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Topic 4 Congruent Triangles TOPIC OVERVIEW VOCABULARY 4-1 Congruent Figures 4-2 Triangle Congruence by SSS and SAS 4-3 Triangle Congruence by ASA and AAS 4-4 Using Corresponding Parts of Congruent Triangles 4-5 Isosceles and Equilateral Triangles 4-6 Congruence in Right Triangles 4-7 Congruence in Overlapping Triangles DIGITAL APPS English/Spanish Vocabulary Audio Online: EnglishSpanish base of an isosceles triangle, p. 168 base de un triángulo isósceles base angles of an isosceles ángulos de la base de un triangle, p. 168 triángulo isósceles congruent polygons, p. 148 polígonos congruentes corollary, p. 168corolario hypotenuse, p. 174hipotenusa legs of an isosceles triangle, p. 168 catetos de un triángulo isósceles legs of a right triangle, p. 174 catetos de un triángulo rectángulo vertex angle of an isosceles ángulo del vértice de un triángulo triangle, p. 174isósceles PRINT and eBook Access Your Homework . . . Online homework You can do all of your homework online with built-in examples and “Show Me How” support! When you log in to your account, you’ll see the homework your teacher has assigned you. Your Digital Resources PearsonTEXAS.com Homework Tutor app Do your homework anywhere! You can access the Practice and Application Exercises, as well as Virtual Nerd tutorials, with this Homework Tutor app, available on any mobile device. STUDENT TEXT AND Homework Helper Access the Practice and Application Exercises that you are assigned for homework in the Student Text and Homework Helper, which is also available as an electronic book. 146 Topic 4 Congruent Triangles 3--Act Math Check It Out! Maybe you’ve played this game before: you draw a picture. Then you try to get a classmate to draw the same picture by giving step-by-step directions but without showing your drawing. Try it with a classmate. Draw a map of a room in your house or a place in your town. Then give directions to a classmate to draw the map that you drew. Compare your drawings. How similar are they? Think about this as you watch this 3-Act Math video. Scan page to see a video for this 3-Act Math Task. If You Need Help . . . Vocabulary Online You’ll find definitions of math terms in both English and Spanish. All of the terms have audio support. Learning Animations You can also access all of the stepped-out learning animations that you studied in class. Interactive Math tools These interactive math tools give you opportunities to explore in greater depth key concepts to help build understanding. Interactive exploration You’ll have access to a robust assortment of interactive explorations, including interactive concept explorations, dynamic activitites, and topiclevel exploration activities. Student Companion Refer to your notes and solutions in your Student Companion. Remember that your Student Companion is also available as an ACTIVebook accessible on any digital device. Virtual Nerd Not sure how to do some of the practice exercises? Check out the Virtual Nerd videos for stepped-out, multi-level instructional support. PearsonTEXAS.com 147 4-1 Congruent Figures TEKS FOCUS VOCABULARY TEKS (6)(C) Apply the definition of congruence, in terms of rigid transformations, to identify congruent figures and their corresponding sides and angles. TEKS (1)(B) Use a problem-solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem-solving process and the reasonableness of the solution. •Congruent polygons – polygons that have congruent corresponding sides and angles •Formulate – create with careful effort and purpose. You can formulate a plan or strategy to solve a problem. •Reasonableness – the quality of being within the realm of common sense or sound reasoning. The reasonableness of a solution is whether or not the solution makes sense. •Strategy – a plan or method for solving a problem Additional TEKS (1)(F), (1)(G) ESSENTIAL UNDERSTANDING You can determine whether two figures are congruent by comparing their corresponding sides and angles. Key Concept Congruent Figures Definition Example Congruent polygons have congruent corresponding sides and angles. When you name congruent polygons, you must list corresponding vertices in the same order. A B F E D C G H AB ≅ EF BC ≅ FG CD ≅ GH DA ≅ HE ∠A ≅ ∠E ∠B ≅ ∠F ∠C ≅ ∠G ∠D ≅ ∠H ABCD ≅ EFGH hsm11gmse_0401_t02405 Theorem 4-1 Third Angles Theorem Theorem If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent. If . . . ∠A ≅ ∠D and ∠B ≅ ∠E D A B Then . . . ∠C ≅ ∠F C E F You will prove Theorem 4-1 in Exercise 37. hsm11gmse_0401_t02416 148 Lesson 4-1 Congruent Figures Problem 1 Finding Congruent Sides and Angles How do you know which sides and angles correspond? The congruence statement HIJK ≅ LMNO tells you which parts correspond. K If HIJK ≅ LMNO, what are the congruent corresponding parts? Sides: HI ≅ LM Angles: ∠H ≅ ∠L IJ ≅ MN JK ≅ NO KH ≅ OL ∠I ≅ ∠M ∠J ≅ ∠N ∠K ≅ ∠O L J O H N I Problem 2 M TEKS Process Standard (1)(B) hsm11gmse_0401_t02407 Using Congruent Sides and Angles You know two angle measures in △ABC. How can they help? In the congruent triangles, ∠D corresponds to ∠A, so you know that ∠D ≅ ∠A. You can find m∠D by first finding m∠A. Multiple Choice The wings of an SR-71 Blackbird aircraft suggest congruent triangles. What is mjD? 30 105 75 150 Use the Triangle AngleSum Theorem to write an equation involving m∠A. Solve for m∠A. ∠A and ∠D are corresponding angles of congruent triangles, so ∠A ≅ ∠D. m∠A + 30 + 75 = 180 m∠A + 105 = 180 m∠A = 75 m∠A = m∠D = 75 The correct answer is B. Problem 3 TEKS Process Standard (1)(G) Finding Congruent Triangles How do you determine whether two triangles are congruent? Compare each pair of corresponding parts. If all six pairs are congruent, then the triangles are congruent. Are the triangles congruent? Justify your answer. AB ≅ ED Given BC ≅ DC BC = 4 = DC AC ≅ EC AC = 6 = EC ∠A ≅ ∠E, ∠B ≅ ∠D Given ∠BCA ≅ ∠DCE Vertical angles are congruent. △ABC ≅ △EDC by the definition of congruent triangles. B A 4 6 E 6 C 4 D hsm11gmse_0401_t02409 PearsonTEXAS.com 149 Problem 4 Proof Proving Triangles Congruent Given: LM ≅ LO, MN ≅ ON, ∠M ≅ ∠O, ∠MLN ≅ ∠OLN HO ME RK O You know four pairs of congruent parts. What else do you need to prove the triangles congruent? You need a third pair of congruent sides and a third pair of congruent angles. NLINE WO M L N Prove: △LMN ≅ △LON O Statements Reasons 1) LM ≅ LO, MN ≅ ON 1) Given 2) LN ≅ LN 2) Reflexive Property of ≅ 3) ∠M ≅ ∠O, ∠MLN ≅ ∠OLN 3) Given 4) ∠MNL ≅ ∠ONL 4) Third Angles Theorem 5) △LMN ≅ △LON 5) Definition of ≅ triangles hsm11gmse_0401_t02421 PRACTICE and APPLICATION EXERCISES For additional support when completing your homework, go to PearsonTEXAS.com. Scan page for a Virtual Nerd™ tutorial video. 1. Apply Mathematics (1)(A) Builders use the king post truss (below left) for the top of a simple structure. In this truss, △ABC ≅ △ABD. List the congruent corresponding sides and angles. A H E G J C D B I F king post truss attic frame truss 2. The attic frame truss (above right) provides open space in the center for storage. In this truss, △EFG ≅ △HIJ. List the congruent corresponding sides and angles. △LMC ≅ △BJK. Complete the congruence statements. J M 3. LC ≅ ? 4.KJ ≅ ? hsm11gmse_0401_t02427 5. ∠K ≅ ? 7. △CML ≅ ? 6.∠M ≅ ? 8.△KBJ ≅ ? L C K B POLY @ SIDE. List each of the following. 9. four pairs of congruent sides 10.four pairs of congruent angles At an archeological site, the remains of two ancient step pyramids are congruent. If ABCD @ EFGH, find each of the following. (Diagrams arehsm11gmse_0401_t02429 not to scale.) 11.AD 13.m∠GHE 15.EF 17.m∠DCB 12.GH 14.m∠BAD 16.BC 18.m∠EFG B 45 ft 128 A 150 Lesson 4-1 Congruent Figures F C 45 ft 52 D HSM11GMSE_0401_a02282 2nd pass 11-19-08 Durke E 52 280 ft G 128 335 ft H HSM11GMSE_0401_a02283 2nd pass 11-19-08 Durke Explain Mathematical Ideas (1)(G) For Exercises 19 and 20, can you conclude that the triangles are congruent? Justify your answers. 