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5.2 Proving Triangles are Congruent: SSS and SAS Goal Show triangles are congruent using SSS and SAS. VOCABULARY Proof A proof is a convincing argument that shows why a statement is true. 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 &** Side NP &* Side PM &* then TMNP Example 1 c c c c QR &*, and RS &*, and SQ &*, T QRS . Q M N P R S Use the SSS Congruence Postulate Does the diagram give enough information to use the SSS Congruence Postulate? Explain. J H L K Solution From the diagram you know that HJ & c LJ & and HK &* c LK &* . By the Reflexive Property, you know that JK & c JK &. Because all three pairs of corresponding sides are congruent, you can use the SSS Congruence Postulate to conclude that THJK c T LJK . 110 Geometry, Concepts and Skills Notetaking Guide • Chapter 5 POSTULATE 13: SIDE-ANGLE-SIDE CONGRUENCE POSTULATE (SAS) 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 If Side PQ &* c WX &**, and Angle aQ c aX, and Side QR &* c XY &*, then TPQR c T WXY . Example 2 Q P X R W Y Use the SAS Congruence Postulate Does the diagram give enough information to use the SAS Congruence Postulate? Explain your reasoning. a. b. D H E G A B C F Solution a. You know that AB & c CB &* and DB &* c DB &* . The included angle between AB & and DB &* is aABD. The included angle between CB &* and DB &* is a CBD . Because the included angles are congruent, you can use the SAS Congruence Postulate to conclude that TABD c T CBD . b. You know that GF &* c GH &* and GE &* c GE &* but the congruent angles a GEH and a GEF are not included between the congruent sides, so you cannot use the SAS Congruence Postulate. Follow-Up In Example 2, part (b), what two angles must be congruent in order to use the SAS Congruence Postulate? aHGE and aFGE Lesson 5.2 • Geometry, Concepts and Skills Notetaking Guide 111 Checkpoint Does the diagram give enough information to determine whether the triangles are congruent? 1. 2. Yes 3. No Yes STEPS FOR WRITING A PROOF p List the given information first. p Decide whether you can use the given information as it is or if you need to make another statement based on the given information. p List your statements in sequential order—in other words, don’t list a conclusion before you list the statement that allows you to make that conclusion. p Give a reason for every statement that you make. p As reasons in a proof, use given information, labeled diagrams, postulates, definitions, and theorems. p End the proof with the statement you are trying to prove . Example 3 Write a Proof Write a two-column proof that shows THJK c TLKJ. J Solution Statements K H Reasons Side 1. HJ & c LK &* 1. Given Side 2. HK &* c LJ & 2. Given Side 3. JK &* c JK &* 3. Reflexive Property of Congruence 4. THJK c TLKJ 112 L Geometry, Concepts and Skills Notetaking Guide • Chapter 5 4. SSS Congruence Postulate Example 4 Prove Triangles are Congruent In the diagram at the right, DR &* ∏ AG &* and RA &* c RG &*. Write a proof to show that TDRA c TDRG. D Solution A 1. Label the diagram with the given information. R G 2. Write what you are given and what you need to prove. Given: DR &* ∏ AG &* , RA &* c RG &* Prove: TDRA c TDRG 3. Write a two-column proof. List the given statements first. Statements Reasons 1. RA &* c RG &* 1. Given 2. DR &* ∏ AG &* 2. Given 3. aDRG and aDRA are right angles. 3. Perpendicular lines form right angles. 4. aDRG c aDRA 4. All right angles are congruent . 5. DR c DR 5. Reflexive Property of Congruence 6. T DRA c T DRG 6. SAS Congruence Postulate Checkpoint Complete the proof. 4. Given: CB &* c CE &, AC & c DC &* Prove: TBCA c TECD Statements B D C A E Reasons 1. CB &* c CE & 1. Given 2. AC &* c DC &* 2. Given 3. aBCA c aECD 3. Vertical Angles Theorem 4. TBCA c TECD 4. SAS Congruence Postulate Lesson 5.2 • Geometry, Concepts and Skills Notetaking Guide 113