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Understanding Triangle Congruence: ASA, AAS, and HL Theorem in Geometry

This presentation covers triangle congruence, specifically focusing on the ASA, AAS, and HL theorems. Students will learn how to apply the Hypotenuse-Leg (HL) theorem to construct triangles and solve problems related to congruence. The lesson includes a warm-up section with key questions, vocabulary, and examples demonstrating how to prove triangles congruent using HL. Additionally, the presentation features quizzes to test understanding and offers a structured approach to proving congruence via two-column proofs.

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Understanding Triangle Congruence: ASA, AAS, and HL Theorem in Geometry

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  1. 4-5 Triangle Congruence: ASA, AAS, and HL Holt Geometry Warm Up Lesson Presentation Lesson Quiz

  2. 4.5 Congruent Triangles by HL 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. 4.5 Congruent Triangles by HL Objectives Apply HL to construct triangles and to solve problems. Prove triangles congruent by using HL.

  4. 4.5 Congruent Triangles by HL Vocabulary included side

  5. 4.5 Congruent Triangles by HL

  6. 4.5 Congruent Triangles by HL Example 1: 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.

  7. 4.5 Congruent Triangles by HL Example 2: 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.

  8. 4.5 Congruent Triangles by HL 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. Check It Out! Example 3 Determine if you can use the HL Congruence Theorem to prove ABC  DCB. If there is, provide a paragraph and two column proof. If not, tell what else you need to know.

  9. 4.5 Congruent Triangles by HL Statements Reasons 1. ABC  DCB 1. Given Right Angle 2. AC  DB. 2. Given Hypotenuse 3. CB  BC 3. Reflexive Prop. Leg 4. ABC  DCB 4. By HL Check It Out! Example 3 Continue

  10. 4.5 Congruent Triangles by HL Lesson Quiz: Part I Identify the postulate or theorem that proves the triangles congruent and create a two column proof. A HL B C D

  11. 4.5 Congruent Triangles by HL Statements Reasons 1. ADB  ADC 1. Given Right Angle 2. AB  AC. 2. Given Hypotenuse 3. CD  BD 3. Given Leg 4. ADC  ADB 4. By HL A Lesson Quiz: Part I Continued C B D

  12. 4.5 Congruent Triangles by HL Lesson Quiz: Part II Identify the postulate or theorem that proves the triangles congruent and create a two column proof. A HL B C D

  13. 4.5 Congruent Triangles by HL Statements Reasons 1. ADB  ADC 1. Given Right Angle 2. AB  AC. 2. Given Hypotenuse 3. CD  BD 3. Given Leg 4. ADC  ADB 4. By HL A Lesson Quiz: Part Ii Continued C B D

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