19.△TRK and △TUK 20.△SPQ and △TUV T 7 S R U V 5 Q 4 P 8 T 6 7 U K 21.Given: AB } DC, ∠B ≅ ∠D, Proof AB ≅ DC, BC ≅ AD Prove: △ABC ≅ △CDA hsm11gmse_0401_t02432 B C hsm11gmse_0401_t02433 A D 22.Evaluate Reasonableness (1)(B) Randall says he can use the information in the figure to prove △BCD ≅ △DAB. Is he correct? Explain. C D hsm11gmse_0401_t02434 B Connect Mathematical Ideas (1)(F) △ABC @ △DEF. Find the measures of the given angles or the lengths of the given sides. A 23.m∠A = x + 10, m∠D = 2x 24.m∠B = 3y, m∠E = 6y - 12 25.BC = 3z + 2, EF = z + 6 26. AC = 7a + 5, DF = 5a hsm11gmse_0401_t02436 +9 27.If △DEF ≅ △LMN, which of the following must be a correct congruence statement? A.DE ≅ LN C.∠N ≅ ∠F B.FE ≅ NL D.∠M ≅ ∠F E L M D F N Connect Mathematical Ideas (1)(F) Find the values of the variables. 28. C 29. M D 3x A 45 4 in. B L 2t in. ABC KLM A 6x 30 C hsm11gmse_0401_t02435 K 30.Complete in two different ways: △JLM ≅ ? . hsm11gmse_0401_t02438 B ACD ACB Z M J N hsm11gmse_0401_t02439 L R hsm11gmse_0401_t02440 PearsonTEXAS.com 151 31.Given: AB # AD, BC # CD, AB ≅ CD, AD ≅ CB, AB } CD A B D C Proof Prove: △ABD ≅ △CDB 32.Analyze Mathematical Relationships (1)(F) Write a congruence statement for two triangles. List the congruent sides and angles. 33.Given: PR } TQ, PR ≅ TQ, PS ≅ QS, PQ bisects RT P hsm11gmse_0401_t02442 Proof Prove: △PRS ≅ △QTS 34.Apply Mathematics (1)(A) The 225 cards in Tracy’s sports card collection are rectangles of three different sizes. How could Tracy quickly sort the cards? T S 35.Connect Mathematical Ideas (1)(F) The vertices of △GHJ are G(-2, -1), H(-2, 3), and J(1, 3), and △KLM ≅ △GHJ. If L and M have coordinates L(3, -3) and M(6, -3), how many pairs of coordinates are possible for K? Find one such pair. R Q 36.a. How many quadrilaterals (convex and concave) with different shapes or sizes hsm11gmse_0401_t02441 can you make on a three-by-three geoboard? Sketch them. One is shown at the right. b. How many quadrilaterals of each type are there? 37.Given: ∠A ≅ ∠D, ∠B ≅ ∠E Proof Prove: ∠C ≅ ∠F B TEXAS Test Practice D A C E F hsm11gmse_0401_t05007 38.△HLN ≅ △GST , m∠H = 66, and m∠S = 42. What is m∠T ? 39.The measure of one angle in a triangle is 80. The other two angles are congruent. What is the measure of each? 40.What is the number of feet in the perimeter of a square with side length 7 ft? 152 Lesson 4-1 Congruent Figures hsm11gmse_0401_t02443 4-2 Triangle Congruence by SSS and SAS TEKS FOCUS VOCABULARY •Number sense – the understanding TEKS (6)(B) Prove two triangles are congruent by applying the Side-AngleSide, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions. of what numbers mean and how they are related TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. Additional TEKS (1)(E), (1)(F), (1)(G), (5)(A), (5)(C) ESSENTIAL UNDERSTANDING You can prove that two triangles are congruent without having to show that all corresponding sides and angles are congruent. In this lesson, you will prove triangles congruent by using (1) three pairs of corresponding sides and (2) two pairs of corresponding sides and one pair of corresponding angles. Postulate 4-1 Side-Side-Side (SSS) Postulate Postulate If the three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. If . . . AB ≅ DE, BC ≅ EF , AC ≅ DF B E A D C Then . . . △ABC ≅ △DEF F Postulate 4-2 Side-Angle-Side (SAS) Postulate Postulate If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent. If . . . hsm11gmse_0402_t05017 AB ≅ DE, ∠A ≅ ∠D, AC ≅ DF B Then . . . △ABC ≅ △DEF E A D C F hsm11gmse_0402_t05020 PearsonTEXAS.com 153 Problem 1 TEKS Process Standard (1)(C) Building Triangles A Select a tool to build a triangle with sides of lengths 3 in., 5 in., and 6 in. Compare your triangle with your classmates’ triangles. What are corresponding sides? Corresponding sides are the sides that match and are in the same position. Corresponding sides of congruent triangles have the same length. To build a triangle with the given lengths, you might select a real object, such as a straw. Cut the straw into pieces of the correct lengths. Thread some string through the three pieces of straw, in any order, as shown. Bring the ends of the string together and tie them to hold your triangle in place. B Make a conjecture about two triangles in which three sides of one triangle are congruent to three sides of the other triangle. Both triangles have the same size and shape. Each triangle fits exactly on top of the other triangle. Conjecture: Triangles with congruent corresponding sides are congruent. Problem 2 TEKS Process Standard (1)(E) Proof Using SSS You have two pairs of congruent sides. What else do you need? You need a third pair of congruent corresponding sides. Notice that the triangles share a common side, LN. N L M Given: LM ≅ NP, LP ≅ NM Prove: △LMN ≅ △NPL LM ≅ NP LN ≅ LN LP ≅ NM Given Reflexive Prop. of ≅ Given △LMN ≅ △NPL SSS 154 P Lesson 4-2 Triangle Congruence by SSS and SAS hsm11gmls_0402_t05019 Problem 3 Using SAS Do you need another pair of congruent sides? Look at the diagram. The triangles share DF. So you already have two pairs of congruent sides. E F What other information do you need to prove △DEF @ △FGD by SAS? Explain. The diagram shows that EF ≅ GD. Also, DF ≅ DF by the Reflexive Property of Congruence. To prove that △DEF ≅ △FGD by SAS, you must have congruent included angles. You need to know that ∠EFD ≅ ∠GDF. D G Problem 4 Identifying Congruent Triangles NLINE HO ME RK O What should you look for first, sides or angles? Start with sides. If you have three pairs of congruent sides, use SSS. If you have two pairs of congruent sides, look for a pair of congruent included angles. WO hsm11gmse_0402_t05022 Would you use SSS or SAS to prove the triangles congruent? If there is not enough information to prove the triangles congruent by SSS or SAS, write not enough information. Explain your answer. A B Use SAS because two pairs of corresponding sides and their included angles are congruent. C hsm11gmse_0402_t05025 There is not enough information; two pairs of corresponding sides are congruent, but one of the angles is not the included angle. D hsm11gmse_0402_t05026 Use SSS because three pairs of corresponding sides are congruent. Use SSS or SAS because all three pairs of corresponding sides and a pair of included angles (the vertical angles) are congruent. hsm11gmse_0402_t05027 hsm11gmse_0402_t05028 PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. 1. Use Representations to Communicate Mathematical Ideas (1)(E) Copy the flow chart and complete the proof. For additional support when completing your homework, go to PearsonTEXAS.com. Given: JK ≅ LM, JM ≅ LK JK ≅ LM K JM ≅ LK Given a. c. L J Prove: △JKM ≅ △LMK M KM ≅ KM b. ≅ d. hsm11gmse_0402_t02639.ai SSS PearsonTEXAS.com hsm11gmse_0402_t02640.ai 155 2. Select Tools to Solve Problems (1)(C) Consider the following conjecture. If two triangles have the same perimeter, then the triangles are congruent. a.Select a real object that you can use to test the conjecture. Explain your choice. b.Is the conjecture true? If not, make a new conjecture based on your results. Explain your reasoning. 3. Explain Mathematical Ideas (1)(G) At least how many triangle measurements must you know in order to guarantee that all triangles built with those measurements will be congruent? Explain your reasoning. 4. Given: IE ≅ GH, EF ≅ HF, 5.Given: WZ ≅ ZS ≅ SD ≅ DW Proof Proof Prove: △WZD ≅ △SDZ F is the midpoint of GI Prove: △EFI ≅ △HFG W G E Z F D H I S What other information, if any, do you need to prove the two triangles congruent by SAS? Explain. 