruent – SSS, SAS G.CO.10 Prove theorems about triangles. G.CO.8 Explain how the criteria for triangle 114 CHAPTER 4 Congruent Triangles D C Copyright © McGraw-Hill Education congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. SMP 7 Write a two-column proof to prove the following. B Given: ABC and CDA ¯ AB ¯ CD Prove: ABC CDA A What You Will Learn Preview Question Lesson 4.4: Proving Triangles Congruent – ASA, AAS G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.CO.10 Prove theorems about triangles. G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. YZ || ¯ QR . X is the midpoint of ¯ YQ. SMP 1 ¯ Prove that XYZ XQR. Y R X Z Q Lesson 4.5: Congruence in Right and Isosceles Triangles G.CO.10 Prove theorems about triangles. G.SRT.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. SMP 3 Jane says you can prove ABC DBC using SAS. Flynn says you can prove ABC DBC using LL. Who is correct? Explain. B Copyright © McGraw-Hill Education A C D Lesson 4.6: Triangles and Coordinate Proof G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. G.CO.10 Prove theorems about triangles. SMP 1 Prove that the triangle with vertices A(1, 6), B(12, 3), and C(2, 1) is a right triangle. CHAPTER 4 Chapter Focus 115 4.1 Modeling: Two-Dimensional Figures STANDARDS Objectives Content: G.MG.1 Practices: 1, 2, 4, 6, 8 Use with Lesson 1–6 • Use two-dimensional figures to model real-world objects and situations on and off the coordinate plane. • Solve problems involving perimeter and area. You can use two-dimensional figures and their properties to model real-world objects and to solve problems. EXAMPLE 1 Model Area Using Two-Dimensional Figures G.MG.1 A design for a medium t-shirt can be modeled as six pieces of material. a. CALCULATE ACCURATELY Find the area of the front and back of the t-shirt. Round to the nearest square centimeter. 20 cm SMP 6 Rectangle area = ( )( 1 semicircle area = ___ π( 2 area of front = 6 cm )= )2 = − cm2; π cm2; π ≈ 8 cm 12 cm 46 cm cm2. 34 cm b. CALCULATE ACCURATELY Find the area of each of the four sleeve pieces as the sum of the areas of a triangle and a rectangle. SMP 6 c. CALCULATE ACCURATELY Find the area of material for each t-shirt. SMP 6 e. PLAN A SOLUTION This design of t-shirt comes in small (S), medium (M), large (L), extra large (XL), and extra-extra large (XXL) sizes. Multiplying or dividing each dimension by 1.1 creates larger or smaller sizes. Explain how to find the amount of material needed for S and XXL sizes of t-shirt. SMP 1 116 CHAPTER 4 Congruent Triangles Copyright © McGraw-Hill Education d. COMMUNICATE PRECISELY Recall that the scale factor is the ratio of the lengths of the corresponding sides of two similar polygons. For a 5 cm square, a scale factor increases the perimeter to 60 cm. How did the area of this square change? Describe in terms of the scale factor. SMP 6 EXAMPLE 2 Model with Two-Dimensional Figures on a G.MG.1 Coordinate Plane Lake Superior has been superimposed on a coordinate grid. Each grid unit represents 27 miles. y 5 O 5 10 x a. USE A MODEL Use points with whole-number coordinates to create a polygon that approximates the outline of Lake Superior. SMP 4 b. CALCULATE ACCURATELY Divide the polygon into shapes whose area you can calculate. Find the approximate area of Lake Superior to the nearest thousand. SMP 6 c. CALCULATE ACCURATELY Using the distance formula where necessary, find the approximate length of the shoreline of Lake Superior. Explain your method. SMP 6 Copyright © McGraw-Hill Education d. USE A MODEL Atma has a motorboat with a 27 gallon fuel tank. His boat gets 15 miles per gallon. Can he travel the perimeter of Lake Superior on a full tank of gas? Show your work or justify your answer. SMP 4 e. EVALUATE REASONABLENESS Which approximation do you think is more reliable, the area or the shoreline length? Explain. SMP 8 4.1 Modeling: Two-Dimensional Figures 117 PRACTICE 1. a. USE A MODEL The model shows the dimensions of a sofa. Draw a diagram to show how to calculate the total surface area of the sofa that would be covered by a fitted G.MG.1, SMP 4 cover. Explain your technique. 26 in. 20 in. 32 in. 8 in. 30 in. 66 in. b. REASON QUANTITATIVELY How much material is needed for a fitted sofa cover? 2. a. REASON QUANTITATIVELY Miguel is planning to renovate his living room. How much finish will he need for the hardwood floor? Assume 1 L of finish covers 4.5 m2 and round to the nearest tenth. G.MG.1, SMP 2 G.MG.1, SMP 2 3.5 m 2.8 m 2.0 m 4.2 m 7.5 m 2.2 m b. CALCULATE ACCURATELY The height of the room is 2.6 m. Approximate how much paint is needed for the walls. Assume 1 L of paint covers 7.5 m2 and round to the nearest tenth. G.MG.1, SMP 6 118 CHAPTER 4 Congruent Triangles G.MG.1, SMP 8 Copyright © McGraw-Hill Education c. EVALUATE REASONABLENESS Why might Miguel adapt your answers in practice? 3. USE A MODEL A two-lane running track is made by connecting two parallel straightaways with semicircular curves on each end. Each lane of the track has width 1.1 meters and the length of each straightaway is 100 meters. G.MG.1, SMP 4 a. If the radius of the semicircle made by the inside of the 100 first lane is ______ π meters, draw a model of the track labeling the information provided. Starting from the same spot on the track, find the distance that a runner must travel to complete a full lap in each of the two lanes. Measure from the inside of each lane. b. Are the distances to run a full lap equal for each lane? If not, how could you make a lap for each runner the same distance without changing their lanes or the finish line? 4. Main Street and 1st Street are perpendicular and intersect at a traffic light. The library is on 1st street and the community center is on Main Street. The distance from the traffic light to the library is 9 miles. The