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Given: π΅πΊ β₯ πΉπ», πΉπΊ β πΊπ» Prove βπΉπ΅πΊ β βπ»πΊπ΅ B Statements Reasons 1. π΅πΊ β₯ πΉπ», πΉπΊ β πΊπ» 1. Given 2. β π΅πΊπΉ πππ β π΅πΊπ» = 90° 2. Definition of β₯ 3. β π΅πΊπΉ β β π΅πΊπ» 3. Right Angle Congruence Theorem 4. π΅πΊ β π΅πΊ 4. Reflexive Property 5. βπΉπ΅πΊ β βπ»πΊπ΅ 5. SAS F G H Chapter 4.5: Triangle Congruence Continues You will learn two more ways to prove two triangles are congruent Activity 1: ASA. β’ Below is a partially drawn triangle. In this case, AB has been drawn and two angles have been created. If you extend two sides from β a and β b, how many different triangles can you create? A B Postulate 21: Angle-Side-Angle (ASA) Congruence β’ If two Angles and the included Side of one triangle are congruent to two Angles and an included Side of another triangle, then the two triangles are congruent. Using ASA in a Proof P Q S R Given: ππ β₯ ππ , ππ β₯ ππ Prove: βπππ β βπ ππ Statements Reasons 1. ππ β₯ ππ , ππ β₯ ππ 1. Given 2. β πππ β β π ππ 2. Alternate Interior Angles 3. β πππ β β π ππ 3. Alternate Interior Angles 4. ππ β ππ 4. Reflexive Property 5. βπππ β βπ ππ 5. ASA Flow Chart Proofs P Q S R Given: ππ β₯ ππ , ππ β₯ ππ Prove: βπππ β βπ ππ Flow Chart Proofs Q P Given: ππ β₯ ππ , ππ β₯ ππ Prove: βπππ β βπ ππ ππ β₯ ππ Given ππ β₯ ππ Given S R β πππ β β π ππ Alternate Interior Angles β πππ β β π ππ Alternate Interior Angles ππ β ππ Reflexive Property βπππ β βπ ππ ASA Postulate 21: Angle-Angle-Side (AAS) Congruence β’ If two Angles and a nonincluded Side of one triangle are congruent to two Angles and the corresponding nonincluded Side of another triangle, then the two triangles are congruent. Using ASA to prove AAS Given: β π β β π, β π β β π, π¨π© β πΏπ Prove: βπ¨π©πͺ β βπΏππ ASA Using ASA to prove AAS Given: β π β β π, β π β β π, π¨π© β πΏπ Prove: βπ¨π©πͺ β βπΏππ β π β β π Given β π β β π Third Angle Theorem β π β β π Given π¨π© β πΏπ Given βπ¨π©πͺ β βπΏππ ASA Triangle Congruence Theorems and Postulates Theorems and Postulates the Prove Triangle Congruence: β’ β’ β’ β’ β’ Side-Side-Side (SSS) Side-Angle-Side (SAS) Hypotenuse-Leg (HL) Angle-Side-Angle (ASA) Angle-Angle-Side (AAS) These do not prove congruence: Do not Use β’ Angle-Angle-Angle (AAA) β’ The Donkey Conjecture: (SSA)