6. hsm11gmse_0402_t02641.ai G N L T 7. hsm11gmse_0402_t02642.ai U T W R M Q V S 8. Evaluate Reasonableness (1)(B) You and a friend are cutting triangles out of felt for an art project. You want all the triangles to be congruent. Your friend tells you that each triangle should have two 5-in.hsm11gmse_0402_t02644.ai sides and a 40° angle. If you follow hsm11gmse_0402_t02643.ai this rule, will all your felt triangles be congruent? Explain. Can you prove the triangles congruent? If so, write the congruence statement and name the postulate you would use. If not, write not enough information and tell what other information you would need. 9. A 10. G N R T Y H W K P D 11. J E S F V T hsm11gmse_0402_t02651.ai hsm11gmse_0402_t02650.ai hsm11gmse_0402_t02649.ai 156 Lesson 4-2 Triangle Congruence by SSS and SAS 12.Use Representations to Communicate Mathematical Ideas (1)(E) Sierpinski’s triangle is a famous geometric pattern. To draw Sierpinski’s triangle, start with a single triangle and connect the midpoints of the sides to draw a smaller triangle. If you repeat this pattern over and over, you will form a figure like the one shown. This particular figure started with an isosceles triangle. Are the triangles outlined in red congruent? Explain. 13.Create Representations to Communicate Mathematical Ideas (1)(E) Use a straightedge to draw any triangle JKL. Then construct △MNP ≅ △JKL using the given postulate. a.SSS b.SAS 14.Analyze Mathematical Relationships (1)(F) Suppose GH ≅ JK , HI ≅ KL, and ∠I ≅ ∠L. Is △GHI congruent to △JKL? Explain. 15.Given: FG } KL, FG ≅ KL Proof Prove: △FGK ≅ △KLF F G 16.Given: AB # CM, AB # DB, CM ≅ DB, Proof M is the midpoint of AB. Prove: △AMC ≅ △MBD D L B C K M A hsm11gmse_0402_t02654.ai TEXAS Test Practice hsm11gmse_0402_t02655.ai Y 17.What additional information do you need to prove that △VWY ≅ △VWZ by SAS? A.YW ≅ ZW C.∠Y ≅ ∠Z B.∠WVY ≅ ∠WVZ D.VZ ≅ VY V 18.The measures of two angles of a triangle are 43 and 38. What is the measure of the third angle? F. 9 G.81 H.99 W Z J.100 hsm11gmse_0402_t02658.ai 19.Which method would you use to find the inverse of a conditional statement? A.Negate the hypothesis only. C.Negate the conclusion only. B.Switch the hypothesis and D.Negate both the hypothesis and the conclusion. the conclusion. PearsonTEXAS.com 157 4-3 Triangle Congruence by ASA and AAS TEKS FOCUS VOCABULARY •Analyze – closely examine objects, ideas, or relationships to TEKS (6)(B) Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, SideSide-Side, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions. learn more about their nature. TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. Additional TEKS (1)(A), (1)(D), (1)(G) ESSENTIAL UNDERSTANDING You can prove that two triangles are congruent without having to show that all corresponding parts are congruent. In this lesson, you will prove triangles congruent by using one pair of corresponding sides and two pairs of corresponding angles. Postulate 4-3 Angle-Side-Angle (ASA) Postulate Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. If . . . ∠A ≅ ∠D, AC ≅ DF , ∠C ≅ ∠F B E A C D Then . . . △ABC ≅ △DEF F Theorem 4-2 Angle-Angle-Side (AAS) Theorem hsm11gmse_0403_t05033.ai Theorem If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent. If . . . ∠A ≅ ∠D, ∠B ≅ ∠E, AC ≅ DF B F C A D Then . . . △ABC ≅ △DEF E You will prove Theorem 4-2 in Exercise 15. hsm11gmse_0403_t05038.ai 158 Lesson 4-3 Triangle Congruence by ASA and AAS Problem 1 TEKS Process Standard (1)(F) Using ASA O Which two triangles are congruent by ASA? Explain. W U From the diagram you know • ∠U ≅ ∠E ≅ ∠T • ∠V ≅ ∠O ≅ ∠W • UV ≅ EO ≅ AW To use ASA, you need two pairs of congruent angles and a pair of included congruent sides. N E S V T A You already have pairs of congruent angles. So, identify the included side for each triangle and see whether it has a congruence marking. hsm11gmse_0403_t05036.ai In △SUV, UV is included between ∠U and ∠V and has a congruence marking. In △NEO, EO is included between ∠E and ∠O and has a congruence marking. In △ATW, TW is included between ∠T and ∠W but does not have a congruence marking. Since ∠U ≅ ∠E, UV ≅ EO, and ∠V ≅ ∠O, △SUV ≅ △NEO. Problem 2 TEKS Process Standard (1)(A) Proof Writing a Proof Using ASA Recreation Members of a teen organization are building a miniature golf course at your town’s youth center. The design plan calls for the first hole to have two congruent triangular bumpers. Prove that the bumpers on the first hole, shown at the right, meet the conditions of the plan. Can you use a plan similar to the plan in Problem 1? Yes. Use the diagram to identify the included side for the marked angles in each triangle. A Given: AB ≅ DE, ∠A ≅ ∠D, ∠B and ∠E are right angles B Prove: △ABC ≅ △DEF Proof: ∠B ≅ ∠E because all right angles are congruent, and you are given that ∠A ≅ ∠D. AB and DE are included sides between the two pairs of congruent angles. You are given that AB ≅ DE. Thus, △ABC ≅ △DEF by ASA. D E C F PearsonTEXAS.com 159 Problem 3 Proof Writing a Proof Using AAS How does information about parallel sides help? You will need another pair of congruent angles to use AAS. Think back to what you learned in Topic 3. WR is a transversal here. M R Given: ∠M ≅ ∠K, WM } RK Prove: △WMR ≅ △RKW W Statements K Reasons 1) ∠M ≅ ∠K 1) Given 2) WM } RK 2) Given 3) ∠MWR ≅ ∠KRW hsm11gmse_0403_t05041.ai 3) If lines are } , then alternate interior ⦞ are ≅. 4) WR ≅ WR 4) Reflexive Property of Congruence 5) △WMR ≅ △RKW 5) AAS Problem 4 Determining Whether Triangles Are Congruent B Multiple Choice Use the diagram at the right. Which of the following statements best represents the answer and justification to the question, “Is △BIF @ △UTO?” Yes, the triangles are congruent by ASA. No, FB and OT are not corresponding sides. Yes, the triangles are congruent by AAS. Can you eliminate any of the choices? Yes. If △BIF @ △UTO then ∠B and ∠U would be corresponding angles. You can eliminate choice D. I F No, ∠B and ∠U are not corresponding angles. U The diagram shows that two pairs of angles and one pair of sides are congruent. The third pair of angles is congruent by the Third Angles Theorem. To prove these triangles congruent, you need to satisfy ASA or AAS. ASA and AAS both fail because FB and TO are not included between the same pair of congruent corresponding angles, so they are not corresponding sides. The triangles are not necessarily congruent. The correct answer is B. O T hsm11gmse_0403_t05044.ai 160 Lesson 4-3 Triangle Congruence by ASA and AAS HO ME RK O NLINE WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. Determine whether the triangles must be congruent. If so, name the postulate or theorem that justifies your answer. If not, explain. For additional support when completing your homework, go to PearsonTEXAS.com. 1. T 2. M 3. W V U P R N O S 4.Given: ∠FJG ≅ ∠HGJ, FG } JH Proof hsm11gmse_0403_t02698.ai △FGJ ≅ △HJG Prove: F Z Y 5.Given: PQ # QS, RS # SQ, Proof T is the midpoint of PR hsm11gmse_0403_t02699.ai hsm11gmse_0403_t02700 Prove: △PQT ≅ △RST G R Q J T H S P 6. Evaluate Reasonableness (1)(B) While helping your family clean out the attic, you find the piece of hsm11gmse_0403_t02702.ai paper shown at the right. The paper contains clues hsm11gmse_0403_t02697.ai to locate a time capsule buried in your backyard. The maple tree is due east of the oak tree in your backyard. Will the clues always lead you to the correct spot? Explain. 7. Connect Mathematical Ideas (1)(F) Anita says that you can rewrite any proof that uses the AAS Theorem as a proof that uses the ASA Postulate. Do you agree with Anita? Explain. 8. Justify Mathematical Arguments (1)(G) Can you prove that the triangles at the right are congruent? Justify your answer. 