length of a direct path between the library and the community center is 17 miles. A city planner wants to put a bike path alongside the streets from the library to the traffic light to the community center and back to the G.MG.1 library. a. USE A MODEL What shape best represents the bike path? Draw a model of the bike path. SMP 4 Copyright © McGraw-Hill Education b. CALCULATE ACCURATELY The bike path is 8 feet wide along the entire route. Using the perimeter of your model, find the number of square feet, to the nearest square foot, of blacktop that the city planner will have to pour to cover the entire bike path. Explain your reasoning. SMP 6 c. CALCULATE ACCURATELY Based on the perimeter of your model, find the area of the city, to the nearest tenth of a mile, that will be enclosed by the bike path. Describe your solution process. SMP 6 4.1 Modeling: Two-Dimensional Figures 119 4.2 Proving Theorems About Triangles STANDARDS Objectives Content: G.CO.10, G.MG.1 Practices: 1, 2, 3, 4, 5, 6, 8 Use with Lesson 4–2 • Prove and apply theorems about the angles of triangles. • Use theorems about the angles of triangles to model real-world situations. B One important characteristic of all triangles is presented in the following theorem. Triangle Angle-Sum Theorem The sum of the measures of the three interior angles of a triangle is 180. In the figure, m∠A + m∠B + m∠C = 180. EXAMPLE 1 A Prove the Triangle Angle-Sum Theorem C G.CO.10 a. USE TOOLS Use tracing paper to verify the Triangle Angle-Sum Theorem. Describe your method and include a sketch in the SMP 5 space provided. b. CONSTRUCT ARGUMENTS Complete the paragraph proof. SMP 3 Given: ABC. Prove: m∠1 + m∠2 + m∠3 = 180 ⟷ BC using the Postulate. ∠4 and ∠BAD form a linear pair. By the Draw AD || ¯ B by the definition of supplementary angles. m∠BAD = by the Angle Addition Postulate, so by the Substitution Property of Equality ∠4 ≈ ∠1 and ∠5 ≈ ∠3, so , and by Definition of Congruent Angles. Therefore, m∠1 + m∠2 + m∠3 = 180 by the Substitution Property of Equality. 120 CHAPTER 4 Congruent Triangles D 2 5 1 3 C , so m∠4 + m∠BAD = m∠4 + m∠2 + m∠5 = 180. By A Copyright © McGraw-Hill Education Supplement Theorem, ∠4 and ∠BAD are 4 EXAMPLE 2 B (2x + 1)° Apply the Triangle Angle-Sum Theorem a. REASON QUANTITATIVELY Use the Vertical Angles Theorem to write an algebraic expression for m∠AEB. Explain. SMP 2 59° A E (3x)° 67° C D b. REASON ABSTRACTLY Use the Triangle Angle-Sum Theorem to write and solve an equation to find the value of x. Justify each step of your solution. G.CO.10, SMP 2 c. CALCULATE ACCURATELY Use your answers to parts a and b to find m∠AEB and m∠CDE. Use properties or theorems to support each step of your solution. SMP 6 A Exterior Angle Theorem The measure of an exterior angle of a triangle equals the sum of the measures of the two remote interior angles. In the figure, m∠A + m∠B = m∠1. Copyright © McGraw-Hill Education EXAMPLE 3 Use the figure above to prove the Exterior Angle Theorem. Statements 1. m∠A + m∠B + m∠ACB = 180 2. 1 C Prove the Exterior Angle Theorem form a linear pair. B G.CO.10, SMP 3 Reasons Triangle Angle-Sum Thm. Def. of a linear pair 3. m∠1 + m∠ACB = 180 4. Substitution 5. m∠1 = m∠A + m∠B 4.2 Proving Theorems About Triangles 121 EXAMPLE 4 Apply Theorems about Triangles G.MG.1 The Flatiron Building in New York City is one of America’s oldest skyscrapers, completed in 1902. Its floorplan is approximately a right triangle. In the figure below, 5th Avenue is perpendicular to East 22nd Street, and m∠B is 10 less than 3 times m∠C. a. REASON QUANTITATIVELY Find the angle measures in the floorplan. Justify your reasoning. SMP 2 A Ea Broad wa 5t h Flatiron Building y Av en ue D C st 2 2n B dS tre e N W E S t b. Find m∠BCD in two ways. Explain each method. A Corollary 1 to Triangle Angle-Sum Theorem The acute angles of a right triangle are complementary. In the figure, ∠C is a right angle, so ∠A and ∠B are complementary. By the definition of complementary angles, m∠A + m∠B = 90. C Corollary 2 to Triangle Angle-Sum Theorem There can be at most one right or obtuse angle in a triangle. In the figure, if ∠D is a right or obtuse angle, then ∠E and ∠F are acute angles. That is, if m∠D ≥ 90, then m∠E < 90 and m∠F < 90. D F EXAMPLE 5 Prove Corollary 2 to the Triangle Angle-Sum Theorem B E G.CO.10, SMP 8 Fill in each missing statement or reason in the flow proof below. Given: In ABC, ∠A is a right or obtuse angle. m∠A ≥ 90 Prove: ∠B is acute. Given m∠A + m∠B + m∠C ≥ 90 + m∠B + m∠C a. Triangle Angle-Sum Thm. m∠C > 0 Property of Angles 180 ≥ 90 + m∠B + m∠C Substitution m∠B + m∠C > m∠B c. Addition Subtraction 90 > m∠B Substitution 122 CHAPTER 4 Congruent Triangles Copyright © McGraw-Hill Education b. The Triangle Angle-Sum Theorem leads to a useful theorem relating the angles of two triangles. Third Angles Theorem If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent. EXAMPLE 6 Prove the Third Angles Theorem G.CO.12, SMP 3 Fill in the reason for each statement in the following 2-column proof. P Given: ∠P ∠X and ∠Q ∠Y Prove: ∠R ∠Z Proof: Z Y X Q Statements R Reasons 1. ∠P ∠X, ∠Q ∠Y 1. 2. m∠P = m∠X, m∠Q = m∠Y 2. 3. m∠P + m∠Q + m∠R = 180 m∠X + m∠Y + m∠Z = 180 3. 4. m∠P + m∠Q + m∠R = m∠X + m∠Y + m∠Z 4. 5. m∠X + m∠Y + m∠R = m∠X + m∠Y + m∠Z 5. 6. m∠R = m∠Z 6. 