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Page 1 of 9 5.2 Goal Show triangles are congruent using SSS and SAS. Key Words Proving Triangles are Congruent: SSS and SAS Geo-Activity Copying a Triangle 1 Use a straightedge to draw a large ● C triangle. Label it TABC. • proof A 2 Open your compass to measure ● 3 Open your compass to measure ● AB &* of TABC. Use this length to construct DE &* so that it is congruent to AB &*. D B AC &*. Use this length to draw an arc with the point of the compass at D. E D 4 Open your compass to measure ● E 5 Label the point of intersection F. ● Then draw TDEF. Measure the angles of the triangles to confirm that TABC c TDEF. BC &*. Use this length to draw an arc centered at E that intersects the arc from Step 3. F F D D E E The Geo-Activity suggests the following postulate. POSTULATE 12 Side-Side-Side Congruence Postulate (SSS) Words If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent. Symbols If Side MN &** c QR &*, and Side NP &* c RS &*, and Side PM &** c SQ &*, then TMNP c TQRS. 5.2 P M N P R S Proving Triangles are Congruent: SSS and SAS 241 Page 2 of 9 EXAMPLE Use the SSS Congruence Postulate 1 Does the diagram give enough information to show that the triangles are congruent? Explain. J L H K Solution &* c LJ &* and HK &** c LK &*. From the diagram you know that HJ &* c JK &*. By the Reflexive Property, you know that JK ANSWER © Yes, enough information is given. Because corresponding sides are congruent, you can use the SSS Congruence Postulate to conclude that THJK c TLJK. POSTULATE 13 Visualize It! Side-Angle-Side Congruence Postulate (SAS) In the triangle below, aB is the included angle between sides AB &* and BC &*. Words If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. Symbols C PQ &* c WX &&, and If Side P X Angle aQ c aX, and A QR &* c XY &**, Side B R P then TPQR c TWXY. included angle EXAMPLE 2 W Y W Use the SAS Congruence Postulate Does the diagram give enough information to use the SAS Congruence Postulate? Explain your reasoning. a. b. D H G A B C E F Solution &* c CB &* and DB &* c DB &*. a. From the diagram, you know that AB &* and DB &* is aABD. The angle included between AB &* and DB &* is aCBD. The angle included between CB Because the included angles are congruent, you can use the SAS Congruence Postulate to conclude that TABD c TCBD. &* c GH &** and GE &* c GE &*. However, the congruent b. You know that GF angles are not included between the congruent sides, so you cannot use the SAS Congruence Postulate. 242 Chapter 5 Congruent Triangles Page 3 of 9 Writing Proofs A proof is a convincing argument that shows why a statement is true. A two-column proof has numbered statements and reasons that show the logical order of the argument. Each statement has a reason listed to its right. HOW TO WRITE A PROOF • List the given information first. • Use information from the diagram. • Give a reason for every statement. • Use given information, definitions, postulates, and theorems as reasons. • List statements in order. If a statement relies on another statement, list it later than the statement it relies on. • End the proof with the statement you are trying to prove. EXAMPLE 3 Write a Proof Write a two-column proof that shows T JKL c TNML. Given © JL &* c NL &* J &**. L is the midpoint of KM Prove © T JKL c TNML K M L N Solution The proof can be set up in two columns. The proof begins with the given information and ends with the statement you are trying to prove. Statements Reasons &* c NL &* S 1. JL 1. Given 2. L is the midpoint Student Help STUDY TIP You can remind yourself about side and angle congruences by writing letters as shown. 