9.Given: ∠N ≅ ∠P, MO ≅ QO Proof Prove: △MON ≅ △QOP Proof M hsm11gmse_0403_t02703.ai Prove: △BDH ≅ △FDH N 10.Given: ∠1 ≅ ∠2, and HSM11GMSE_0403_a02291 DH bisects ∠BDF 3rd pass 12-22-08 O Q 1 P B hsm11gmse_0403_t02701.ai Durke D H 2 F PearsonTEXAS.com hsm11gmse_0403_t02706 161 11.Given: AB } DC, AD } BC A B Proof Prove: △ABC ≅ △CDA C D 12.Create Representations to Communicate Mathematical Ideas (1)(E) Draw two noncongruent triangles that have two pairs of congruent angles and one pair of congruent sides. hsm11gmse_0403_t02708.ai 13.Given AD } BC and AB } DC, name as many pairs of congruent triangles as you can. B C E 14.Create Representations to Communicate Mathematical Ideas (1)(E) Use a straightedge to draw a triangle. Label it △JKL. Construct △MNP ≅ △JKL so that the triangles are congruent by ASA. A D 15.Prove the Angle-Angle-Side Theorem (Theorem 4-2). Use the diagram next to it on page 158. 16.In △RST at the right, RS = 5, RT = 9, and m∠T = 30. Show that there is no SSA congruence rule by constructing △UVW with UV = RS, UW = RT , and m∠W = m∠T , but with △UVW R △RST . R hsm11gmse_0403_t02709.ai 9 5 30 S TEXAS Test Practice hsm11gmse_0403_t02710.ai 17.Suppose RT ≅ ND and ∠R ≅ ∠N. What additional information do you need to prove that △RTJ ≅ △NDF by ASA? A.∠T ≅ ∠D C.∠J ≅ ∠D B.∠J ≅ ∠F D.∠T ≅ ∠F 18.You plan to make a 2 ft-by-3 ft rectangular poster of class trip photos. Each photo is a 4 in.-by-6 in. rectangle. If the photos do not overlap, what is the greatest number of photos you can fit on your poster? F. 4 H.32 G.24 J.36 19.Write the converse of the true conditional statement below. Then determine whether the converse is true or false. If you are less than 18 years old, then you are too young to vote in the United States. 162 Lesson 4-3 Triangle Congruence by ASA and AAS T Technology Lab Use With Lesson 4-3 Exploring AAA and SSA teks (5)(A), (1)(F) So far, you know four ways to conclude that two triangles are congruent—SSS, SAS, ASA, and AAS. It is good mathematics to wonder about the other two possibilities. 1 > > Construct Use geometry software to construct AB and AC . Construct BC to > > form △ABC. Construct a line parallel to BC that intersects AB and AC at points D and E to form △ADE. B Investigate Are the three angles of △ABC congruent to the three angles of △ADE? Manipulate the figure to change the positions of DE and BC. Do the corresponding angles of the triangles remain congruent? Are the two triangles congruent? Can the two triangles be congruent? 2 > D E A C > Construct Construct AB . Draw a circle with center C that intersects AB at two hsm11gmse_04fa_t05104.ai points. Construct AC. Construct point E on the circle and construct CE. > Investigate Move point E around the circle until E is on AB and forms > △ACE. Then move E on the circle to the other point on AB to form another △ACE. D C Compare AC, CE, and m∠A in the two triangles. Are two sides and a nonincluded angle of one triangle congruent to two sides and a nonincluded angle of the other triangle? Are the triangles congruent? If you change the measure of ∠A and the size of the circle, do you get the same results? A E B hsm11gmse_04fa_t05105.ai Exercises 1.Make a Conjecture Based on your first investigation above, can you prove triangles congruent using AAA? Explain. For Exercises 2–4, use what you learned in your second investigation above. 2.Make a Conjecture Can you prove triangles congruent using SSA? Explain. > 3.Manipulate the figure so that ∠A is obtuse. Can the circle intersect AB twice to form two triangles? Would SSA work if the congruent angles were obtuse? Explain. 4.Suppose you are given CE, AC, and ∠A. What must be true about CE, AC, and m∠A so that you can construct exactly one △ACE? (Hint: Consider cases.) PearsonTEXAS.com 163 4-4 Using Corresponding Parts of Congruent Triangles TEKS FOCUS VOCABULARY •Justify – explain with logical reasoning. You can justify a TEKS (6)(B) Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and HypotenuseLeg congruence conditions. mathematical argument. •Argument – a set of statements put forth to show the truth or falsehood of a mathematical claim TEKS (1)(G) Display, explain, and justify mathematical ideas and arguments using precise mathematical language in written or oral communication. ESSENTIAL UNDERSTANDING If you know two triangles are congruent, then you know that every pair of their corresponding sides and angles is also congruent. Problem 1 Proof Proving Parts of Triangles Congruent Given: ∠KBC ≅ ∠ACB, ∠K ≅ ∠A Prove: KB ≅ AC K In the diagram, which congruent pair is not marked? The third angles of both triangles are congruent. But there is no AAA congruence rule. So, find a congruent pair of sides. 164 B KBC ACB Given C A BC BC Reflexive Property of hsm11gmse_0404_t02447 K A Given KBC ACB AAS Theorem Lesson 4-4 Using Corresponding Parts of Congruent Triangles KB AC Corresp. parts of are . Problem 2 TEKS Process Standard (1)(G) Proof Proving Triangle Parts Congruent to Measure Distance Which congruency rule can you use? You have information about two pairs of angles. Guess-andcheck AAS and ASA. STEM Measurement Thales, a Greek philosopher, is said to have developed a method to measure the distance to a ship at sea. He made a compass by nailing two sticks together. Standing on top of a tower, he would hold one stick vertical and tilt the other until he could see the ship S along the line of the tilted stick. With this compass setting, he would find a landmark L on the shore along the line of the tilted stick. How far would the ship be from the base of the tower? Given: ∠TRS and ∠TRL are right angles, ∠RTS ≅ ∠RTL Prove: RS ≅ RL T S L R Statements Reasons 1) ∠RTS ≅ ∠RTL 1) Given 2) TR ≅ TR 2) Reflexive Property of Congruence 3) ∠TRS and ∠TRL are right angles. 3) Given 4) ∠TRS ≅ ∠TRL 4) All right angles are congruent. 5) △TRS ≅ △TRL 5) ASA Postulate 6) RS ≅ RL s are ≅. 6) Corresponding parts of ≅ △ The distance between the ship and the base of the tower would be the same as the distance between the base of the tower and the landmark. PearsonTEXAS.com 165 HO ME RK O NLINE WO PRACTICE and APPLICATION EXERCISES For additional support when completing your homework, go to PearsonTEXAS.com. Scan page for a Virtual Nerd™ tutorial video. 1. Explain Mathematical Ideas (1)(G) Tell why the two triangles are congruent. Give the congruence statement. Then list all the other corresponding parts of the triangles that are congruent. L K 2.Given: ∠ABD ≅ ∠CBD, Proof ∠BDA ≅ ∠BDC 3. Given: OM ≅ ER, ME ≅ RO O N Prove: ∠M ≅ ∠R O hsm11gmse_0404_t02457 B C A J Proof Prove: AB ≅ CB M R M E D 4. Justify Mathematical Arguments (1)(G) A balalaika is a stringed instrument. Prove that the bases of the hsm11gmse_0404_t02463 balalaikas are congruent. hsm11gmse_0404_t02461 Given: RA ≅ NY , ∠KRA ≅ ∠JNY , ∠KAR ≅ ∠JYN R Prove: KA ≅ JY Proof: It is given that two angles and the included side of one triangle are congruent to two angles and the included side of the other. So, a. ? ≅ △JNY by b. ? . KA ≅ JY because c. ? . 