7. ∠R ∠Z 7. PRACTICE J 1. REASON QUANTITATIVELY Find the angle measures in KLM. Justify G.CO.10, SMP 2 your calculations. K 81° a. Find m∠KML. N 67° M L Copyright © McGraw-Hill Education b. Find m∠L. P 2. REASON QUANTITATIVELY Find the angle measures in PQR . Justify your calculations. G.CO.10, SMP 2 77° 22° S a. Find m∠PRQ. 37° R Q b. Find m∠QPR. 4.2 Proving Theorems About Triangles 123 3. PLAN A SOLUTION Find the measure of the indicated angle. Show your work, and state any theorems you use. G.CO.10, SMP 1 A a. Find m∠C. C B (5x)° (2x - 10)° (x + 15)° 27° E D b. Find m∠A. 4. CONSTRUCT ARGUMENTS Prove Corollary 1 to the Triangle Angle-Sum Theorem. Given: ABC with ∠C a right angle. Prove: m∠A + m∠B = 90 G.CO.10, SMP 3 5. USE A MODEL Cassie, a real estate agent, is assessing a triangular plot of land next to a ravine. She has determined that m∠1 = 64 and m∠4 = 154. Find m∠2. Which theorem did you G.MG.1, SMP 4 use? 1 Ravine 36° 3 4 A a. Find m∠D and m∠ACB. Justify each step. D 124 CHAPTER 4 Congruent Triangles C B Copyright © McGraw-Hill Education 6. CONSTRUCT ARGUMENTS In ACD, m∠DAC = 36 and ∠D ∠ACD. G.CO.10, SMP 3 In ABC, m∠B > 18. 2 b. Give a reason why ∠B is acute. Then write and solve an inequality describing the measure of ∠B 7. REASON QUANTITATIVELY To navigate around a peninsula a ship sails from Port A at a bearing 58° west of due north and when it reaches point B, it changes course to the bearing 20° west of due south to reach Port C. The ship’s route is shown in the figure. G.MG.1, SMP 2 B 20° 58° E C D A a. Find m∠DAB, ∠DBA, and m∠BCD. Explain your reasoning. b. Find m∠BCE. State any theorems you use to determine your answer. 8. REASON QUANTITATIVELY A wall panel needs to be cut to fill a space in transition to a staircase as shown in the figure. If m∠DFE = 60, find m∠AFD and m∠FDC, the angles at which the panel should be cut. A B F G.MG.1, SMP 2 staircase Copyright © McGraw-Hill Education E D C 9. REASON QUANTITATIVELY In the two triangles ABC and DEF, ∠A ∠D and ∠C ∠F. Find the value of x if m∠B = 3x - 5 and m∠E = x + 27. Justify each step of your solution. G.CO.10, SMP 2 4.2 Proving Theorems About Triangles 125 4.3 Proving Triangles Congruent—SSS, SAS STANDARDS Objectives Content: G.CO.7, G.CO.8, G.CO.10, G.SRT.5 Practices: 1, 2, 3, 4, 6, 7 Use with Lesson 4-4 • Show that the SSS and SAS criteria for triangle congruence follow from the definition of congruence in terms of rigid motions. • Use congruence criteria for triangles to prove relationships in figures. Recall that two figures are congruent if each pair of corresponding parts is congruent. The converse is also true. If corresponding parts of two different figures are congruent, then the figures are congruent. In this lesson you will explore triangle congruence in terms of rigid motion using two postulates: SSS (side-side-side) Congruence and SAS (side-angle-side) Congruence. To show that SSS is sufficient to show triangle congruence, the Perpendicular Bisector Theorem and its converse will be used. These theorems will be explored further in a later chapter. KEY CONCEPT Perpendicular Bisector Theorem Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. EXAMPLE 1 Explore SSS Congruence EXPLORE In ABC and DEF, ¯ AB ¯ DE, ¯ BC ¯ EF, ¯ ¯ and CA FD. Use rigid motion transformations to show that ABC DEF. G.CO.8 A E B a. PLAN A SOLUTION Describe a sequence of right motion transformations that would map ¯ CA to ¯ FD as shown in the figure at the right. Label the figure with all known information. SMP 1 C F D E F C D A c. REASON ABSTRACTLY Which rigid motion transformation maps E to B? 126 CHAPTER 4 Congruent Triangles B SMP 2 Copyright © McGraw-Hill Education b. REASON ABSTRACTLY Given that ¯ EF ¯ BC, what can you conclude ¯ EB? Explain. about the relationship between FD and ¯ SMP 2 d. CONSTRUCT ARGUMENTS Explain how your observations complete the argument that ABC DEF. SMP 3 EXAMPLE 2 Use SSS to Determine Triangle Congruence G.CO.7 The triangles ABC, DEF, and GHI are placed on the coordinate grid shown to the right. a. CALCULATE ACCURATELY Use the distance formula to find the lengths of the sides of ABC and DEF. Show your work. SMP 6 (-2, 5) C 6 (-6, 4) A F (3, 4) D (1, 1) (-4, 1) B -6 y E (5, 0) O 6x H (2, -2) (-3, -2) G I (0, -5) -6 b. REASON QUANTITATIVELY Determine which sides of the two triangles are congruent. Explain your reasoning. SMP 2 c. CONSTRUCT ARGUMENTS Use the results from part b to conclude that ABC DEF. SMP 3 Copyright © McGraw-Hill Education d. USE STRUCTURE Use the distance formula to find the lengths of the sides of GHI. Is ABC GHI? Explain your reasoning. SMP 7 In Example 1, rigid motion transformations are used to show that SSS is sufficient to prove triangle congruence. A similar argument can be used to show that SAS is a valid congruence criterion. This argument uses the Angle Bisector Theorem and its converse. These theorems will be explored further in a later chapter. KEY CONCEPT Angle Bisector Theorem Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle. Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. 4.3 Proving Triangles Congruent—SSS, SAS 127 EXAMPLE 3 Explore SAS Congruence In ABC and DEF ¯ AC ¯ DF, ¯ BC ¯ EF, and ∠BCA ∠EFD. Use rigid motion transformations to show that ABC DEF. G.CO.8 A a. PLAN A SOLUTION Describe a sequence of rigid motion transformations that would map ¯ CA to ¯ FD as shown in the figure at the right. Label the figure with all known information. SMP 1 D E B F C D A E b. REASON ABSTRACTLY Given that ¯ EF ¯ BC, what can you conclude about ¯ ¯ the relationship between FD and EB ? Explain. SMP 2 B F C c. REASON ABSTRACTLY Which rigid motion transformation maps E to B and ¯ DE to ¯ AB? SMP 2 d. REASON ABSTRACTLY Since ∠BCA ∠EFD, what can be concluded about ¯ FD? SMP 2 e. REASON ABSTRACTLY What can you conclude about ∠EDF and ∠BAC? Explain. SMP 2 f. CONSTRUCT ARGUMENTS Explain how your observations complete the argument that ABC DEF. SMP 3 Determine Triangle Congruence G.CO.10, G.SRT.5 In the figure, ¯ AC ¯ AD. a. INTERPRET PROBLEMS Suppose you know ∠C ∠D. Can you prove SMP 1 that ABC ABD? Why or why not? C A B D 128 CHAPTER 4 Congruent Triangles Copyright © McGraw-Hill Education EXAMPLE 4 ⟶ b. CONSTRUCT ARGUMENTS Suppose you are given that AB bisects ∠CAD. Write a paragraph proof to show that ABC ABD. SMP 3 KEY CONCEPT SSS and SAS Congruence Postulates Complete each congruence postulate. Then mark each figure to show an example of given information that would allow you to use the postulate to prove the triangles are congruent. Postulate Example D A Side-Side-Side (SSS) Congruence If F C , E B then the triangles are congruent. Side-Angle-Side (SAS) Congruence D A If F C , E B then the triangles are congruent. PRACTICE J 1. CONSTRUCT ARGUMENTS In the figure, point P is the JL and the midpoint of ¯ KM. midpoint of ¯ M P G.CO.10, G.SRT.5, SMP 3 K L Copyright © McGraw-Hill Education a. Write a paragraph proof to show that ∠K ∠M. b. What can you conclude about JPK and LPM regarding rigid motions? 