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Geometry 5.7 Using Congruent Triangles Essential Question How can you use indirect triangle to find an indirect measurement? January 8, 2016 5.7 Using Congruent Triangles Goals Use congruent triangles to solve problems. Use CPCTC. January 8, 2016 5.7 Using Congruent Triangles How do we prove triangles congruent? SSS SAS ASA AAS HL(Rt. January 8, 2016 ) 5.7 Using Congruent Triangles We’ve been using: If corresponding parts of two triangles are congruent, then the triangles are congruent. January 8, 2016 5.7 Using Congruent Triangles Now we add: If two triangles are congruent, then their corresponding parts are congruent. Corresponding Parts of Congruent Triangles are Congruent. January 8, 2016 5.7 Using Congruent Triangles The Basic Idea: Given Information •SSS •SAS •ASA •AAS Prove Triangles Congruent CPCTC January 8, 2016 5.7 Using Congruent Triangles Shows Corresponding Parts Congruent Example 1 B A K C J Is ABC JKL? YES What’s the reason? SAS January 8, 2016 5.7 Using Congruent Triangles L Example 1 continued B K ABC JKL A C J What other angles are congruent? B K and C L What other side is congruent? BC KL January 8, 2016 5.7 Using Congruent Triangles L Example 1 continued B K ABC JKL A C J What other angles are congruent? B K and C L What other side is congruent? BC KL January 8, 2016 5.7 Using Congruent Triangles L Example 2 Explain how you can use the given information to prove the hang glider parts are congruent. January 8, 2016 The triangles are congruent by ______. AAS This means parts like QT and ST are CPCTC congruent because ________. 5.7 Using Congruent Triangles Example 3 SSS CPCTC January 8, 2016 5.7 Using Congruent Triangles Example 4 Surveyors use this strategy to measure difficult distances. Explain how you can use the given information to find the distance across the river. ASA CPCTC January 8, 2016 5.7 Using Congruent Triangles Proofs Show that two triangles are congruent. Then show corresponding parts are congruent. CPCTC January 8, 2016 5.7 Using Congruent Triangles Example 5 Given: HJ || LK and JK || HL Prove: H K Plan: Show JHL LKJ by ASA, then use CPCTC. J H Statements Reasons 1. HJL KLJ (Alt Int s) L K 2. LJ LJ 3.HLJ KJL 4. JHL LKJ 5. H K QED January 8, 2016 5.7 Using Congruent Triangles (Reflexive) (Alt Int s) (ASA) (CPCTC) Example 6 Given: MS || TR and MS TR Prove: A is the midpoint of MT. M Since MS || TR, M T (Alt. Int. s) SAM RAT (Vert. s) R MS TR (Given) A SAM RAT (AAS) S T Midpoint Definition Plan: Show the triangles are MA AT (CPCTC) congruent using AAS, then If M is the midpoint of MA =AT. By definition, A is the AB, then AM MT. MB. midpoint of segment A is the midpoint of MT (Def. midpoint) January 8, 2016 5.7 Using Congruent Triangles Example 7 Given: MP bisects LMN and LM NM L LM NM PM PM PMN PML LP NP QED M January 8, 2016 (Given) NMP LMP (def. bis) Prove: LP NP P N MP bis. LMN 5.7 Using Congruent Triangles (Given) (Ref) (SAS) (CPCTC) Your Turn Try the next two proofs on your own. January 8, 2016 5.7 Using Congruent Triangles Your Turn Proof 1 Given: AB DC, AD BC Prove: A C A B Statements Reasons 1. AB DC 1. Given 2. AD BC 2. Given 3. BD BD 3. Reflexive 4. ABD CDB 4. SSS D C January 8, 2016 5. A C 5.7 Using Congruent Triangles 5. CPCTC Your Turn Proof 2 A E C B D 1. AC DC 2. A D (given) (given) 3. ACB DCE (vert s) 4. ACB DCE (ASA) (CPCTC) 5. B E January 8, 2016 5.7 Using Congruent Triangles Proofs Ask: to show angles or segments congruent, what triangles must be congruent? Then, how do you prove triangles congruent? (SSS, SAS, ASA, AAS) Prove triangles congruent, then use CPCTC. January 8, 2016 5.7 Using Congruent Triangles Assignment January 8, 2016 5.7 Using Congruent Triangles