5.Given: ∠SPT ≅ ∠OPT , Proof SP ≅ OP T K A J Y 6.Given: YT ≅ YP, ∠C ≅ ∠R, ∠T ≅ ∠P Proof Prove: ∠S ≅ ∠O S N Prove: CT ≅ RP O R C Y P P T Analyze Mathematical Relationships (1)(F) Copy and mark the figure to show the given information. Explain how you wouldhsm11gmse_0404_t02464 prove jP @ jQ. K hsm11gmse_0404_t02465 7. Given: PK ≅ QK , KL bisects ∠PKQ 8. Given: KL is the perpendicular bisector of PQ. 9. Given: KL # PQ, KL bisects ∠PKQ 166 Lesson 4-4 Using Corresponding Parts of Congruent Triangles P L Q C 10.Justify Mathematical Arguments (1)(G) The construction of a line perpendicular to line / through point < P> on line / is shown. Explain why you can conclude that CP is perpendicular to /. P A 11.The construction of ∠B congruent to given ∠A is shown. AD ≅ BF because they are congruent radii. DC ≅ FE because both arcs have the same compass settings. Explain why you can conclude that ∠A ≅ ∠B. E C hsm11gmse_0404_t02468 D A Prove: AB ≅ CD K C E A F D F B 13.Given: JK } QP, JK ≅ PQ 12.Given: BE # AC, DF # AC, Proof BE ≅ DF , AF ≅ CE B B Proof Prove: KQ bisects JP hsm11gmse_0404_t02474 P M J Q 14.Apply Mathematics (1)(A) Rangoli is a colorful design pattern drawn outside houses in India, especially during festivals. Vina plans to use the pattern at the right as the base of her design. In this pattern, RU , SV , and QT bisect eachhsm11gmse_0404_t02479 other at O. hsm11gmse_0404_t02476 RS = 6, RU = 12, RU ≅ SV , ST } RU , and RS } QT . What is the perimeter of the hexagon? In the diagram at the right, BA @ KA and BE @ KE. 15.Prove: S is the midpoint of BK . Proof 16. Prove: BK # AE A K S E Proof B TEXAS Test Practice hsm11gmse_0404_t02481 For Exercises 17 and 18, use the diagram at the right. TM # BD and TM bisects jBTD and j ATC. B A M C D 17.Suppose BD = 17 and AM = 5. What is the length of CD? 18.Suppose m∠ATC = 64, and m∠BTA = 16. What is m∠B? 19.Two parallel lines q and s are cut by a transversal t. ∠1 and ∠2 are a pair of alternate interior angles and m∠2 = 38. ∠1 and ∠3 are vertical angles. What is m∠3? T 20.△ABC has vertices A(1, 9), B(4, 3), and C(x, 6). For what value of x is △ABC a right triangle with right ∠B? hsm11gmse_0404_t02482 PearsonTEXAS.com 167 4-5 Isosceles and Equilateral Triangles TEKS FOCUS VOCABULARY TEKS (5)(C) Use the constructions of congruent segments, congruent angles, angle bisectors, and perpendicular bisectors to make conjectures about geometric relationships. TEKS (1)(C) Select tools, including real objects, manipulatives, paper and pencil, and technology as appropriate, and techniques, including mental math, estimation, and number sense as appropriate, to solve problems. Additional TEKS (5)(A), (6)(B), (6)(D) •Base of an isosceles triangle – the non-congruent side of an isosceles triangle •Base angles of an isosceles triangle – the two angles in an isosceles triangle that are formed by the intersection of a leg and the base •Corollary – a theorem that can be proved easily using another theorem •Legs of an isosceles triangle – the congruent sides of an isosceles triangle •Vertex angle of an isosceles triangle – the angle in an isosceles triangle formed by the two congruent legs •Number sense – the understanding of what numbers mean and how they are related ESSENTIAL UNDERSTANDING The sides and angles of isosceles and equilateral triangles have special relationships. Theorem 4-3 Isosceles Triangle Theorem Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent. If . . . AC ≅ BC Then . . . ∠A ≅ ∠B C A C B A B For a proof of Theorem 4-3, see Problem 2. hsm11gmse_0405_t02727 Theorem 4-4 Converse ofhsm11gmse_0405_t02726 the Isosceles Triangle Theorem Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent. If . . . ∠A ≅ ∠B Then . . . AC ≅ BC C A C A B B You will prove Theorem 4-4 in Exercise 16. hsm11gmse_0405_t02729 168 Lesson 4-5 Isosceles and Equilateral Triangles hsm11gmse_0405_t02730 Theorem 4-5 Theorem If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base. If . . . AC ≅ BC and ∠ACD ≅ ∠BCD Then . . . CD # AB and AD ≅ BD C A D B C A D B You will prove Theorem 4-5 in Exercise 10. Corollary to Theorem 4-3 hsm11gmse_0405_t02734.ai hsm11gmse_0405_t02 Corollary If a triangle is equilateral, then the triangle is equiangular. Y If . . . XY ≅ YZ ≅ ZX Then . . . ∠X ≅ ∠Y ≅ ∠Z Z X Y Z X Corollary to Theorem 4-4 Corollary If a triangle is equiangular, then the triangle is equilateral. If . . . ∠X ≅ ∠Y ≅ ∠Z Then . . . Y Z X Problem 1 Y hsm11gmse_0405_t02736.ai XY ≅ YZ ≅ ZX hsm11gmse_0405_t027 X Z TEKS Process Standard (1)(C) hsm11gmse_0405_t02737.ai hsm11gmse_0405_t027 Using Constructions of Congruent Segments Construct congruent segments to make a conjecture about the angles opposite the A congruent sides in an isosceles triangle. Step 1 Construct an isosceles ∆ABC on tracing paper, with AC ≅ BC. C A B A B A Step 2 Fold the paper so that the two congruent sides fit exactly one on top of the other. Crease the paper. Notice that hsm11gmse_04fa_t02587.ai ∠A and ∠B appear to be congruent. Conjecture: Angles opposite the congruent sides in an isosceles triangle are congruent. B B How can folding a piece of paper help you tell if two angles are congruent? When folding the paper, congruent angles will fit exactly one on top of the other. C A continued on next page ▶ PearsonTEXAS.com 169 hsm11gmse_04fa_t02588.ai Problem 1 continued Construct congruent angles to make a conjecture about the sides opposite B congruent angles in a triangle. Step 1Draw ∠ABC on tracing paper. Then construct ∠CBD congruent to ∠ABC so that CD intersects AB, resulting in a triangle. A B D C A B C Step 2Fold the paper so that the two congruent angles fit exactly on top of each other. Notice that the sides of the triangle opposite the congruent angles appear to be congruent. Conjecture: Sides opposite the congruent angles in a triangle are congruent. Problem 2 Proof How are the sides and angles of an isosceles triangle related? The congruent sides of an isosceles triangle are its legs, and the third side is its base. The two congruent legs form the vertex angle, while the other two angles are the base angles. Proving the Isosceles Triangle Theorem X Begin with isosceles △XYZ with XY ≅ XZ. Draw XB, the bisector of vertex angle jYXZ. 1 2 Given: XY ≅ XZ, XB bisects ∠YXZ Y Prove: ∠Y ≅ ∠Z Statements B Z Reasons 1) XY ≅ XZ 1) Given 2) ∠1 ≅ ∠2 2) Definition of angle bisector 3) XB ≅ XB 3) Reflexive Property of Congruence 4) △XYB ≅ △XZB 4) SAS Postulate 5) ∠Y ≅ ∠Z s are ≅. 5) Corresponding parts of ≅ △ hsm11gmse_0405_t02728 Problem 3 What are you looking for in the diagram? To use the Isosceles Triangle theorems, you need a pair of congruent angles or a pair of congruent sides. 170 Using the Isosceles Triangle Theorem and its Converse A Is AB congruent to CB? Explain. B Yes. Since ∠C ≅ ∠A, AB ≅ CB by the Converse of the Isosceles Triangle Theorem. D B Is jA congruent to jDEA? Explain. Yes. Since AD ≅ ED, ∠A ≅ ∠DEA by the Isosceles Triangle Theorem. Lesson 4-5 Isosceles and Equilateral Triangles A E C hsm11gmse_0405_t02731.ai Problem 4 Using Algebra What does the diagram tell you? Since AB ≅ CB, △ABC is isosceles. Since ∠ABD ≅ ∠CBD, BD bisects the vertex angle of the isosceles triangle. A What is the value of x? x D Since BD bisects ∠ABC, you know by Theorem 4-5 that BD # AC. So m∠BDC = 90. m∠C + m∠BDC + m∠DBC = 180 54 + 90 + x = 180 x = 36 C 3 6 Triangle Angle-Sum Theorem Substitute. Subtract 144 from each side. Problem 5 B 54 Since AB ≅ CB, by the Isosceles Triangle Theorem, ∠A ≅ ∠C. So m∠C = 54. . . . . . . . 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 0 0 0 1 1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 hsm11gmse_0405_t02735.ai TEKS Process Standard (1)(G) Finding Angle Measures Design What are the measures of jA, jB, and jADC in the photo image at the right? D The triangles are equilateral, so they are also equiangular. Find the measure of each angle of an equilateral triangle. Let a = measure of one angle. 3a = 180 a = 60 ∠A and ∠B are both angles in an equilateral triangle. m∠A = m∠B = 60 Use the Angle Addition Postulate to find the measure of ∠ADC. m∠ADC = m∠ADE + m∠CDE Both ∠ADE and ∠CDE are angles in an equilateral triangle. So m∠ADE = 60 and m∠CDE = 60. Substitute into the above equation and simplify. A C E B m∠ADC = 60 + 60 m∠ADC = 120 PearsonTEXAS.com 171 HO ME RK O NLINE WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. Complete each statement. Explain why it is true. V 1. VT ≅ ? For additional support when completing your homework, go to PearsonTEXAS.com. 2. UT ≅ ? ≅ YX 3. VU ≅ ? U 4. ∠VYU ≅ ? T Y X W A 5. Justify Mathematical Arguments (1)(G) A builder using the truss shown at the right claims that ∠ACB will have the same measure as ∠ADB. AC and AD represent hsm11gmse_0405_t02743.ai identical beams, and AB bisects ∠CAD. Is the builder correct? Justify your answer. C B king post truss Connect Mathematical Ideas (1)(F) Find the values of x and y. 6. 