4.3 Proving Triangles Congruent—SSS, SAS 129 2. CONSTRUCT ARGUMENTS Write a two-column proof for the following. G.CO.10, G.SRT.5, SMP 3 C B Given: ¯ AB is the perpendicular bisector of ¯ CD. D A Prove: ∠C ∠D Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. INTERPRET PROBLEMS In Exercises 3– 6, explain whether there is enough information given in the figure to prove that the triangles are congruent using SAS or SSS. G.CO.10, G.SRT.5, SMP 1 3. 4. H Q R G L J T S K 5. 6. A D C 130 CHAPTER 4 Congruent Triangles Copyright © McGraw-Hill Education B 7. USE A MODEL An engineer is designing a new cell phone tower. Part of the tower is shown in the figure. The engineer makes sure that line m is parallel to AB ¯ CD. Can she prove that ABC DCB? Explain why or line n and that ¯ why not. G.CO.10, G.SRT.5, SMP 4 m n A C B D 8. REASON ABSTRACTLY For each pair of triangles, describe a sequence of rigid motions attempting to map ABC to DEF that could be used to determine whether or not the triangles are congruent. G.CO.7, SMP 2 a. D b. A D A B C E B E F F 9. CALCULATE ACCURATELY Triangles ABC and DEF are placed on the coordinate grid with vertices A(-5, 2), B(-1, -1), C(1, 3), D(-3, -2), E(1, 2), and F(3, -3). G.CO.7, SMP 6 y (1, 3) C (-5, 2) A a. Use the distance formula to find the lengths of the sides of ABC and DEF. Copyright © McGraw-Hill Education C E (1, 1) O (-3, -2) D x B (-1, -1) (3, -3) F b. Which sides of the two triangles are congruent? c. Are the two triangles congruent? Explain your reasoning. d. What can you conclude about ABC and DEF regarding rigid motions? 4.3 Proving Triangles Congruent—SSS, SAS 131 4.4 Proving Triangles Congruent—ASA, AAS STANDARDS Objectives Content: G.CO.7, G.CO.8, G.CO.10, G.SRT.5 Practices: 1, 2, 3, 4, 5, 7 Use with Lesson 4-5 • Show that the ASA criterion for triangle congruence follows from the definition of congruence in terms of rigid motions. • Use congruence criteria for triangles to prove relationships in figures. An included side is the side located between two consecutive angles of a PQ is the included side between polygon. In the triangle shown at the right, ¯ ∠P and ∠Q. If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent by the Angle-Side-Angle (ASA) Congruence Postulate. EXAMPLE 1 Explore ASA Congruence Postulate included side between ∠P and ∠Q P Q R G.CO.8 Follow these steps to show how the ASA Congruence Postulate follows from the rigid-motion definition of congruence. a. PLAN A SOLUTION In the figure at right, ∠A ∠D, ∠B ∠E, and ¯ AB ¯ DE. Mark this information on the figure, then describe a AB sequence of rigid-motion transformations that would map ¯ ¯ to DE as shown in the figure below. SMP 1 C A B F E b. CONSTRUCT ARGUMENTS Julia states that if¯ AB ¯ DE there will AB always exist a sequence of rigid-motion transformations that map ¯ DE even if the transformations are not easy to determine. Do you agree? to ¯ Explain your reasoning. SMP 3 D F E B d. CONSTRUCT ARGUMENTS What rigid transformation maps C to F? Justify your answer. SMP 3 132 CHAPTER 4 Congruent Triangles C Copyright © McGraw-Hill Education c. REASON ABSTRACTLY In the figure at right, what is the relationship between ¯ AB and ∠FEC? ¯ AB and ∠FDC? SMP 2 D A ⟶ f. USE STRUCTURE What can you say about the image of BC under this reflection? Why? SMP 7 g. USE REASONING Explain how your answer to parts e and f complete the argument that ABC DEF. SMP 3 KEY CONCEPT ASA Congruence Postulate Complete the congruence postulate. Then mark the figure to show an example of given information that would allow you to use the postulate to prove the triangles are congruent. Postulate Example Angle-Side-Angle (ASA) Congruence D A If F C , EXAMPLE 2 E B then the triangles are congruent. Use ASA to Determine Triangle Congruence G.CO.10, G.SRT.5 Pamela is studying Native American arrowheads. She has drawn a diagram to model aspects of the shape of a certain arrowhead and wants to know if AB AD. She drew a dashed line AC through the diagram and found that AC bisects ∠BAD and ∠BCD. ‾ ‾ a. PLAN A SOLUTION Mark the congruent angles on the figure. SMP 1 b. CONSTRUCT ARGUMENTS Write a two-column proof using the ASA Congruence Postulate. SMP 3 Copyright © McGraw-Hill Education A ‾ ‾ B C D Given: ¯ AC bisects ∠BAD and ∠BCD. Prove: ¯ AB ¯ AD Statements Reasons 1. 2. 3. 4. 5. 4.4 Proving Triangles Congruent—ASA, AAS 133 c. REASON ABSTRACTLY Describe a rigid motion that would map ABC to ADC. SMP 2 EXAMPLE 3 Prove the Angle-Angle-Side (AAS) Congruence Theorem G.CO.10, G.SRT.5 M Follow these steps to prove the AAS Congruence Theorem. J Given: ∠J ∠M, ∠K ∠N, and ¯ JL ¯ MP Prove: JKL MNP P L a. PLAN A SOLUTION Mark the given information on the figure. Then explain how you can prove the AAS Congruence Theorem by using one of the three congruence postulates you have SMP 1 already established. N K b. CONSTRUCT ARGUMENTS Write a two-column proof of the AAS Congruence SMP 3 Theorem. Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. KEY CONCEPT AAS Congruence Theorem Theorem Copyright © McGraw-Hill Education Complete the congruence theorem. Then mark the figure to show an example of given information that would allow you to use the theorem to prove the triangles are congruent. Example Angle-Angle-Side (AAS) Congruence A D If C , then the triangles are congruent. 134 CHAPTER 4 Congruent Triangles B F E EXAMPLE 4 Use the AAS Congruence Theorem G.CO.10, G.SRT.5, SMP 3 Althea used a kit to build a picnic table for her yard. The side view is shown in the figure. Althea made sure the tabletop is parallel to the ground and she checked that BC DC. She wants to know if she can conclude that AC EC. ‾ ‾ A B ‾ ‾ C a. CONSTRUCT ARGUMENTS Write a two-column proof using the AAS Congruence Theorem. D Given: ¯ AB || ¯ DE, ¯ BC ¯ DC E Prove: ¯ AC ¯ EC Statements Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. b. CRITIQUE REASONING Althea said she proved that ¯ AC ¯ EC using a different congruence criterion. Do you think this is possible? If so, explain how. If not, explain why not. PRACTICE 1. In the figure, ¯ JK is perpendicular to ¯ LM, and ¯ JK bisects ∠LKM. L Copyright © McGraw-Hill Education G.CO.10, G.SRT.5 a. CONSTRUCT ARGUMENTS Write a paragraph proof to show that LKJ MKJ. SMP 3 J K M b. REASON ABSTRACTLY What rigid-motion transformation maps LKJ to MKJ? SMP 2 4.4 Proving Triangles Congruent—ASA, AAS 135 2. CONSTRUCT ARGUMENTS Write a two-column proof for the following. Given: ¯ AC is parallel to ¯ BD. Point D is the midpoint CE. ∠CAD ∠DBE of ¯ G.CO.10, G.SRT.5, SMP 3 A AD ¯ BE Prove: ¯ C B D Statements E Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 3. USE TOOLS Use a compass and straightedge and the ASA Congruence Postulate to construct a triangle congruent to PQR. Show your work in the space at the right. G.SRT.5, SMP 5 P Q R INTERPRET PROBLEMS In Exercises 4 and 5, explain whether there is enough information given in the figure to prove that the triangles are congruent. If so, describe a sequence of rigid motions mapping one triangle to the other. G.CO.10, G.SRT.5, SMP 1 4. 5. F P D G E 136 CHAPTER 4 Congruent Triangles M N Q Copyright © McGraw-Hill Education H 6. USE A MODEL Dylan came to a river during a hike and he wanted AB , to estimate the distance across it. He held his walking stick, ¯ vertically on the ground at the edge of the river and sighted along the top of the stick across the river to the base of a tree, T. Then he turned, without changing the angle of his head, and sighted along the top of the stick to a rock, R, located on his side of the river. G.CO.10, G.SRT.5, SMP 4 A B T R a. Explain why ABT ABR. b. Dylan finds that it takes 27 paces for him to walk from his current location to the rock. He also knows that each of his paces is 14 inches long. Explain how he can use this information to estimate the distance across the river. 7. CRITIQUE REASONING Raj says that he can draw two triangles that have two sides and a nonincluded angle congruent and that the two triangles are congruent. G.CO.10, G.SRT.5, SMP 3 a. Raj says that there must be a SSA Congruence Theorem to justify the triangle he constructed. Do you agree? Explain. Copyright © McGraw-Hill Education b. Provide a counterexample to disprove Raj’s conjecture. c. Two triangles have two sides and a nonincluded angle congruent. Prove that if any other pair of angles of the two triangles is congruent, then the two triangles are congruent. 4.4 Proving Triangles Congruent—ASA, AAS 137 4.5 Congruence in Right and Isosceles Triangles STANDARDS Objectives Content: G.CO.10, G.SRT.5 Practices: 3, 4, 5, 6, 7, 8 Use with Extend 4–5, Lesson 4–6 • Use congruence criteria for right triangles. • Prove that base angles of an isosceles triangle are congruent. • Apply the Isosceles Triangle Theorem and its converse. In Lessons 4.3 and 4.4, you developed and applied the SSS, SAS, ASA, and AAS Congruence Criteria. Now you will investigate SSA congruence. EXAMPLE 1 Investigate SSA Congruence G.SRT.5 EXPLORE Follow these steps to use a ruler, protractor, and compass to create triangles. Compare the triangles you draw with those of other students. . Then use the protractor a. USE TOOLS In the space at the left below, draw a ray, AX , so that m∠A = 30°. Mark point B on AY so that AB = 6 cm. Finally, to draw a ray, AY −− draw BC so that BC = 4 cm. To do this, open your compass to 4 cm, place the point on B, . Label the triangle. Is there more than one way to and draw an arc that intersects AX draw ABC? If so, draw it a second way in the space at the right below. SMP 5 b. MAKE A CONJECTURE Is there an SSA Congruence Criterion? Explain. 138 CHAPTER 4 Congruent Triangles Copyright © McGraw-Hill Education c. USE TOOLS Repeat part a in the space at the right. Use the same dimensions, but this time draw the triangle so that BC = 3 cm. SMP 5 SMP 3 d. COMMUNICATE PRECISELY Describe how the situation in part c is different from the SMP 6 situation in part a. e. MAKE A CONJECTURE Is there ever a time when the SSA Congruence Criterion works? If so, when? SMP 3 The congruence criteria that you worked with in Lessons 4.3 and 4.4 (SSS, SAS, ASA, AAS) all hold for right triangles, and can be given special names using the parts of a right triangle. In addition, there is an SSA Congruence Theorem for right triangles. KEY CONCEPT Right Triangle Congruence Complete each congruence theorem. Then mark the figure to show an example of given information that would allow you to use the theorem to prove the right triangles are congruent. Theorem Example A Leg-Leg (LL) Congruence If , C B D then the triangles are congruent. F E A Hypotenuse-Angle (HA) Congruence If C B D , then the triangles are congruent. F E A Leg-Angle (LA) Congruence If C B Copyright © McGraw-Hill Education D , then the triangles are congruent. F E A Hypotenuse-Leg (HL) Congruence If C B D , then the triangles are congruent. E F 4.5 Congruence in