Triangle Congruency Proofs

New York State Common Core Math Geometry, Module 1, Lesson 26 & Lesson 27

Worksheets for Geometry

Student Outcomes

• Students complete proofs requiring a synthesis of the skills learned in the last four lessons.

Triangle Congruency Proofs

Classwork

Exercises for Lesson 26

1. Given: 𝐴𝐵 ⊥ 𝐵𝐶, 𝐵𝐶 ⊥ 𝐷𝐶
   𝐷𝐵 bisects ∠𝐴𝐵𝐶, 𝐴𝐶 bisects ∠𝐷𝐶𝐵
   𝐸𝐵 = 𝐸𝐶
   Prove: △ 𝐵𝐸𝐴 ≅ △ 𝐶𝐸𝐷

2. Given: 𝐵𝐹 ⊥ 𝐴𝐶, 𝐶𝐸 ⊥ 𝐴𝐵
   𝐴𝐸 = 𝐴𝐹
   Prove: △ 𝐴𝐶𝐸 ≅△ 𝐴𝐵𝐹
   

3. Given: 𝑋𝐽 = 𝑌𝐾, 𝑃𝑋 = 𝑃𝑌, 𝑚∠𝑍𝑋𝐽 = 𝑚∠𝑍𝑌𝐾
   Prove: 𝐽𝑌 = 𝐾𝑋

4. Given: 𝐽𝐾 = 𝐽𝐿, 𝐽𝐾 ∥ 𝑋𝑌
   Prove: 𝑋𝑌 = 𝑋𝐿

5. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4
   Prove: 𝐴𝐶 ≅ 𝐵𝐷

6. Given: 𝑚∠1 = 𝑚∠2, 𝑚∠3 = 𝑚∠4, 𝐴𝐵 = 𝐴𝐶
   Prove: (a) △ 𝑨𝑩𝑫 ≅ △ 𝑨𝑪𝑫
   (b) 𝒎∠𝟓 = 𝒎∠6

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Exercises for Lesson 27

1. Given: 𝐴𝐵 = 𝐴𝐶, 𝑅𝐵 = 𝑅𝐶
   Prove: 𝑆𝐵 = 𝑆𝐶

2. Given: Square 𝐴𝐵𝐶𝑆 ≅ Square 𝐸𝐹𝐺𝑆,
   𝑅𝐴𝐵, 𝑅𝐸𝐹
   Prove: △ 𝐴𝑆𝑅 ≅ △ 𝐸𝑆𝑅

3. Given: 𝐽𝐾 = 𝐽𝐿,𝐽𝑋 = 𝐽𝑌
   Prove: 𝐾𝑋 = 𝐿𝑌

4. Given: 𝐴𝐷 ⊥ 𝐷𝑅, 𝐴𝐵 ⊥ 𝐵𝑅,
   𝐴𝐷 ≅ 𝐴𝐵
   Prove: ∠𝐷𝐶𝑅 ≅ ∠𝐵𝐶𝑅

5. Given: 𝐴𝑅 = 𝐴𝑆, 𝐵𝑅 = 𝐶𝑆,
   𝑅𝑋 ⊥ 𝐴𝐵, 𝑆𝑌 ⊥ 𝐴𝐶
   Prove: 𝐵𝑋 = 𝐶𝑌

6. Given: 𝐴𝑋 = 𝐵𝑋, 𝑚∠𝐴𝑀𝐵 = 𝑚∠𝐴𝑌𝑍 = 90°
   Prove: 𝑁𝑌 = 𝑁𝑀

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