7. x 100 50 y 8. 52 x 4 x hsm11gmse_0401_t02427 y 110 y Mathematics (1)(A) Each face of the Great Pyramid at Giza is an isosceles triangle with a 76° vertex angle. What are the measures of the base angles? STEM9.Apply hsm11gmse_0405_t02746.ai hsm11gmse_0405_t02745.ai 10.Prove Theorem 4-5. Use the diagram next to it on page 169. hsm11gmse_0405_t02744.ai Given isosceles △JKL with base JL, find each value. K 11.If m∠L = 58, then m∠LKJ = ? . 12.If JL = 5, then ML = ? . 13.If m∠JKM = 48, then m∠J = ? . 14.If m∠J = 55, then m∠JKM = ? . J L M 15.Analyze Mathematical Relationships (1)(F) A triangle has angle measures x + 15, 3x - 35, and 4x. What type of triangle is it? Be as specific as possible. Justify your answer. hsm11gmse_0405_t02748.ai 16.Supply the missing information in this statement of the Converse of the Isosceles Proof Triangle Theorem. Then write a proof. R Begin with △PRQ with ∠P ≅ ∠Q. Draw a. ? , the bisector of ∠PRQ. Given: ∠P ≅ ∠Q, b. ? bisects ∠PRQ Prove: PR ≅ QR 172 P S Q Lesson 4-5 Isosceles and Equilateral Triangles hsm11gmse_0405_t02750.ai E G D F attic fra STEM 17. a.Apply Mathematics (1)(A) In the diagram at the right, what type of triangle is formed by the cables of the same height and the ground? b.What are the two different base lengths of the triangles? c.How is the tower related to each of the triangles? 800 ft 600 ft Cables 18.Analyze Mathematical Relationships (1)(F) The length of the base of an isosceles triangle is x. The length of a leg is 2x - 5. The perimeter of the triangle is 20. Find x. For each pair of points, there are six points that could be the third vertex of an isosceles right triangle. Find the coordinates of each point. 19.(4, 0) and (0, 4) Radio tower 1009 ft tall 1000 ft 400 ft 200 ft 0 ft 450 ft 20.(0, 0) and (5, 5) 21.(2, 3) and (5, 6) 550 ft HSM11GMSE_0405_a02297 C pass 12-22 -08 22.Create Representations to Communicate Mathematical Ideas (1)(E) 3rd Durke Use △ABC at the right. A a.Construct a right triangle with one leg congruent to AB and hypotenuse congruent to BC . b.Construct a right triangle with one leg congruent to AC and hypotenuse congruent to BC . c.Draw a new right triangle with different side lengths than △ABC. Repeat parts (a) and (b) for your new right triangle. d.Use your results to make a conjecture about congruence criteria for right triangles. B TEXAS Test Practice 23.In isosceles △ABC, the vertex angle is ∠A. What can you prove? A.AB = CB B.m∠B = m∠C C.∠A ≅ ∠B D. BC ≅ AC 24.What is the exact area of the base of a circular swimming pool with diameter 16 ft? F. 1018.29 ft2 G.1018.3 ft2 H.64p ft2 J.256p ft2 25.Suppose △ABC and △DEF are nonright triangles. If ∠B ≅ ∠E and AB ≅ DE, what else do you need to know to prove △ABC ≅ △DEF ? Explain. PearsonTEXAS.com 173 4-6 Congruence in Right Triangles TEKS FOCUS VOCABULARY •Hypotenuse – the side opposite the right angle •Legs of a right triangle – the two sides other than the hypotenuse TEKS (6)(B) Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Side-Side-Side, Angle-Angle-Side, and HypotenuseLeg congruence conditions. in a right triangle •Analyze – closely examine objects, ideas, or relationships to learn TEKS (1)(F) Analyze mathematical relationships to connect and communicate mathematical ideas. more about their nature ESSENTIAL UNDERSTANDING You can prove that two triangles are congruent without having to show that all corresponding sides and angles are congruent. In this lesson, you will prove right triangles congruent by using one pair of right angles, a pair of hypotenuses, and a pair of legs. Theorem 4-6 Hypotenuse-Leg (HL) Theorem Theorem If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. If . . . s, △PQR and △XYZ are right △ PR ≅ XZ, and PQ ≅ XY X P Q R Then . . . △PQR ≅ △XYZ Z Y For a proof of Theorem 4-6, see the Reference section on page 683. Key Concept hsm11gmse_0406_t02499.ai Conditions for HL Theorem To use the HL Theorem, the triangles must meet three conditions. Conditions • There are two right triangles. • The triangles have congruent hypotenuses. • There is one pair of congruent legs. 174 Lesson 4-6 Congruence in Right Triangles Problem 1 TEKS Process Standard (1)(F) Proof Using the HL Theorem On the basketball backboard brackets shown below, ∠ADC and ∠BDC are right angles and AC ≅ BC. Are △ADC and △BDC congruent? Explain. A How can you visualize the two right triangles? Imagine cutting △ABC along DC. On either side of the cut, you get triangles with the same leg DC. D C B • You are given that ∠ADC and ∠BDC are right angles. So, △ADC and △BDC are right triangles. • The hypotenuses of the two right triangles are AC and BC. You are given that AC ≅ BC. • DC is a common leg of both △ADC and △BDC. DC ≅ DC by the Reflexive Property of Congruence. Yes, △ADC ≅ △BDC by the HL Theorem. PearsonTEXAS.com 175 Problem 2 Proof Writing a Proof Using the HL Theorem D B Given: BE bisects AD at C, AB # BC, DE # EC, AB ≅ DE How can you get started? Identify the hypotenuse of each right triangle. Prove that the hypotenuses are congruent. C E A Prove: △ABC ≅ △DEC BE bisects AD. AC ≅ DC Given Def. of bisector hsm11gmse_0406_t02545 ∠ABC and ∠DEC are right ⦞. AB ⊥ BC DE ⊥ EC Given Def. of ⊥ lines △ ABC and △ DEC are right . △ABC ≅ △DEC Def. of right triangle HL Theorem AB ≅ DE NLINE HO ME RK O Given WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. hsm11gmse_0406_t02546.ai 1. Justify Mathematical Arguments (1)(G) Copy the flow chart and complete the proof. For additional support when completing your homework, go to PearsonTEXAS.com. R S T Given: PS ≅ PT , ∠PRS ≅ ∠PRT Prove: △PRS ≅ △PRT ∠PRS and ∠PRT are ≅. Given P ∠PRS and ∠PRT are right ⦞. a. ∠PRS and ∠PRT are supplementary. ⦞ that form a linear pair are supplementary. PS ≅ PT △PRS and △PRT are right . b. hsm11gmse_0406_t02550.ai c. PR ≅ PR △PRS ≅ △PRT e. d. 2. Study Exercise 1. Can you prove that △PRS ≅ △PRT without using the HL Theorem? Explain. 3. Explain Mathematical Ideas (1)(G) Complete the paragraph proof. B hsm11gmse_0406_t02551.ai D Given: ∠A and ∠D are right angles, AB ≅ DE Prove: △ABE ≅ △DEB A Proof: It is given that ∠A and ∠D are right angles. So, a. ? by the definition of right triangles. b. ? , because of the Reflexive Property of Congruence. It is also given that c. ? . So, △ABE ≅ △DEB by d. ? . 176 Lesson 4-6 Congruence in Right Triangles E 4.Given: HV # GT , GH ≅ TV , Proof I is the midpoint of HV 5.Given: PM ≅ RJ , PT # TJ , RM # TJ , M is the midpoint of TJ Proof Prove: △IGH ≅ △ITV Prove: △PTM ≅ △RMJ G P V I H T T J M R Connect Mathematical Ideas (1)(F) For what values of x and y are the triangles congruent by HL? 6. hsm11gmse_0406_t02553.ai x x3 3y 7. y1 3y x hsm11gmse_0406_t02554x 5 yx y5 8. Apply Mathematics (1)(A) △ABC and △PQR are right triangular sections of a fire escape, as shown. Is each story of the building the same height? hsm11gmse_0406_t02555 hsm11gmse_0406_t02556 Explain. 9. Connect Mathematical Ideas (1)(F) “Aha!” exclaims your classmate. “There must be an HA Theorem, sort of like the HL Theorem!” Is your classmate correct? Explain. 10.Given: △LNP is isosceles with base NP, Proof MN # NL, QP # PL, ML ≅ QL C B A R Prove: △MNL ≅ △QPL L M Q N P P Q Create Representations to Communicate Mathematical Ideas (1)(E) Copy the triangle and construct a triangle congruent to it using the givenhsm11gmse_0406_t02558 method. 11.SAS 12.HL 13.ASA 14.SSS hsm11gmse_0406_t02559 PearsonTEXAS.com 177 15.Given: △GKE is isosceles with Proof base GE, ∠L and ∠D are right angles, and K is the midpoint of LD. Prove: LG ≅ DE K L 16. Given: LO bisects ∠MLN , OM # LM, ON # LN Proof Prove: △LMO ≅ △LNO M O D L G N E 17.Justify Mathematical Arguments (1)(G) Are the triangles at the right congruent? Explain. hsm11gmse_0406_t02560 C F 5 hsm11gmse_0406_t02561 13 B 5 E 13 A Analyze Mathematical Relationships (1)(F) For Exercises 18 and 19, use the figure at the right. 