Right and Isosceles Triangles 139 EXAMPLE 2 Use Right Triangle Congruence G.CO.10, G.SRT.5 −− −− −− −− In the figure, AC AD and AB is perpendicular to CD . A a. CONSTRUCT ARGUMENTS Mark the given information on the figure. Then write a paragraph proof that SMP 3 ABC ABD. C D B b. COMMUNICATE PRECISELY What type of triangle is CAD? What must be true about ∠C and ∠D? Why? SMP 6 vertex angle In an isosceles triangle, the congruent sides are called the legs of the triangle. The angle whose sides are the legs of the triangle is the vertex angle. The side opposite the vertex angle is the base of the triangle. The two angles formed by the base and the congruent sides are the base angles. leg leg base base angles The Key Concept box summarizes a relationship that you may have discovered in Example 2. KEY CONCEPT Isosceles Triangles Complete each example. Theorem Example Isosceles Triangle Theorem P R −− −− Example: If PQ PR , then Converse of the Isosceles Triangle Theorem If two angles of a triangles are congruent, then the sides opposite those angles are congruent. K L J Example: If ∠K ∠L, then 140 CHAPTER 4 Congruent Triangles . . Copyright © McGraw-Hill Education If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Q Prove the Isosceles Triangle Theorem EXAMPLE 3 G.CO.10, G.SRT.5 A Follow these steps to prove the Isosceles Triangle Theorem. −− −− Given: AB AC B Prove: ∠B ∠C a. USE STRUCTURE The first steps of the proof are to let P be the midpoint of −− −− BC and to draw the auxiliary line segment AP . Why are these steps justified? b. CONSTRUCT ARGUMENTS Write a two-column proof for the theorem. Statements SMP 7 SMP 3 Reasons 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. EXAMPLE 4 C Use the Isosceles Triangle Theorem G.CO.10 Julia works for a company that makes lounge chairs. As shown in the figure, the back of each chair is an isosceles triangle that can be adjusted so the person sitting on the chair can recline. Q Copyright © McGraw-Hill Education P Q R S P R S a. CONSTRUCT ARGUMENTS Suppose the chair is adjusted so that m∠Q = 50. What is m∠QRS? Write a paragraph proof to justify your answer. SMP 3 4.5 Congruence in Right and Isosceles Triangles 141 b. DESCRIBE A METHOD Julia would like a general method that she can use to find m∠QRS if she knows m∠Q. Write an expression for m∠QRS when m∠Q = x. Explain. SMP 8 c. CRITIQUE REASONING Manuel says that he can use the Exterior Angle Theorem to get the result shown in part b. Is he correct? Explain. SMP 3 PRACTICE −− 1. CONSTRUCT ARGUMENTS In the figure, BD is the perpendicular −− −− −− bisector of AC , and AB CD . Write a paragraph proof to show that AEB CED. G.CO.10, G.SRT.5, SMP 3 B A C E D M 2. CONSTRUCT ARGUMENTS Write a two-column proof of the converse G.CO.10, G.SRT.5, SMP 3 of the Isosceles Triangle Theorem. Given: ∠N ∠P −−− −− Prove: MN MP N Statements Reasons 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. Copyright © McGraw-Hill Education 1. 142 CHAPTER 4 Congruent Triangles P 3. Each of the triangles shown below is isosceles. G.CO.10, G.SRT.5 a. USE TOOLS Use a ruler to find the midpoint of each side of each triangle. Then draw the triangle formed by connecting the midpoints of each side. SMP 5 b. MAKE A CONJECTURE Look for patterns in your drawings. Make a conjecture about what you notice. SMP 3 A c. CONSTRUCT ARGUMENTS In the isosceles triangle at the −− −− right, AB AC . Use the figure to help you explain why the SMP 3 conjecture you made in part b is true. F E B 4. A boat is traveling at 25 mi/h parallel to a straight section −− of the shoreline, XY, as shown. An observer in a lighthouse L spots the boat when the angle formed by the boat, the lighthouse, and the shoreline is 35° and again when this angle is 70°. G.CO.10 Copyright © McGraw-Hill Education a. USE STRUCTURE Explain how you can prove that BCL is isosceles. SMP 7 B D C C 70° 35° X L Y Shoreline b. USE A MODEL It takes the boat 15 minutes to travel from point B to point C. When the boat is at point C, what is its distance to the lighthouse? SMP 4 5. CRITIQUE REASONING Anisa says that if two exterior angles of a triangle are G.CO.10, SMP 3 congruent, then the triangle is isosceles. Do you agree? Explain. 4.5 Congruence in Right and Isosceles Triangles 143 4.6 Triangles and Coordinate Proof STANDARDS Objectives • Use coordinates to prove simple geometric theorems algebraically. Content: G.CO.10, G.GPE.4 Practices: 1, 2, 3, 4, 6, 7, 8 Use with Lesson 4-8 • Prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length of the third side. EXAMPLE 1 Investigate a Triangle Property G.CO.10 EXPLORE A town is preparing for a 5K run. The race will start at city hall, C. The course will take runners along straight streets to the library, L, to the science museum, S, and back to city hall for the finish. A city employee has been asked to develop a different route for runners who want a shorter race. a. CALCULATE ACCURATELY The employee decides to create a shorter −− −− route by locating the midpoint X of CL and the midpoint Y of CS . The runners will go from C to X to Y and back to C. Use the Midpoint Formula −− −− −− SMP 6 to locate the midpoints of CL and CS . Draw XY on the figure. y C 8 6 4 S 2 L O 2 4 6 8 10 x 2 4 6 x b. CALCULATE ACCURATELY Use the Slope Formula and Distance Formula to −− −− compare XY to LS . What do you notice? SMP 6 c. MAKE A CONJECTURE Draw and label your own triangle, PQR, in the space at the right. Then find the midpoints, M and N, of two sides and draw the segment joining these midpoints. Use the Midpoint Formula, Slope Formula, and Distance Formula to check if the relationship you noticed in part b holds for this figure. State your observations as a conjecture. SMP 3 y 6 4 2 d. USE A MODEL What is the length of the race course from C to X to Y and back to