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Triangle Congruence by ASA and AAS 4-3 4-3 1. Plan GO for Help What You’ll Learn Check Skills You’ll Need • To prove two triangles In k JHK, which side is included between the given pair of angles? congruent using the ASA Postulate and the AAS Theorem 1. &J and &H JH In k NLM, which angle is included between the given pair of sides? 3. LN and LM lL 4. NM and LN lN To prove that the two sides of a lacrosse goal are congruent triangles, as in Example 2 Give a reason to justify each statement. 5. PR > PR P 6. &A > &D Q R Reflexive Prop. of O A 1 2 3 4 D Angle-Side-Angle (ASA) Postulate P B B A 1 4 5 A A B B D C Multiple Choice Which triangle is congruent to #CAT by the ASA Postulate? E D C See p. 196E for a list of the resources that support this lesson. Using ASA E D C B EXAMPLE Lesson Planning and Resources H E D C B A 3 C B A 2 1 E D More Math Background: p. 196C N K G #HGB > #NKP C Using ASA Real-World Connection Planning a Proof Writing a Proof ASA is presented in this lesson as a postulate, but it could be established as a theorem (whose proof requires constructing congruent segments) that follows from the SAS postulate, much as SSS also could be established as a theorem that follows from the SAS Postulate. The proof of the AAS Theorem follows from the ASA Postulate and the Triangle Angle-Sum Theorem. In Lesson 4-2 you learned that two triangles are congruent if two pairs of sides are congruent and the included angles are congruent (SAS). The Activity Lab on page 204 suggests that two triangles are also congruent if two pairs of angles are congruent and the included sides are congruent (ASA). 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. To prove two triangles congruent using the ASA Postulate and the AAS Theorem Math Background Using the ASA Postulate and the AAS Theorem Postulate 4-3 1 Examples If 2 ' of a k are O to C B F E 2 ' of another k, the third ' are O. S Key Concepts Objectives 2. &H and &K HK . . . And Why 1 Lesson 4-2 D E E Test-Taking Tip #DOG #GDO In a triangle congruence statement, remember to list corresponding vertices in the same order. PowerPoint #INF #FNI D O &C > &G, CA > GD, and &A > &D. #CAT > #GDO by ASA. Choice C is correct. Quick Check Bell Ringer Practice C F G A T Check Skills You’ll Need For intervention, direct students to: N I 1 Can you conclude that #INF is congruent to either of the other two triangles? Explain. No; the O side is not the included side. Using the SAS and SSS Postulates Lesson 4-2: Example 2 Extra Skills, Word Problems, Proof Practice, Ch. 4 Proving Triangles Congruent Chapter 4 Triangle Congruence by ASA and AAS Special Needs L1 Have students mark congruent sides and angles as shown to distinguish between AAS and ASA. A A A A learning style: visual learning style: verbal S A S A S Lesson 4-1: Example 4 Extra Skills, Word Problems, Proof Practice, Ch. 4 L2 Students may substitute specific measures for the congruent angles in the flow proof of the AAS Theorem to see the dependence on the ASA Postulate before proceeding to the general proof. S A Below Level A 213 213 2. Teach Real-World Guided Instruction 2 EXAMPLE The Iroquois Nationals compete for the World Lacrosse Championship every four years. 2 EXAMPLE Real-World Connection C Lacrosse Study what you are given and what you are to prove about the lacrosse goal. Then write a proof that uses ASA. Diversity D A B E Front view of lacrosse goal Given: &CAB > &DAE, AB > AE, &ABC and &AED are right angles. Lacrosse originated as “baggataway,” a Native American game with up to 200 players on each side. Its modern name comes from French Canadians. Prove: #ABC > #AED Proof: &ABC > &AED because all right angles are congruent. You are given that AB > AE and &CAB > &DAE. Thus, #ABC > #AED by ASA. Teaching Tip Examine the flow proof of the AAS Theorem as a class. Point out the three arrows leading to the conclusion. Ask: Why are there three arrows? Each arrow is from one of the congruence statements needed to prove the AAS Theorem. 