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4.5 – Prove Triangles Congruent by ASA and AAS Angle-Side-Angle (ASA) Congruence Postulate E A 4cm A C AB CD B D F B C 4cm D If two angles and the ______________ included side of congruent to two angles one triangle are _____________ included and the ____________ side of a second triangle, then the two triangles are ________________. congruent Angle-Angle-Side (AAS) Congruence Theorem E A F B C D OR Angle-Angle-Side (AAS) Congruence Theorem E A F B C D nonincluded If two angles and a _____________________ congruent side of one triangle are _______________ to two angles and the corresponding nonincluded ____________________ side of a second triangle, then the two triangles are congruent ________________. Does AAA work? Does AAA work? NO! Does ASS work? No! There is no ASS in geometry! Given: AB CD AC BD Prove: ∆ABC ∆DCB Statements Reasons 1. AB CD 1. 2. BCD CBA 3. AC BD 2. 3. 4. 5. 6. 4. BCA CBD 5. BC BC 6. ∆ABC ∆DCB Given alternate interior angles Given alternate interior angles Reflexive Prop. ASA Given: AE bisects DC at B A E Prove: ABD EBC Statements Reasons 1. AE bisects DC at B 1. Given 2. Def. of segment bisector 2. DB BC 3. A E 4. ABD EBC 5. ∆ABD ∆EBC 3. Given 4. vertical angles 5. AAS Decide whether the triangles are congruent. Explain your reasoning. Yes, ASA Decide whether the triangles are congruent. Explain your reasoning. Yes, AAS Decide whether the triangles are congruent. Explain your reasoning. Yes, SAS Decide whether the triangles are congruent. Explain your reasoning. No, ASS doesn’t work! Decide whether the triangles are congruent. Explain your reasoning. Yes, AAS