C? Explain how you know. SMP 4 144 CHAPTER 4 Congruent Triangles O Copyright © McGraw-Hill Education −2 e. DESCRIBE A METHOD Describe how the employee could find another route that is half as long as the original route. Explain your reasoning. SMP 8 A coordinate proof uses figures in the coordinate plane and algebra to prove geometric relationships. Use can use a coordinate proof to prove the conjecture you made in the previous exploration. The Key Concept box provides suggestions for placing triangles on the coordinate plane when writing a coordinate proof. KEY CONCEPT Placing Triangles on the Coordinate Plane Step 1 Use the origin as a vertex or center of the triangle. Step 2 Place at least one side of the triangle on an axis. Step 3 Keep the triangle within the first quadrant if possible. Step 4 Use coordinates that make computations as simple as possible. EXAMPLE 2 Write a Coordinate Proof G.CO.10, G.GPE.4 P Follow these steps to write a coordinate proof for the following. −− Given: PQR, where M is the midpoint of PQ and N is the −− midpoint of PR ‾ ‾ ‾ N 1 Prove: MN is parallel to QR, and MN = ___ QR. 2 a. REASON ABSTRACTLY Place PQR on the coordinate plane and assign coordinates to the vertices of the triangle. For −− convenience, place vertex Q at the origin and place QR along the positive x-axis. Since the proof will involve midpoints, it makes sense to assign coordinates that are multiples of two. What are appropriate coordinates for R in terms of a? What are appropriate coordinates for P in terms of b and c? Label the coordinates of the vertices P, Q, and R in the figure. Copyright © McGraw-Hill Education M Q R P M N Q R SMP 2 b. CALCULATE ACCURATELY Show how to find the coordinates of M and N. SMP 6 −−− c. PLAN A SOLUTION What theorem or postulate can you use to prove that MN is −− SMP 1 parallel to QR ? 4.6 Triangles and Coordinate Proof 145 −− −−− d. CONSTRUCT ARGUMENTS Prove that MN is parallel to QR . 1 e. CONSTRUCT ARGUMENTS Prove that MN = __2 QR. EXAMPLE 3 Write a Coordinate Proof SMP 3 SMP 3 G.CO.10, G.GPE.4 Follow these steps to prove that the midpoint of the hypotenuse of a right triangle is equidistant from the vertices of the triangle. Given: JKL is a right triangle with a right angle at ∠K, and M is the midpoint of JL. ‾ J M K L Prove: JM = KM = LM a. REASON ABSTRACTLY Place the triangle above on the coordinate plane. Label the coordinates of the vertices J, K, and L. SMP 2 b. CALCULATE ACCURATELY Show how to find the coordinates SMP 6 of M. c. PLAN A SOLUTION What property, theorem, or formula can you use to complete the proof? SMP 1 Explain. d. CALCULATE ACCURATELY Show how to find each of the following distances. SMP 6 JM = LM = e. USE STRUCTURE The segment that joins K and M divides JKL into two smaller triangles. What types of triangles are these? Explain how you know. SMP 7 146 CHAPTER 4 Congruent Triangles Copyright © McGraw-Hill Education KM = Some coordinate proofs are based on specific values for the coordinates of the figures. As in the more general proofs, you can use the Distance Formula, Midpoint Formula, Slope Formula, and/or congruence criteria to write the proof. EXAMPLE 4 Write a Coordinate Proof G.CO.10, G.GPE.4 Andrew is using a coordinate plane to design a quilt. Two of the triangular patches for the quilt are shown in the figure. Andrew wants to be sure that ∠A and ∠D have the same measure. 5 A B Follow these steps to prove that ∠A ∠D. C O -5 a. PLAN A SOLUTION Describe the main steps you can use to prove that ∠A ∠D? SMP 1 5x D E -5 b. CONSTRUCT ARGUMENTS Write a paragraph proof that ∠A ∠D. y F SMP 3 PRACTICE 1. CONSTRUCT ARGUMENTS PQR is a right triangle with a right angle at ∠Q, and M is the midpoint of ¯ PR. Draw a figure and assign coordinates to prove that the area of QMR is one-half the area of PQR. Copyright © McGraw-Hill Education G.CO.10, G.GPE.4, SMP 3 4.6 Triangles and Coordinate Proof 147 2. Complete the following proof. A G.CO.10, G.GPE.4 Given: ABC is isosceles with ¯ AB ¯ AC. D is the midpoint of ¯ AB, E is ¯ AC. the midpoint of BC , and F is the midpoint of ¯ D B Prove: DEF is isosceles. F E C a. USE STRUCTURE In the space at the right, show how to place ABC on a coordinate plane. Show how to assign coordinates to the vertices of ABC. SMP 7 b. REASON ABSTRACTLY What are the coordinates of SMP 2 D, E, and F? c. CONSTRUCT ARGUMENTS Explain how to complete the proof. SMP 3 3. CRITIQUE REASONING A student was asked to prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side. He set up the figure and coordinates shown at right. Then he gave the argument shown below the figure. G.CO.10, G.GPE.4, SMP 3 a. Is the student’s proof correct? If not, explain why not and explain what the student would need to do differently to write a correct proof. R (0, 2b) M S (0, 0) N T(2a, 0) −− Let M be the midpoint of RS . Then the coordinates of M are M(0, b). −− Let N be the midpoint of ST. Then the coordinates of N are N(a, 0). −−− 0 - b b __ The slope of MN is ________ a - 0 = -a. −− 0 - 2b b __ The slope of RT is __________ 2a - 0 = - a . 148 CHAPTER 4 Congruent Triangles Copyright © McGraw-Hill Education Since the slopes are equal, b. Sharon argues that the proof in Example 2 makes an assumption −−− −− MN || RT. about the triangle and is therefore invalid. She claims that we assumed that the triangle lies on the x-axis with one of its vertices at the origin. In order to prove this theorem in general, we would have to assume that the triangle has three vertices at (a, b), (c, d), and (e, f). Is Sharon correct? 