3 Here is how you can use the ASA Postulate in a proof. Connection Quick Check 2 Write a proof. L P Given: NM > NP, &M > &P N M O It is given that NM O NP and lM O lP. lMNL O lPNO because vert. ' are O. kNML O kNPO by ASA. You can use the ASA Postulate to prove the Angle-Angle-Side Congruence Theorem. A flow proof is shown below. Prove: #NML > #NPO EXAMPLE Key Concepts Point out that an effective Plan for Proof may involve the strategy working backward. In this Plan for Proof, the conclusion of the conditional statement is what must be proved. The second sentence of the Plan for Proof tells why the hypothesis is true. Theorem 4-2 Angle-Angle-Side (AAS) 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. C M #CDM > #XGT PowerPoint Proof Additional Examples G Prove: #ABC > #XYZ Suppose that &F is congruent to &C and &I is not congruent to &C. Name the triangles that are congruent by the ASA Postulate. kFNI O kCAT O kGDO A X Given B Y Given 2 Write a paragraph proof. A BC YZ Given Y X Proof of the Angle-Angle-Side Theorem Given: &A > &X, &B > &Y, BC > YZ 1 Use the triangles in Example 1. T D P X A C Z If 2 s of one are to 2 s of another , then the 3rd s are . Y B Z C ABC XYZ ASA Postulate X A proof starts in your head with a plan. It may help to jot down your plan. You will find this especially helpful as proofs become more complex. As you become skilled at proof, you will find that a good plan looks a lot like a paragraph proof. B Given: &A &B, AP BP Prove: APX BPY lAPX O lBPY by the Vertical Angles Theorem, so kAPX O kBPY by ASA. 214 Chapter 4 Congruent Triangles Advanced Learners English Language Learners ELL L4 After students read the ASA Postulate and the AAS Theorem, ask: If you could use only one in the remainder of this course, which would you choose and why? 214 learning style: verbal Point out that AAS, or Angle-Angle-Side Theorem, is a theorem and not a postulate. A postulate is an accepted statement of fact, but theorems are conjectures that are proven. learning style: verbal 3 EXAMPLE 3 Write a Plan for Proof that uses AAS. Planning a Proof A Study what you are given and what you are to prove. Then plan a proof that uses AAS. Given: &S > &Q, RP bisects &SRQ. P D Prove: #SRP > #QRP S Prove: ABC CDA Because AB n CD, lBAC O lDCA (alt. int. angles). kABC O kCDA by AAS if AB O CD, BC O DA, or AC O AC. By Reflexive Prop., AC O AC. R The second statement is true by the Reflexive Property of Congruence. Proof 3 Use the plan from Example 3 and write a proof. See back of book. 4 EXAMPLE 4 Write a two-column proof of Additional Example 3. 1. lB O lD, AB n CD (Given) 2. lBAC O lDCA (If lines are n, then alt. int. angles are O.) 3. AC O AC (Reflexive Prop. of O) 4. kABC O kCDA (AAS) Writing a Proof Study what you are given and what you are to prove. Then write a proof that uses AAS. nline Q Given: XQ 6 TR, XR bisects QT. Prove: #XMQ > #RMT Plan: Visit: PhSchool.com Web Code: aue-0775 X XQ 6 TR gives two pairs of congruent alternate interior angles. XR bisects QT means QM > TM. Use AAS. Statements Quick Check M R Resources • Daily Notetaking Guide 4-3 L3 • Daily Notetaking Guide 4-3— L1 Adapted Instruction T Reasons 1. XQ 6 TR 1. Given 2. &Q > &T, &X > &R 2. 9 3. XR bisects QT. 3. Given 4. QM > TM 4. Definition of segment bisector 5. #XMQ > #RMT 5. AAS Closure 4 a. Supply the reason that justifies Step 2. If n lines, then alt. int. ' are O. b. Critical Thinking Explain how you could prove #XMQ > #RMT by ASA. kXMQ O kRMT because vert. ' are O. EXERCISES C Given: &B &D, AB 6 CD Q Plan: #SRP > #QRP by AAS if SP > QP or RP > RP. Quick Check B For more exercises, see Extra Skill, Word Problem, and Proof Practice. Explain why the letters of ASA and AAS are written in a different order. ASA compares triangles in which O sides are between the two pairs of O angles, and AAS compares triangles in which O sides are not between pairs of O angles. Practice and Problem Solving A Practice by Example Example 1 GO for Help Name two triangles that are congruent by the ASA Postulate. 