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Triangle Congruence: ASA, AAS, and HL

This lesson presentation introduces the concepts of ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and HL (Hypotenuse-Leg) to prove triangle congruence. Examples and practice problems are provided for better understanding.

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Triangle Congruence: ASA, AAS, and HL

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  1. 4-6 Triangle Congruence: ASA, AAS, and HL Holt Geometry Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry

  2. AC • Warm Up • 1.What are sides AC and BC called? Side AB? • 2. Which side is in between A and C? • 3. Given DEF and GHI, if D  G and E  H, why is F  I? legs; hypotenuse Third s Thm.

  3. Objectives Apply ASA, AAS, and HL to construct triangles and to solve problems. Prove triangles congruent by using ASA, AAS, and HL.

  4. Vocabulary included side

  5. COPY THIS SLIDE: An included side is the common side of two consecutive angles in a polygon. The following postulate uses the idea of an included side.

  6. COPY THIS SLIDE:

  7. Example 2: Applying ASA Congruence COPY THIS SLIDE: Determine if you can use ASA to prove the triangles congruent. Explain. No, 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.

  8. 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. Check It Out! Example 2 COPY THIS SLIDE: Determine if you can use ASA to prove NKL LMN. Explain.

  9. COPY THIS SLIDE:

  10. Example 3: Using AAS to Prove Triangles Congruent COPY THIS SLIDE: Prove the triangles congruent. Given:X  V, YZW  YWZ, XY  VY Prove: XYZ  VYW  XYZ  VYW by AAS

  11. COPY THIS SLIDE:

  12. Example 4A: Applying HL Congruence COPY THIS SLIDE: 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. The hypotenuses are congruent b/c of the tick marks. They share a leg, so by the reflexive property, the leg is congruent to itself. Therefore the triangles are congruent by HL.

  13. Example 4B: Applying HL Congruence COPY THIS SLIDE: 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.

  14. Yes; AC DB b/c of congruency marks. BC  CB by the Reflexive Property. Since ABC and DCB are right triangles, ABC  DCB by HL. Check It Out! Example 4 Determine if you can use the HL Congruence Theorem to prove ABC  DCB. If not, tell what else you need to know.

  15. Examples: Identify the postulate or theorem that proves the triangles congruent. HL ASA SAS or SSS

  16. Classwork/Homework • 4.6 SSS, SAS, ASA, AAS W/S

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