2. Given These are the given statements. &**. of KM A 3. aJLK c aNLM 3. Vertical Angles Theorem &* c ML &** S 4. KL 4. Definition of midpoint 5. TJKL c TNML 5. SAS Congruence Postulate 5.2 This information is from the diagram. Statement 4 follows from Statement 2. Statement 5 follows from the congruences of Statements 1, 3, and 4. Proving Triangles are Congruent: SSS and SAS 243 Page 4 of 9 IStudent Help EXAMPLE 4 Prove Triangles are Congruent ICLASSZONE.COM You are making a model of the window shown in the photo. &* ∏ AG &* and RA &* c RG &*. Write a proof to show You know that DR that TDRA c TDRG. MORE EXAMPLES More examples at classzone.com D A R G Solution D 1 Make a diagram and label it with ● the given information. 2 Write the given information and ● the statement you need to prove. &* ∏ AG &*, RA &* c RG &* Given © DR Prove © TDRA c TDRG A R G 3 Write a two-column proof. List the given statements first. ● Student Help Statements Reasons STUDY TIP &* c RG &* 1. RA 1. Given Think about what you can say given that DR &* ∏ AG &*. &* ∏ AG &* 2. DR 2. Given 3. aDRA and aDRG are 3. ∏ lines form right angles. Then think about what other information you can deduce from the diagram. right angles. 4. aDRA c aDRG 4. Right angles are congruent. &* c DR &* 5. DR 5. Reflexive Property of Congruence 6. TDRA c TDRG 6. SAS Congruence Postulate Prove Triangles are Congruent 1. Fill in the missing statements and reasons. D B &* c CE &*, AC &* c DC &** Given © CB Prove © TBCA c TECD 244 Chapter 5 C A Statements Reasons &* c CE &* 1. CB 1. _________?_________ 2. _________?_________ 2. Given 3. aBCA c aECD 3. _________?_________ 4. TBCA c TECD 4. _________?_________ Congruent Triangles E Page 5 of 9 5.2 Exercises Guided Practice Vocabulary Check Use TJKL to name the angle included between the two sides. &* and KM &** 1. JK &* and MJ &* 2. JK K &* and JL & 3. KL &** and LM &** 4. KM J L M &* and KM &** 5. LK Skill Check Decide whether enough information is given to show that the triangles are congruent. If so, tell which congruence postulate you would use. 6. TABC, TDEC A 7. TFGH, TJKH B P P F G C E 8. TPQR, TSRQ H D K S R J Practice and Applications Extra Practice See p. 683. Naming Included Angles Use the diagram shown to name the angle included between the two sides. &* and BD &* 9. AB &* and BC &* 10. CD &* and CB &* 11. AC &* and AD &* 12. BA &** and BD &* 13. DC &* and BC &* 14. BD B A D C Using SSS Decide whether enough information is given to use the SSS Congruence Postulate. Explain your reasoning. Homework Help 15. 16. C E K L D J M Example 1: Exs. 15–17, 21–31 Example 2: Exs. 18–31 Example 3: Exs. 34–36 Example 4: Exs. 34–36 A B 17. 5.2 P P S Proving Triangles are Congruent: SSS and SAS R 245 Page 6 of 9 Using SAS Congruence Decide whether enough information is given to use the SAS Congruence Postulate. Explain your reasoning. 18. P 19. B C 20. J D K N P S R A L E M You be the Judge Decide whether enough information is given to show that the triangles are congruent. If so, tell which congruence postulate you would use. 21. B A G 22. H 23. M N P P F C D E 24. K 25. R X W J S T D 26. Y A B C U Z Textiles In Exercises 27 and 28, use the photo of the Navajo rug. In &* c DE &* and AC &* c CE &*. the triangles on the rug, BC 27. What additional information is needed to use the SSS Congruence Postulate to show that TABC c TCDE ? 28. What additional information is needed to use the SAS Congruence Postulate to show that TABC c TCDE ? Missing Information Determine what single piece of information you need to know in order to use either the SSS or SAS Congruence Postulate to show that the triangles are congruent. 