4 6 Triangle Congruence ASA AAS and HL

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 Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Vocabulary included side Holt Mc. Dougal 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 Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL ASA Postulate Holt Mc. Dougal 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 Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Check It Out! Example 2 Given: Prove: NKL LMN. Holt Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Statements Reasons 1. Given 2. Alternate Interior Angles Theorem 3. Reflexive Property of 4. ASA Postulate Holt Mc. Dougal Geometry

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-Side Theorem, or AAS Theorem. Holt Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Holt Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Check It Out! Example 3 Given: JL bisects KLM, K M Prove: JKL JML Holt Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Holt Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Statements Holt Mc. Dougal Geometry Reasons

4 -6 Triangle Congruence: ASA, AAS, and HL Holt Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Example 4 A: 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 Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Example 4 B: 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 Mc. Dougal 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 Mc. Dougal Geometry

4 -6 Triangle Congruence: ASA, AAS, and HL Homework: Pg 264 – 265, #4 – 17, 19, 22, 23 Holt Mc. Dougal Geometry