1. P Q (page 213) S X W 2. E F B H D U V C G A T kACB O kEFD kPQR O kVXW Answer each question without drawing the triangle. R I 3. Which side is included between &R and &S in #RST? RS 4. Which angles include NO in #NOM? lN and lO Lesson 4-3 Triangle Congruence by ASA and AAS 215 215 5. Developing Proof Complete the proof by filling in the blanks. Example 2 3. Practice (page 214) Given: &LKM > &JKM, &LMK > &JMK Assignment Guide K Prove: #LKM > #JKM L J M Proof: &LKM > &JKM and &LMK > &JMK are 1 A B 1-26 C Challenge 27-32 Test Prep Mixed Review 33-36 37-43 given. KM > KM by the a. 9 Property of Congruence. Reflexive #LKM > # JKM by the b. 9 Postulate. ASA Proof Homework Quick Check 6. Given: &BAC > & DAC AC ' BD 7. Given: QR > TS, QR 6 S T Prove: #QRT > #TSQ Prove: #ABC > #ADC 6–7. See back of book. B To check students’ understanding of key skills and concepts, go over Exercises 5, 11, 17, 22, 26. A R Q C T S D Teaching Tip Students practice several different methods of writing and planning proofs in this lesson. Point out that a good plan uses only the necessary information and clearly states the reasons. The goal is to plan and write clear and correct proofs. Learning multiple techniques will help students reach that goal. 8. Developing Proof Complete the proof plan by filling in the blanks. Example 3 (page 215) Given: &UWT and &UWV are right angles, &T > &V. U Prove: #UWT > #UWV T Plan: #UWT > #UWV by AAS if &T > &V, &UWT > a. 9, and UW > b. 9. lUWV; UW V W &UWT > &UWV because all c. 9 angles are congruent. right UW > UW by the d. 9 Property of Congruence. Reflexive Proof 9. Use your plan from Exercise 8 and write a proof. See back of book. 10. Developing Proof Complete the proof by filling in the blanks. Example 4 (page 215) Given: &N > &S, line / bisects TR at Q. ᐉ N Prove: #NQT > #SQR R Q T GPS Guided Problem Solving Statements L3 L4 Enrichment 1. 2. 3. c. 5. L2 Reteaching L1 Adapted Practice Practice Name Class L3 Date Practice 4-3 Triangle Congruence by ASA and AAS Tell whether the ASA Postulate or the AAS Theorem can be applied directly to prove the triangles congruent. If the triangles cannot be proved congruent, write not possible. L 1. R Q T 2. U Proof 3. W X M N S B 4. D Z Y P V A H G 5. N 6. P R J E F M K U 7. V W Q L 8. Z 9. A H &N > &S &NQT > &SQR O bisects TR at Q. 9 TQ O QR #NQT > #SQR 1. 2. 3. 4. 5. Given 9 Vert. ' are O. 9 Given Definition of bisect 9 AAS 11. Given: &V > &Y, WZ bisects &VWY. Prove: #VWZ > #YWZ 11–12. See back of book. W S C D Y C X B G R Q F © Pearson Education, Inc. All rights reserved. 11. Write a flow proof. 10. Write a two-column proof. Given: K M, KL ML Given: Q S, TRS RTQ Prove: JKL PML Prove: QRT STR K J Q R T S V L M P What else must you know to prove the triangles congruent for the reason shown? 12. ASA 13. AAS A B D C E 14. ASA K J L M G 216 Q N F H 12. Given: PQ ' QS, RS ' QS, T is the midpoint of PR. Prove: # PQT > #RST E T S Reasons P 216 Chapter 4 Congruent Triangles Z T Y P S Alternative Method B Apply Your Skills Write a congruence statement for each pair of triangles. Name the postulate or theorem that justifies your statement. kZVY O kWVY; AAS V W M T 13. 14. 15. U P R N O kPMO O kNMO; ASA Real-World 16. Multiple Choice For the triangles shown, &D > &T, &E > &U, and EO > UX. Which of the following statements is true? D #TUX > #DOE by ASA #UTX > #DEO by AAS #TXU > #ODE by ASA #TUX > #DEO by AAS Connection 23. kBDH O kFDH by ASA since lBDH O lFDH by def. of l bis. and DH O DH by the Reflexive Prop. of O. 26. Answers may vary. Sample: C A D Proof nline Homework Help Visit: PHSchool.com Web Code: aue-0403 D E X Exercise 11 Students sometimes Prove: #MON > #QOP Prove: #FGJ > #HJG M F N triangle must be flipped over in order for the triangles to match exactly. G