18.Given: BE # EA, BE # EC, △ABC is equilateral Proof Prove: △AEB ≅ △CEB D B hsm11gmse_0406_t10988.ai E A 19.Given: △AEB ≅ △CEB, BE # EA, BE # EC C Can you prove that △ABC is equilateral? Explain. hsm11gmse_0406_t02562 TEXAS Test Practice 20.You often walk your dog around the neighborhood. Based on the diagram at the right, which one of the following statements about distances is true? A.SH = LH C.SH 7 LH B.PH = CH D.PH 6 CH School (S) Park (P) Home (H) Café (C ) Library (L) X 21.In equilateral △XYZ, name four pairs of congruent right triangles. Explain why they are congruent. hsm11gmse_0406_t02563 Q P Y 178 Lesson 4-6 Congruence in Right Triangles S R Z hsm11gmse_0406_t02564 4-7 Congruence in Overlapping Triangles TEKS FOCUS VOCABULARY •Representation – a way to display or describe information. TEKS (6)(B) Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, SideSide-Side, Angle-Angle-Side, and Hypotenuse-Leg congruence conditions. You can use a representation to present mathematical ideas and data. TEKS (1)(E) Create and use representations to organize, record, and communicate mathematical ideas. Additional TEKS (1)(F), (1)(G) ESSENTIAL UNDERSTANDING You can sometimes use the congruent corresponding parts of one pair of congruent triangles to prove another pair of triangles congruent. This often involves overlapping triangles. Problem 1 Identifying Common Parts What common angle do △ACD and △ECB share? E A B How can you see an individual triangle in order to redraw it? Use your finger to trace along the lines connecting the three vertices. Then cover up any untraced lines. D C Separate and redraw △ACD and △ECB. A E D B hsm11gmse_0407_t05049.ai C C The common angle is ∠C. hsm11gmse_0407_t05050.ai PearsonTEXAS.com 179 Problem 2 TEKS Process Standard (1)(E) Proof Using Common Parts Z Y Given: ∠ZXW ≅ ∠YWX, ∠ZWX ≅ ∠YXW Prove: ZW ≅ YX W • ∠ZXW ≅ ∠YWX and ∠ZWX ≅ ∠YXW • The diagram shows that △ZWX and △YXW are overlapping triangles. hsm11gmse_0407_t05052 Show △ZWX ≅ △YXW . Then use corresponding parts of congruent triangles to prove ZW ≅ YX. A diagram of the triangles separated Z W Y XW X X ∠ZXW ≅ ∠YWX Given hsm11gmse_0407_t05053 WX ≅ WX △ZWX ≅ △YXW ZW ≅ YX Reflexive Prop. of ≅ ASA Corresp. parts of ≅ are ≅. ∠ZWX ≅ ∠YXW Given Problem 3 Proof Using Two Pairs of Triangles hsm11gmse_0407_t05054 How do you choose another pair of triangles to help in your proof? Look for triangles that share parts with △GED and △JEB and that you can prove congruent. In this case, first prove △AED ≅ △CEB. Given: In the origami design, E is the midpoint of AC and DB. Prove: △GED ≅ △JEB Proof: E is the midpoint of AC and DB, so AE ≅ CE and DE ≅ BE. ∠AED ≅ ∠CEB because vertical angles are congruent. Therefore, △AED ≅ △CEB by SAS. ∠D ≅ ∠B because corresponding parts of congruent triangles are congruent. ∠GED ≅ ∠JEB because vertical angles are congruent. Therefore, △GED ≅ △JEB by ASA. A G D 180 Lesson 4-7 Congruence in Overlapping Triangles B E J C Problem 4 TEKS Process Standard (1)(G) Proof Separating Overlapping Triangles C Given: CA ≅ CE , BA ≅ DE Prove: BX ≅ DX Which triangles are useful here? If △BXA ≅ △DXE, then BX ≅ DX because they are corresponding parts. If △BAE ≅ △DEA, you will have enough information to show △BXA ≅ △DXE. B D X E A B D X A B E D A Statements E E A Reasons 1) BA ≅ DE 1) Given 2) CA ≅ CE 2) Given 3) ∠CAE ≅ ∠CEA 3) Base ⦞ of an isosceles △ are ≅. 4) AE ≅ AE 4) Reflexive Property of ≅ 5) △BAE ≅ △DEA 5) SAS 6) ∠ABE ≅ ∠EDA s are ≅. 6) Corresp. parts of ≅ △ 7) ∠BXA ≅ ∠DXE 7) Vertical angles are ≅. 8) △BXA ≅ △DXE 8) AAS 9) BX ≅ DX s are ≅. 9) Corresp. parts of ≅ △ NLINE HO ME RK O hsm11gmse_0407_t05057 WO PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd™ tutorial video. In each diagram, the red and blue triangles are congruent. Identify their common side or angle. For additional support when completing your homework, go to PearsonTEXAS.com. 1. K 2. P L E 3. X D N T W G F Z Y M Separate and redraw the indicated triangles. Identify any common sides or angles. 4. △PQS and △QPR 5.△ACB and △PRB 6.△JKL and △MLK hsm11gmse_0407_t02768 P A Q hsm11gmse_0407_t02767 P hsm11gmse_0407_t02769 K L O C R T S B J M R PearsonTEXAS.com 181 hsm11gmse_0407_t02772 hsm11gmse_0407_t02771 hsm11gmse_0407_t02770 7. Justify Mathematical Arguments (1)(G) Complete the flow proof. P Given: ∠T ≅ ∠R, PQ ≅ PV Prove: ∠PQT ≅ ∠PVR Q V ∠T ≅ ∠R a. S R T ∠TPQ ≅ ∠RPV △TPQ ≅ △RPV b. ∠PQT ≅ ∠PVR e. d. hsm11gmse_0407_t02773 PQ ≅ PV c. 8.Given: RS ≅ UT , RT ≅ US 9.Given: QD ≅ UA, ∠QDA ≅ ∠UAD Prove: △RST ≅ △UTS S T Prove: △QDA ≅ △UAD U Q Proof hsm11gmse_0407_t02774 M R D V 10.Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4 Proof Prove: △QET ≅ △QEU T hsm11gmse_0407_t02775 3 4 R U W Q Proof 1 2 E 11.Given: AD ≅ ED, D is the midpoint of BF Proof Prove: △ADC ≅ △EDG hsm11gmse_0407_t02776 A G B U A F B D E C 12.Explain Mathematical Ideas (1)(G) In the diagram at the right, ∠V ≅ ∠S, VU ≅ ST, and PS ≅ QV. Which two triangles are congruent by SAS? Explain. hsm11gmse_0407_t02777.ai Q P X W hsm11gmse_0407_t02778.ai V 13.Identify a pair of overlapping congruent triangles in the diagram. Then use the given information to write a proof to show that the triangles are congruent. Given: AC ≅ BC, ∠A ≅ ∠B U R S T A B F E hsm11gmse_0407_t02779.ai D C hsm11gmse_0407_t02784 182 Lesson 4-7 Congruence in Overlapping Triangles Mathematics (1)(A) The figure at the right is part of a clothing design pattern, and it has the following relationships. STEM14.Apply G E B J 4 H 8 9 I • GC # AC • AB # BC • AB } DE } FG • m∠A = 50 A D 1 F 2 7 5 3 6 C • △DEC is isosceles with base DC. a.Find the measures of all the numbered angles in the figure. b.Suppose AB ≅ FC. Name two congruent triangles and explain how you can prove them congruent. 15.Given: AC ≅ EC , CB ≅ CD 16.Given: QT # PR, QT bisects PR, Proof QT bisects ∠VQS Q Prove: VQ ≅ SQ P Proof Prove: ∠A ≅ ∠E C B A D F V E R S T 17.Create Representations to Communicate Mathematical Ideas (1)(E) Draw a Proof quadrilateral ABCD with AB } DC, AD } BC, and diagonals AC and DB intersecting at E. Label your diagram to indicate the parallel sides. hsm11gmse_0407_t02780.ai a.List all the pairs of congruent segments in your diagram. hsm11gmse_0407_t02781.ai b.Explain how you know that the segments you listed are congruent. TEXAS Test Practice 18.According to the diagram at the right, which statement is true? A.△DEH ≅ △GFH by AAS C.△DEF ≅ △GFE by AAS B.△DEH ≅ △GFH by SAS D.△DEF ≅ △GFE by SAS 19.△ABC is isosceles with base AC. If m∠C = 37, what is m∠B? F. 37 G.74 H.106 J.143 G F H E 20.Which word correctly completes the statement “All ? angles are D congruent”? A.adjacent B.supplementary C.right D.corresponding J G 21.In the figure, LJ } GK and M is the midpoint of LG. a.Copy the diagram. Then mark your diagram with the given information. hsm11gmse_0407_t05015 b.Prove △LJM ≅ △GKM. M L c.Can you prove that △LJM ≅ △GKM another way? Explain. K PearsonTEXAS.com 183 hsm11gmse_0407_t05018 Topic 4 Review TOPIC VOCABULARY • base of an isosceles triangle, p. 168 • base angles of an isosceles • congruent polygons, p. 148 • hypotenuse, p. 174 • legs of an isosceles triangle, p. 168 • legs of a right triangle, • corollary, p. 168 • vertex angle of an p. 174 triangle, p. 168 isosceles triangle, p. 168 Check Your Understanding Choose the correct term to complete each sentence. 1. The two congruent sides of an isosceles triangle are the 2. The side opposite the right angle of a right triangle is the 3. A 4. ? ? ? . ? . to a theorem is a statement that follows immediately from the theorem. have congruent corresponding parts. 