Triangle Congruence Proofs 4 Objectives: G.CO.8: Explain how the criteria for triangle congruence (ASA,SAS, SSS, and AAS) follow from the definition of congruence in terms of rigid motions. G.CO.7: Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.SRT.5: Use congruence and similarity criteria for triangles to solve problems and prove relationships in geometric figures. For the Board: You will be able to use congruent triangles to prove segments and angles congruent. Bell Work: State whether or not you can use the following information to prove two triangles are congruent 1. ASA 2. SSA 3. SAS 4. SSS 5. AAS 6. AAA Anticipatory Set: Definition of Congruent Triangles (CPCTC) Two triangles are congruent if and only if all three angle and all three sides of one triangle are congruent to the corresponding three angles and three sides of another triangle. Instruction: B Given: AB||CD, BC||DA C Prove: AB CD A D Proof: Statements Reasons 1. AB||CD, BC||DA 1. Given 2. <CBD <ADB 2. Alternate Interior Angles Theorem 3. <ABD <CDB 3. Alternate Interior Angles Theorem 4. BD BD 4. Reflexive Property of Congruence 5. ΔBAC ΔDCB 5. ASA 6. AB CD 6. CPCTC M Given: A midpoint of MT and SR R Prove: <M <T A Proof: Statements Reasons 1. A midpoint MT and SR 1. Given S 2. MA TA, SA RA 2. Definition of Midpoint T 3. <MAS <TAR 3. Vertical Angle Theorem 4. ΔMAS ΔTAR 4. SAS 5. <SMA <RTA 5. CPCTC Base Angles Theorem A If two sides of a triangle are congruent then the angles opposite them are congruent. Given: AB AC Prove: <B <C. Proof: Statements Reasons B C 1. Let X be the midpoint of BC 1. Every segment has a unique midpoint. 2. Draw AX 2. Through two points there is exactly one segment. 3. AB AC 3. Given 4. BX CX 4. Definition of Midpoint 4. AD AD 5. Reflexive Property of Congruence 6. ΔABD ΔACD 6. SSS 6. <B <C 7. CPCTC Base Angles Converse Theorem If two angles of a triangle are congruent then the sides opposite them are congruent. Given: <B <C Prove: AB AC. Proof: Statements Reasons 1. Draw the bisector of <A 1. Every angle has a unique bisector. 2. The bisector intersects BC at X 2. Two lines intersect in at most 1 point. 3. <B <C 3. Given 4. <BAX <CAX 4. Definition of Angle Bisector 5. AD AD 5. Reflexive Property of Congruence 5. ΔABX ΔACX 5. AAS 6. AB AC 6. CPCTC Assessment: Question student pairs. Independent Practice: Handout 4 .