29. B A D 246 Chapter 5 30. J C Congruent Triangles K F 31. P S E L M P R Page 7 of 9 Quilting Quilting Use the labels on the diagrams to explain how you know that the triangles are congruent. &* i CD &* 33. AB 32. WXYZ is a square. X TRIANGLES To create the W Y A B C D quilt pattern in Ex. 33, squares are cut out of two rectangular strips of fabrics sewn together. Application Links Z CLASSZONE.COM Reasoning In Exercises 34 and 35, fill in the missing statements and reasons. H E &* c GH &* 34. Given © EF &* c HE &** FG Prove © TEFG c TGHE G IStudent Help ICLASSZONE.COM HOMEWORK HELP Extra help with problem solving in Exs. 34–35 is at classzone.com 35. F Statements Reasons &* c GH &* 1. EF 1. Given &* c HE &** 2. FG 2. _________?_________ &* c GE &* 3. GE 3. _________?_________ 4. TEFG c TGHE 4. _________?_________ Given © &* &* SP c TP P &* bisects aSPT. PQ S T Prove © TSPQ c TTPQ P Statements Reasons &* 1. &* SP c TP 1. Given &* bisects aSPT. 2. PQ 2. _________?_________ 3. aSPQ c aTPQ 3. _________?_________ 4. _________?_________ 4. Reflexive Prop. of Cong. 5. TSPQ c TTPQ 5. _________?_________ 5.2 Proving Triangles are Congruent: SSS and SAS 247 Page 8 of 9 36. Reasoning Fill in the missing statements and reasons. Given © AC &* c BC &* C &*. M is the midpoint of AB Prove © TACM c TBCM A Student Help STUDY TIP It is helpful to label congruent sides and congruent angles in steps of a proof. Statements Reasons &* c BC &* S 1. AC 1. Given &*. 2. M is the midpoint of AB B M 2. _________?_________ S 3. _________?_________ 3. Definition of midpoint &** c CM &** S 4. CM 4. _________?_________ 5. TACM c TBCM 5. _________?_________ 37. Error Analysis Using the diagram below, Maria was asked whether it can be shown that TABC c TDEF. Explain her error. C Maria A B E D TABC c TDEF by the SAS Congruence Postulate. F Challenge Write a proof to show that the triangles are congruent. 38. TABD and TCBD &* c YZ &*, and ZM &** 39. XZ are equilateral. bisects aYZX. Y B A M C X D Standardized Test Practice Z &* c AB &*, ST &* c BC &*, and 40. Multiple Choice In TRST and TABC, RS &* c CA &*. Which angle is congruent to aT ? TR A X aA B aR X C aC X D aB X 41. Multiple Choice In the diagram below, TDEF is equilateral &*. Which of the statements and G is the midpoint of DE is not true? F X H X &* c EF &* DF &** c EG &* DG &** c DF &* G DG X J TDFG c TEFG X D 248 Chapter 5 Congruent Triangles F G E Page 9 of 9 Mixed Review Classifying Angles Use the diagram to determine whether the angles are corresponding, alternate interior, alternate exterior, or same-side interior angles. (Lesson 3.3) 42. a1 and a5 43. a2 and a6 44. a7 and a2 45. a5 and a8 46. a9 and a4 47. a5 and a9 1 3 4 5 2 6 7 8 9 Using the Triangle Inequality Determine whether it is possible to draw a triangle with the given side lengths. (Lesson 4.7) 48. 14, 8, 25 Algebra Skills 49. 20, 10, 28 50. 16, 14, 30 Evaluating Square Roots Evaluate. Give the exact value if possible. If not, approximate to the nearest tenth. (Skills Review, p. 669) 51. Ï3 w 52. Ï1 w2 w 53. Ï4 w0 w 54. Ï1 w5 w9 w 55. Ï1 w4 w.7 w6 w 56. Ï0 w.8 w7 w 57. Ï1 w.1 w2 w 58. Ï4 w0 w.8 w5 w Quiz 1 In the diagram, TABC c TQPR. Complete the statement with the corresponding congruent part. (Lesson 5.1) C R &* 1. aR c __?__ 2. AB c __?__ 3. TBAC c __?__ 4. TRPQ c __?__ A 5. Write a congruence statement for B Z the congruent triangles shown at the right. (Lesson 5.1) P P Y F X E G Decide whether enough information is given to show that the triangles are congruent. If so, tell which congruence postulate you would use. Explain your reasoning. (Lesson 5.2) 7. D 6. A B C Y X G E 8. F F J Z 5.2 K M Proving Triangles are Congruent: SSS and SAS N 249