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andHL HL 4-6 TriangleCongruence: Congruence: ASA, ASA, AAS, AAS, and 4-6 Triangle Warm Up Lesson Presentation Lesson Quiz Holt HoltGeometry McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Warm Up 1. What are sides AC and BC called? Side AB? legs; hypotenuse 2. Which side is in between A and C? AC 3. Given DEF and GHI, if D G and E H, why is F I? Third s Thm. Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Objectives Apply ASA, AAS, and HL to construct triangles and to solve problems. Prove triangles congruent by using ASA, AAS, and HL. Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Can someone explain what the “included” angle means? 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 Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Example 1: Applying ASA Congruence Determine if you can use ASA to prove the triangles congruent. Explain. Two congruent angle pairs are given, but the included sides are not given as congruent. So, we cannot use ASA. Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Check It Out! Example 1 Determine if you can use ASA to prove NKL LMN. Explain. By the Alternate Interior Angles Theorem. KLN MNL. NL LN by the Reflexive Property. No other congruence relationships can be determined, so ASA cannot be applied. Holt McDougal 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-Angle-Side (AAS). Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Check It Out! Example 2 Prove the triangles congruent. Given: JL bisects KLM, K M Prove: JKL JML A A S S A A Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL ✔ ✔✔ USE: AAS Statements Reasons 1. ÐK @ ÐM 1. Given A 2. JL bisects ÐKLM 2. Given 3. ÐKLJ @ ÐMLJ 4. JL @ JL 3. An angle bisector cuts an A angle into 2 congruent angles 4. Reflexive Property S 5. DJKL @ DJML 5. SAS Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL The bad word one… • SSA or ASS….. • We cannot use this to prove two triangles are congruent. • It’s a bad word, it’s bad news, and it doesn't’t work… so NEVER use it! Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL I think of this one as rASS Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL • In a HL proof you need 3 things The hypotenuses The legs Tell us that the triangles have a right angle, thus making them right triangles. Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Example 3A: Applying HL Congruence Determine if you can use the HL Congruence 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 3B: Applying HL Congruence This conclusion cannot be proved by HL. 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 3 Determine if you can use the HL Congruence Theorem to prove ABC DCB. If not, tell what else you need to know. Yes, so lets prove it! Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL ✔ ✔ ✔ USE: HL-RT Statements Reasons 1. AC @ BD 2. CB @ CB 1. Given 3. ÐABCandÐDCB 3. Given H 2. Reflexive Property areright angles 4. DABCand DDCB are right triangles 4. A right triangle RT has 1 right angle 5. DABC @ DDCB 5. HL Holt McDougal Geometry L 4-6 Triangle Congruence: ASA, AAS, and HL Lesson Quiz: Part I Identify the postulate or theorem that proves the triangles congruent. ASA HL SAS or SSS Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Lesson Quiz: Part II 4. Given: FAB GED, ABC DCE, AC EC Prove: ABC EDC Holt McDougal Geometry 4-6 Triangle Congruence: ASA, AAS, and HL Lesson Quiz: Part II Continued Statements Reasons 1. FAB GED 1. Given 2. BAC is a supp. of FAB; DEC is a supp. of GED. 2. Def. of supp. s 3. BAC DEC 3. Supp. Thm. 4. ACB DCE; AC EC 4. Given 5. ABC EDC 5. ASA Holt McDougal Geometry