Proof J P 22. Given: AE 6 BD, AE > BD, &E > &D GPS Prove: #AEB > #BDC E Exercise 22 Because the parallel segments terminate at the transversal AC , students may have difficulty spotting corresponding angles. Have them copy just the parallel segments AE and BD and the transversal AC to help them find the corresponding angles more easily. Prove: #BDH > #FDH D 1 2 H C B Visual Learners H 23. Given: DH bisects &BDF, &1 > &2. D B F Connection to Discrete Math 24. Constructions Using a straightedge, draw a triangle. Label it # JKL. Construct #MNP > # JKL so you know that the triangles are congruent by ASA. See back of book. A B Proof 25. Given: AB 6 DC, AD 6 BC D think an angle bisector also bisects the opposite side or is a perpendicular bisector. Ask: From the Given, can you conclude &PQS &RQS or PQ RQ? no Review how angle bisectors, segment bisectors, and perpendicular bisectors are different. Exercise 16 Point out that one O Q to find a second way to complete the proof. Ask: Suppose that you did not know the Vertical Angles Theorem. How could you find another pair of congruent angles? Because lN O lS, NT n RS by the Converse of the Alt. Int. Angles Thm., so lT O lR because they are alt. int. angles. QT O QR by definition of a bisector, and kNQT O kSQR by AAS. Error Prevention! O 21. Given: &F > &H, FG 6 JH Prove: #ABC > #CDA See back of book. GO T U 20. Given: &N > &P, MO > QO A B Y 19. Reasoning Can you prove the triangles at the right congruent using ASA or AAS? Justify your answer. No; also need one pair of corres. sides O. 20. kMON O kQOP by AAS; lMON and lQOP are O vert. '. 22. kAEB O kBDC by ASA, since lEAB O lDBC because n lines have O corr. '. Z 17. Writing 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. See margin. 18. &E > &I and FE > GI. What else must you know to prove #FDE > #GHI by AAS? By ASA? lFDE O lGHI; lDFE O lHGI Congruent lapel and collar triangles help you look sharp. 21. kFGJ O kHJG by AAS since lFGJ O lHJG because when lines are n, then alt. int. ' are O and GJ O GJ by the Reflexive Prop. of O. S kUTS O kRST; AAS Exercise 10 Challenge students Exercise 31 There are 20 ways because 6C3 = 63 ?? 52 ?? 41 = 20. To list the sets of three efficiently, suggest that students rename the congruence statements 1, 2, 3, 4, 5, and 6. C 26. Reasoning If possible, draw two noncongruent triangles that have two pairs of congruent angles and one pair of congruent sides. If this is not possible, explain why. See left. Lesson 4-3 Triangle Congruence by ASA and AAS 217 17. Yes; if 2 ' of a k are O to 2 ' of another k, then the 3rd ' are O. So, an AAS proof can be rewritten as an ASA proof. 217 4. Assess & Reteach C Challenge PowerPoint Lesson Quiz 1. Which side is included between &R and &F in FTR? RF 2. Which angles in STU include US? lS and lU Tell whether you can prove the triangles congruent by ASA or AAS. If you can, state a triangle congruence and the postulate or theorem you used. If not, write not possible. 3. G Real-World Connection The two triangles above are congruent if just one additional condition is met. H I P Q R P L U 32. 9 5 30° Y A What additional information about KQ and NR will allow you to conclude that J # JKQ > #MNR? Explain. Q They are l bisectors; ASA. 31. Probability Here are six congruence statements about the triangles at the right. 13 20 &A > &X &B > &Y &C > &Z AB > XY AC > XZ BC > YZ L M W V P R A 32. #RST at the right has RS = 5, RT = 9, and m&T = 30. Show that there is no SSA congruence rule by constructing #UVW with UV = 5, UW = 9, and m&W = 30, but with #UVW R #RST. See left. C B X There are 20 ways to choose a group of three statements from these six. What is the probability that three statements chosen at random from the six will guarantee that the triangles are congruent? kGHI O kPQR; AAS 4. 