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Understanding Proof-Oriented Triangle Theorems: Congruent Angles and AAS

In this lesson, we explore two important triangle theorems. Theorem 53 states that if two angles of one triangle are congruent to two angles of another triangle, then their corresponding third angles are also congruent. Theorem 54 asserts that if there is a correspondence between two triangles such that two angles and a non-included side of one triangle are congruent to those of another, the triangles themselves are congruent (AAS). We'll demonstrate these concepts through examples and solve related problems.

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Understanding Proof-Oriented Triangle Theorems: Congruent Angles and AAS

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  1. Two Proof-Oriented Triangle Theorems Lesson 7.2

  2. Theorem 53: If 2 angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent. (no-choice Theorem) F C B A D E If <A congruent <D <B congruent <E Then <C congruent <F Since the sum = 180 subtract and get <C congruent <F The triangles do not have to be congruent, the angles do!

  3. Theorem 54: If there exists a correspondence between the vertices of two triangles such that two angles and a non-included side of one triangle are congruent to the corresponding parts of the other, then the triangles are congruent. (AAS)

  4. J Given: JM  GM GK  KJ Conclude: <G  <J K G H M 1. JM  GM, GK  KJ 2. GMJ, JKG rt s 3. GMJ  JKG 4. GHM, JHK vert s 5. GHM  JHK 6. G  J • Given •  lines from rt s • Rt s are  • Assumed from diagram • Vert. s are  • No Choice Theorem

  5. Given: Triangle as marked. Find the m 1. 60 3x-5 2x+5 1 By Ext  Theorem 3x – 5 = 60 + (2x + 5) 3x – 5 = 65 + 2x x = 60 1 is supp to (3x – 5) Then 1 + (3x – 5) = 180 1 + 3(60) – 5 = 180 1 + 180 – 5 = 180 1 + 175 = 180 1 = 5

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