4-1 Congruent Figures Quick Review Exercises Congruent polygons have congruent corresponding sides and angles. When you name congruent polygons, always list corresponding vertices in the same order. RSTUV @ KLMNO. Complete the congruence statements. Example HIJK @ PQRS. Write all possible congruence statements. The order of the parts in the congruence statement tells you which parts correspond. Sides: HI ≅ PQ, IJ ≅ QR, JK ≅ RS, KH ≅ SP 5.TS ≅ 6.∠N ≅ ? 7.LM ≅ 8.VUTSR ≅ ? W Z 80 145 X 3 Y R Q 100 8.6 S 35 10 5 P 9.m∠P 10.QR 11.WX 12. m∠Z 13.m∠X 14.m∠R hsm11gmse_04cr_t05080 Topic 4 Review ? WXYZ ≅ PQRS. Find each measure or length. Angles: ∠H ≅ ∠P, ∠I ≅ ∠Q, ∠J ≅ ∠R, ∠K ≅ ∠S 184 ? 4-2 and 4-3 Triangle Congruence by SSS, SAS, ASA, and AAS Quick Review Exercises You can prove triangles congruent with limited information about their congruent sides and angles. 15. In △HFD, what angle is included between DH and DF ? Postulate or Theorem You need Side-Side-Side (SSS) three sides 16. In △OMR, what side is included between ∠M and ∠R? Side-Angle-Side (SAS)two sides and an included angle Angle-Side-Angle (ASA)two angles and an included side Angle-Angle-Side (AAS)two angles and a nonincluded side Which postulate or theorem, if any, could you use to prove the two triangles congruent? If there is not enough information to prove the triangles congruent, write not enough information. 17. 18. 19. 20. Example What postulate would you use to prove the triangles congruent? You know that three sides are congruent. Use SSS. hsm11gmse_04cr_t05082hsm11gmse_04cr_t05083 4-4 Using Corresponding Parts of Congruent Triangles hsm11gmse_04cr_t05084 hsm11gmse_04cr_t05081 hsm11gmse_04cr_t05085 Exercises Quick Review Once you know that triangles are congruent, you can make conclusions about corresponding sides and angles because, by definition, corresponding parts of congruent triangles are congruent. You can use congruent triangles in the proofs of many theorems. How can you use congruent triangles to prove the statement true? 21. TV ≅ YW 22.BE ≅ DE V C W B Example How can you use congruent triangles to prove jQ @ jD? W K D T E X Y 23. ∠B ≅ ∠D D 24.KN ≅ ML C K L Q E V Since △QWE ≅ △DVK by AAS, you know that ∠Q ≅ ∠D because corresponding parts of congruent triangles are congruent. hsm11gmse_04cr_t05087hsm11gmse_04cr_t05088 B E D N M hsm11gmse_04cr_t05086 hsm11gmse_04cr_t05089hsm11gmse_04cr_t05090 PearsonTEXAS.com 185 4-5 Isosceles and Equilateral Triangles Quick Review Exercises If two sides of a triangle are congruent, then the angles opposite those sides are also congruent by the Isosceles Triangle Theorem. If two angles of a triangle are congruent, then the sides opposite those angles are congruent by the Converse of the Isosceles Triangle Theorem. Algebra Find the values of x and y. Equilateral triangles are also equiangular. 25. 26. x 50 4 x y 27. 28. 25 y Example y y 125 x hsm11gmse_04cr_t05093 hsm11gmse_04cr_t05092 x What is mjG? 25 7 E Since EF ≅ EG, ∠F ≅ ∠G by the Isosceles Triangle Theorem. So m∠G = 30. F 30 G hsm11gmse_04cr_t05094hsm11gmse_04cr_t05095 4-6 Congruence in Right Triangles hsm11gmse_04cr_t05091 Exercises Quick Review If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent by the Hypotenuse-Leg (HL) Theorem. Write a proof for each of the following. 29. Given: LN # KM, KL ≅ ML Prove: △KLN ≅ △MLN L Example Which two triangles are congruent? Explain. M B N A C Z L Y X Since △ABC and △XYZ are right triangles with congruent legs, and BC ≅ YZ, △ABC ≅ △XYZ by HL. K N M P 30. Given: PS # SQ, RQ # QS, PQ ≅ RS Q hsm11gmse_04cr_t05097 Prove: △PSQ ≅ △RQS S R hsm11gmse_04cr_t05096 hsm11gmse_04cr_t05098 186 Topic 4 Review 4-7 Congruence in Overlapping Triangles Quick Review Exercises To prove overlapping triangles congruent, you look for the common or shared sides and angles. Name a pair of overlapping congruent triangles in each diagram. State whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL. Example 31. E Separate and redraw the overlapping triangles. Label the vertices. F C E F E D 33. P hsm11gmse_04cr_t05100 C G I H B Q A D 32. F F D E A R S hsm11gmse_04cr_t05102 T hsm11gmse_04cr_t05099 hsm11gmse_04cr_t05101 C hsm11gmse_04cr_t05103 PearsonTEXAS.com 187 Topic 4 TEKS Cumulative Practice Multiple Choice 4.Given: ∠1 ≅ ∠2, AB ≅ AC Read each question. Then write the letter of the correct answer on your paper. A 1.Given: DE } CB, ∠ADE ≅ ∠AED What additional information do you need to prove △ABD ≅ △ACE by AAS? A Prove: AC ≅ AB E D Proof: Since DE } CB, 4 62 15 3 ∠ACB ≅ ∠ADE and C B E D ∠AED ≅ ∠ABC by the B C Corresponding Angles F. AD ≅ AEH. ∠5 ≅ ∠6 Theorem. Since ∠ADE ≅ ∠AED, G. BD ≅ ECJ. BE ≅ DC ∠ACB ≅ ∠ABC by the Transitive Property of Congruence. Which theorem or definition 5.Which of the following statements is true? hsm11gmse_04cu_t05071 proves that AC ≅ AB? hsm11gmse_04cu_t05070 A. Point, line, and plane are undefined terms. A. Isosceles Triangle Theorem B. A theorem is an accepted statement of fact. B. Converse of Isosceles Triangle Theorem C. “Vertical angles are congruent” is a definition. C. Alternate Interior Angles Theorem D. A postulate is a conjecture that is proven. D. Definition of congruent segments 6.Which condition allows you to prove that / } m? 2.Which statement must be true for two polygons to be congruent? 1 8 F. All the corresponding sides should be congruent. 2 3 G. All the corresponding sides and angles should be congruent. 4 5 m H. All the corresponding angles should be congruent. J. All sides in each polygon should be congruent. 3.If △ABC ≅ △CDA, which of the following must be true? A B F. ∠1 ≅ ∠8H. ∠3 ≅ ∠4 G. ∠2 ≅ ∠8J. ∠3 ≅ ∠5 7.hsm11gmse_04cu_t05072 The measure of one base angle of an isosceles triangle is 23. What is the measure of the vertex angle? A. 113C. 23 B. 134D. 67 D C A. AB ≅ CAC. ∠CAB ≅ ∠ACD B. BC ≅ DCD. ∠ABC ≅ ∠CAD hsm11gmse_04cu_t05069 188 Topic 4 TEKS Cumulative Practice Gridded Response 13.Write a proof for the following. 8.What is the value of x in the figure below? D A S Prove: △GAT ≅ △TSG 75 T L 14.Write a proof for the following. Given: LN bisects ∠OLM and ∠ONM. (5x 6) 44 E Given: AT ≅ SG, AT } GS G F 9.What is the value of x in the figure below? Prove: ON ≅ MN hsm11gmse_04ct_t02583 M O N (2x 5) 15. Isosceles △ABC, with right ∠B, has a point D on AC such that BD # AC. What is the relationship between △ABD and △CBD? Explain. hsm11gmse_04cu_t05074 135 hsm11gmse_04ct_t02584 10. ABCD ≅ WXYZ. What is WX? A 3 Y B 2 hsm11gmse_04cu_t05075 4 D C X W Constructed Response 11. Is GK ≅ HK ? Explain. hsm11gmse_04cu_t05076 G H K hsm11gmse_04cu_t05079 b.How confident are you about your conjecture? Explain. J 12. Write a proof for the following. Given: AE ≅ DE, EB ≅ EC hsm11gmte_0405_t83180.ai Prove: △AEB ≅ △DEC A Halley’s Comet can be seen periodically at its perihelion, the shortest distance from the sun during its orbit. Mark Twain was born two weeks after the comet’s perihelion. In his biography he said, “I came in with Halley’s Comet in 1835. It is coming again next year, and I expect to go out with it.” Twain died in 1910, the day after the comet’s perihelion. The most recent sighting of Halley’s Comet was in 1986. Its next appearance is expected in 2061. a.Make a conjecture about the year in which Halley’s Comet will appear after 2061. Explain your reasoning. L D E B Z 2 4 16. Read this excerpt from an online news article. 17. The coordinates of the vertices of rectangle LMNK are L( -2, 5), M(2, 5), N(2, 3), and K( -2, 3). The coordinates of the vertices of rectangle PQRS are P(3, 0), Q(3, -3), R(1, -3), and S(1, 0). Are these two rectangles congruent? Explain why or why not. If not, how could you change the vertices of one of the rectangles to make them congruent? C hsm11gmse_04cu_t05078 PearsonTEXAS.com 189