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Proving Statements About Angles: Angle Congruence and Properties

In this geometry lesson, students will learn and apply geometric concepts related to angle congruence. The objectives include using properties of angle congruence and proving properties about special pairs of angles. Students will explore theorems such as the reflexive, symmetric, and transitive properties of angle congruence. They will also understand the concepts of congruent supplements and complements. Through examples and proofs, students will gain a comprehensive understanding of angle relationships and congruence.

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Proving Statements About Angles: Angle Congruence and Properties

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  1. 2.6 Proving Statements about Angles Mrs. Spitz Geometry Fall 2004

  2. Standards/Objectives Standard 3: Students will learn and apply geometric concepts. Objectives: • Use angle congruence properties • Prove properties about special pairs of angles.

  3. Theorem 2.2 Properties of Angle Congruence • Angle congruence is reflexive, symmetric, and transitive. • Examples: • Reflexive: For any angle A, A ≅ A. • Symmetric: If A ≅ B, then B ≅ A • Transitive: If A ≅ B and B ≅ C, then A ≅ C.

  4. Ex. 1: Transitive Property of Angle Congruence • Prove the Transitive Property of Congruence for angles Given: A ≅ B, B ≅ C Prove: A ≅ C C B A

  5. Statement: A ≅ B, B ≅ C mA = mB mB = mC mA = mC A ≅ C Reason: Given Def. Cong. Angles Def. Cong. Angles Transitive property Def. Cong. Angles Ex. 1: Transitive Property of Angle Congruence

  6. Ex. 2: Using the Transitive Property Given: m3 ≅ 40, 1 ≅ 2, 2 ≅ 3 Prove: m1 =40 1 4 2 3

  7. Statement: m3 ≅ 40, 1 ≅ 2, 2 ≅ 3 1 ≅ 3 m1 = m3 m1 = 40 Reason: Given Trans. Prop of Cong. Def. Cong. Angles Substitution Ex. 2:

  8. Theorem 2.3 All right angles are congruent. Example 3: Proving Theorem 2.3 Given: 1 and 2 are right angles Prove: 1 ≅ 2

  9. Statement: 1 and 2 are right angles m1 = 90, m2 = 90 m1 = m2 1 ≅ 2 Reason: Given Def. Right angle Transitive property Def. Cong. Angles Ex. 3:

  10. Properties of Special Pairs of Angles • Theorem 2.4: Congruent Supplements. If two angles are supplementary to the same angle (or to congruent angles), then they are congruent. If m1 +m2 = 180 AND m2 +m3 = 180, then 1 ≅ 3.

  11. Congruent Complements Theorem • Theorem 2.5: If two angles are complementary to the same angle (or congruent angles), then the two angles are congruent. If m4 +m5 = 90 AND m5 +m6 = 90, then 4 ≅ 6.

  12. Proving Theorem 2.4 Given: 1 and 2 are supplements, 3 and 4 are supplements, 1 ≅ 4 Prove: 2 ≅ 3 3 4 1 2

  13. Statement: 1 and 2 are supplements, 3 and 4 are supplements, 1 ≅ 4 m 1 + m 2 = 180; m 3 + m 4 = 180 m 1 + m 2 =m 3 + m 4 m 1 = m 4 m 1 + m 2 =m 3 + m 1 m 2 =m 3 2 ≅ 3 Reason: Given Def. Supplementary angles Transitive property of equality Def. Congruent Angles Substitution property Subtraction property Def. Congruent Angles Ex. 4:

  14. Postulate 12: Linear Pair Postulate If two angles form a linear pair, then they are supplementary. 1 2 m 1 + m 2 = 180

  15. Example 5: Using Linear Pairs In the diagram, m8 = m5 and m5 = 125. Explain how to show m7 = 55 7 8 5 6

  16. Solution: • Using the transitive property of equality m8 = 125. The diagram shows that m 7 + m 8 = 180. Substitute 125 for m 8 to show m 7 = 55.

  17. Vertical Angles Theorem • Vertical angles are congruent. 2 3 1 4 1≅ 3; 2≅ 4

  18. Proving Theorem 2.6 Given: 5 and 6 are a linear pair, 6 and 7 are a linear pair Prove: 5 7 5 7 6

  19. Statement: 5 and 6 are a linear pair, 6 and 7 are a linear pair 5 and 6 are supplementary, 6 and 7 are supplementary 5 ≅ 7 Reason: Given Linear Pair postulate Congruent Supplements Theorem Ex. 6: Proving Theorem 2.6

  20. Your Assignment • In-Class--2.6 pp. 112-114 #1-28 Coming attractions: • Test Ch. 2 and Binder Check Wednesday • Friday—Chapter 3 Definitions on page 128 and Postulates/Theorems • Chapter 2 Review/Algebra Review may be completed for extra credit. SHOW ALL WORK!

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