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4-6 Triangle Congruence: ASA, AAS, and HL Learning Targets I will apply the ASA Postulate, the AAS Theorem, and the HL Theorem to construct triangles and to solve problems. I will prove triangles congruent by using the ASA Postulate, AAS Theorem, and HL Theorem. Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Vocabulary included side Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side. Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL ASA Postulate Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Example: Applying ASA Congruence Determine if you can use the ASA Postulate to prove the triangles congruent. Explain. Two congruent angle pairs are give, but the included sides are not given as congruent. Therefore ASA cannot be used to prove the triangles congruent. Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Check It Out! Example 2 Given: KNL MLN , KL || MN Prove: NKL LMN. Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Statements Reasons 1. KNL MLN , KL || MN 1. Given 2. Alternate Interior Angles Theorem 2. KLN MNL 3. NL NL 4. NKL LMN Holt McDougal Geometry 3. Reflexive Property of 4. ASA Postulate 4-6 Triangle Congruence: ASA, AAS, and HL You can use the Third Angles Theorem to prove another congruence relationship based on ASA. This theorem is Angle-Angle-Side Theorem, or AAS Theorem. Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Check It Out! Example 3 Given: JL bisects KLM, K M Prove: JKL JML Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Statements 1. JL bisects KLM , K M 2. KLJ MLJ 3. JL JL 4. JKL JML Holt McDougal Geometry Reasons 1. 2. 3. 4. Given Def. Angle Bisector Reflexive Property of AAS Theorem 4-6 Triangle Congruence: ASA, AAS, and HL Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Example 4A: Applying HL Theorem Determine if you can use the HL Theorem to prove the triangles congruent. If not, tell what else you need to know. According to the diagram, the triangles are right triangles that share one leg. It is given that the hypotenuses are congruent, therefore the triangles are congruent by HL. Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Example 4B: Applying HL Theorem This conclusion cannot be proved by the HL Theorem. According to the diagram, the triangles are right triangles and one pair of legs is congruent. You do not know that one hypotenuse is congruent to the other. Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Check It Out! Example 4 Determine if you can use the HL Theorem to prove ABC DCB. If not, tell what else you need to know. Yes; it is given that AC DB. BC CB by the Reflexive Property of Congruence. Since ABC and DCB are right angles, ABC and DCB are right triangles. ABC DCB by HL. Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Homework: Pg 264 – 265, #4 – 17, 19, 22, 23 Holt McDougal Geometry