4. USE A MODEL A landscape architect is using a coordinate plane to design a triangular community garden. The fence that will surround the garden is modeled by ABC. 5 y A The architect wants to know if the any of the three angles in the fence will be congruent. Determine the answer for the architect and give a coordinate proof to justify your response. G.CO.10, G.GPE.4, SMP 4 O -5 5x C B -5 5. Complete the following proof. G.CO.10, G.GPE.4 JK, Q is the midpoint Given: JKL where P is the midpoint of ¯ KL, and R is the midpoint of ¯ JL. of ¯ Prove: The area of JKL is 4 times the area of PQR. a. USE STRUCTURE In the space at the right, show how to place JKL on a coordinate plane. Show how to assign coordinates to the vertices of JKL. SMP 7 b. CONSTRUCT ARGUMENTS Write the proof. SMP 3 Copyright © McGraw-Hill Education c. CRITIQUE REASONING Jane approaches this proof differently. She argued that the four triangles KPQ, QRL, PJR, and RQP are all congruent. Since they are all congruent, each of them is one fourth of JKL. Is Jane correct? Outline how this proof would work making use of Example 2. 6. CONSTRUCT ARGUMENTS Write the following proof. AB ¯ AC Given: ¯ AB, Y is the midpoint of ¯ AC. X is the midpoint of ¯ BY ¯ CX Prove: ¯ G.CO.10, G.GPE.4, SMP 3 A X B Y C 4.6 Triangles and Coordinate Proof 149 Performance Task Designing a Park Provide a clear solution to the problem. Show all of your work, including relevant drawings. Justify your answers. A town is building a skateboarding park within the region bounded by the pentagon shown on the map. Each unit on the grid represents 10 meters. The town wants the designer to do the following: • Create the park in the shape of an equilateral triangle. • Have one side be parallel to Jones Avenue. • Make the perimeter of the park 180 meters. 12 10 y West Street Jones Avenue A 8 Bay Road 6 4 2 O 2 4 6 8 10 12 x Part A You are told Jones Avenue makes a 120º angle with West Street. Use this information to construct the side of the park parallel to Jones Avenue from point A as accurately as possible. Explain your reasoning. Label the other endpoint as point B. Copyright © McGraw-Hill Education 150 CHAPTER 4 Congruent Triangles Part B Construct the rest of the park. Explain what you did. Label the final vertex of the park as point C. Part C Copyright © McGraw-Hill Education The mayor is impressed with your drawing, but would like to know the coordinates of points B and C. Give approximate values for each point based on your drawing. Then find the exact coordinate of point B. CHAPTER 4 Performance Task 151 Performance Task Kites and Congruence Provide a clear solution to the problem. Be sure to show all of your work, include all relevant drawings, and justify your answers. In a kite design, four triangular pieces of fabric are sewn together to form a quadrilateral. Two rods are attached to the kite so that one rod bisects the angles, as shown. B E A C D Part A The manufacturer wants to create templates for the triangular shapes that need to be AC bisects ¯ BD where cut. If m∠DCA = 50, AB = 20 in., EC = 10 in., m∠ABC = 93, and ¯ BD = 24 in., solve for all angles and side lengths for each triangle in the figure. Label the angle measures in the figure. Name all the congruent triangles shown within the design. B A E 152 CHAPTER 4 Congruent Triangles Copyright © McGraw-Hill Education D C Part B One worker notices that the rods intersect at right angles. Write a paragraph proof to prove that the rods will always intersect at right angles. Part C Copyright © McGraw-Hill Education The manufacturer will use a bolt of fabric measuring 60 inches wide and 130 yards long. Draw a model and find how many kite patterns can be cut from this bolt of fabric. CHAPTER 4 Performance Task 153 Standardized Test Practice 1. The coordinates of the vertices of a triangle are (0, 0), (a, 0), and (b, c). The area of this triangle square units. is 5. The triangles below can be shown to be congruent . by G.CO.8 G.GPE.4 2. Shelley has drawn two triangles, XYZ and TUV. She knows that ∠X ∠T and ∠Y ∠U. In order to prove that XYZ TUV using AAS, she also needs to know that . or G.CO.8 3. Archeologists have found two triangular building foundations. The diagram shows some measurements from the foundations. Which of the following additional measurements would be sufficient to prove that the two triangles are congruent? G.MG.1 6. Which of the following statements are true for the following diagram? G.SRT.5 N M E B 36˚ P 67 ft A 65 ft 41 ft O Q C D MQN MQP PQO OQN MNP OPN MPO MNO 69˚ 41 ft F ONQ OPQ QPM QPO m∠E = 36 and DE = 67 ft 7. Consider the following diagram. BC = 65 ft and DE = 67 ft G.CO.8 L M K m∠E = 36 and m∠A = 69 J 4. William draws triangle PQR. He rotates, translates, and reflects the triangle to create a new triangle with vertices S, T, and V. He finds that ∠P ∠T, ∠Q ∠S, and ∠R ∠V. From this, William determines that he can write the congruence statement . G.CO.7 154 CHAPTER 4 Congruent Triangles JKN JN means that ¯ N by or . This . 8. The HL Congruence Theorem states that in two triangles, if the are are congruent and corresponding congruent, the triangles are congruent. G.CO.10 Copyright © McGraw-Hill Education m∠A = 69 and DE = 67 ft 9. Complete the following proof. G.CO.10 AB ¯ BC Given: ¯ B Prove: ∠A ∠C A C Statement Reason ¯ AB ¯ BC ¯ Draw BD bisecting ∠ABC Definition of angle bisector ¯ BD ¯ BD SAS ∠A ∠C 10. Consider the following diagram G.SRT.5 B D C A E a. What can you conclude about ABE and EDA? Justify your answer. b. What can you conclude about BCA and DCE? Justify your answer. c. If AE = 8 and the perimeter of EDA is 22, what is the perimeter of DCE? Explain how you know. Copyright © McGraw-Hill Education 11. The vertices of ABC are A(-2, -4), B(-1, 1) and C(3, -2). The vertices of DEF are D(-3, 2), E(2, 3) and F(6, 0). G.GPE.4, G.CO.7 a. Are the triangles congruent? Justify your answer. b. The vertices of GHI are G(-2, 0), H(1, 4), and I(3, -1). Is GHI congruent to either of ABC or DEF? Justify your answer. CHAPTER 4 Standardized Test Practice 155