27. a. Open-Ended Draw a triangle. Draw a second triangle that shares a common side with the first one and is congruent to it. Check students’ work. b. Think about how you drew your second triangle. What postulate or theorem did you use to make the second triangle congruent to the first one? most likely ASA Use the figure at the right. Name as many pairs of B C congruent triangles as you can for the information given. E 28. ABCD is a parallelogram. 28–29. See margin. 29. ABCD is a rectangle. A D K N 30. Reasoning #JKL > #MNP. Z Y R 9 5 30 T S not possible A 5. Test Prep Multiple Choice B X C 34. Suppose RT > ND and &R > &N. What additional information is needed to prove #RTJ > #NDF by ASA? F F. &T > &D G. &R > &N H. & J > &D J. &T > &F kABX O kACX; AAS Short Response Alternative Assessment Have students explain the four ways they have learned to prove triangles congruent—SSS, SAS, ASA, and AAS—and include a diagram with each method. Extended Response 218 35. PQ bisects &RPS and &RQS. Justify each answer. a. Which pairs of angles, if any, are congruent? b. By what theorem or postulate can you prove that #PRQ > #PSQ? See margin. 36. LJ 6 KG and M is the midpoint of LG. a. Why is LM > GM? a–c. See margin, p. 219. b. Can the two triangles be proved congruent by ASA? Explain. c. Can the two triangles be proved congruent by AAS? Explain. R P Q S L K M J G Chapter 4 Congruent Triangles 28. kAEB O kCED, kBEC O kDEA, kABC O kCDA, kBAD O kDCB 218 33. Which of the following is NOT a method used to prove triangles congruent? D A. AAS B. ASA C. SAS D. SSA 29. kAEB O kCED, kBEC O kDEA, kABC O kCDA, kABD O kDCA, kBAD O kDCB, kABD O kDCB, kCBA O kDAB, kBCD O kADC 35. [2] a. lRPQ O lSPQ, lRQP O lSQP (Def. of l bisector) b. ASA [1] one part correct Test Prep Mixed Review GO for Help Lesson 4-2 In Exercises 37 and 38, decide whether you can use the SSS Postulate or the SAS Postulate to prove the triangles congruent. If so, write the congruence statement and name the postulate. If not, write not possible. L 37. P 38. M T O Lesson 3-2 Lesson 1-9 R Q Resources For additional practice with a variety of test item formats: • Standardized Test Prep, p. 253 • Test-Taking Strategies, p. 248 • Test-Taking Strategies with Transparencies S not possible N kONL O lMLN; SAS 39. For any #ABC, which sides are not included between &A and &B?AC and CB 40. State the theorem or postulate a that justifies the statement: 1 2 If &1 > &3, then a 6 b. b 3 If corr. ' are O, then the lines are n. Photography You want to arrange class-trip photos without overlap to make a 2 ft-by-3 ft poster. You collect 3 in.-by-5 in. and 4 in.-by-6 in. photos. What is the greatest number of each type of photo that you can fit on your poster? 41. 3 in.-by-5 in. 56 Use this Checkpoint Quiz to check students’ understanding of the skills and concepts of Lessons 4-1 through 4-3. Resources Grab & Go • Checkpoint Quiz 1 42. 4 in.-by-6 in. 36 43. What percent more paper is used for a large photo than a regular photo? 60% more paper Checkpoint Quiz 1 Lessons 4-1 through 4-3 1. #RST > # JKL. List the three pairs of congruent corresponding sides and the three pairs of congruent corresponding angles. RS O JK ; ST O KL ; RT O JL ; lR O lJ; lS O lK; lT O lL State the postulate or theorem you can use to prove the triangles congruent. If the triangles cannot be proven congruent, write not possible. 2. ASA 3. 4. SAS SSS 36. [4] a. Def. of midpt. 5. 6. 7. AAS not possible not possible Use the information given in the diagram. Tell why each statement is true. 8. &H > &K L K c. Yes; if two ' of one k are O to 2 ' of another k, the third ' are O. N If n lines, then alt. int. ' O. 9. &HNL > &KNJ Vert. ' are O. 10. #HNL > #KNJ ASA or AAS lesson quiz, PHSchool.com, Web Code: aua-0403 H J Lesson 4-3 Triangle Congruence by ASA and AAS b. Yes; lJLM O lKGM because they are alt. int. ' of n lines, and lLMJ O lGMK because vertical ' are O. So the > are O by ASA. 219 [3] incorrect ' for part b or c, but otherwise correct [2] correct conclusions but incomplete explanations for parts b and c [1] at least one part correct 219