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Exploring Proof-Oriented Triangle Theorems: Angles Congruence and Triangle Congruence

In this lesson on triangle theorems, we cover two essential principles: Theorem 53, which states that if two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent. Theorem 54 presents the Angle-Angle-Side (AAS) criterion for triangle congruence, asserting that if two angles and a non-included side of one triangle correspond to the other triangle, the triangles are congruent. These theorems help students understand the relationships between angles and sides in triangles.

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Exploring Proof-Oriented Triangle Theorems: Angles Congruence and Triangle Congruence

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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 x+5 1 By Ext  Theorem 3x – 5 = 60 + (x + 5) 3x – 5 = 65 + x 2x = 70 x = 35 1 is supp to (3x – 5) Then 1 + (3x – 5) = 180 1 + 3(35) – 5 = 180 1 + 105 – 5 = 180 1 + 100 = 180 1 = 80

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