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Transcript
Geometry Lesson 2.7 NOTES Prove Angle Pair Relationships Rode2010 Theorem 2.3 Right Angle Congruence Theorem All right angles are congruent. PROOF Given ∠1 and ∠2 are right angles Prove ∠1 ≅ ∠2 Statements 1. ∠1 and ∠2 are right angles 2. m∠1 = 90o , ∠2 = 90o 3. m∠1 = m∠2 4. ∠1 ≅ ∠2 Reasons 1. Given 2. Definition of right angle 3. Transitive property of equality 4. Definition of congruent angles Example 1 Write a proof. ̅̅̅̅ ⊥ 𝐵𝐶 ̅̅̅̅ , 𝐷𝐶 ̅̅̅̅ ⊥ 𝐵𝐶 ̅̅̅̅ Given: 𝐴𝐵 Prove: ∠𝐵 ≅ ∠𝐶 Statements ̅̅̅̅ ⊥ 𝐵𝐶 ̅̅̅̅ , 𝐷𝐶 ̅̅̅̅ ⊥ 𝐵𝐶 ̅̅̅̅ 1. 𝐴𝐵 2. ∠𝐵 and ∠𝐶 are right angles 3. ∠𝐵 ≅ ∠𝐶 2 1 D C A Reasons 1. Given 2. Definition of perpendicular lines 3. Right angle congruence theorem B Theorem 2.4 Congruent Supplements Theorem If two angles are supplementary to the same angle then they are congruent. Theorem 2.5 Congruent Complements Theorem If two angles are complementary to the same angle then they are congruent. Example 2 Prove Theorem 2.4: Two angles are supplementary to the same angle are congruent. Given: ∠1 and ∠2 are supplements ∠3 and ∠2 are supplements 1 Prove: ∠1 ≅ ∠3 Statements 1. ∠1 and ∠2 are supplements ∠3 and ∠2 are supplements 2. m∠1 + m∠2 =180o m∠3 + m∠2 =180o 3. m∠1 + m∠2 = m∠3 + m∠2 4. m∠1 = m∠3 5. ∠1 ≅ ∠3 Reasons 1. Given 2. Definition of supplementary angles 3. Transitive Property of Equality 4. Subtraction Property of Equality 5. Definition of congruent angles 2 3 Guided Practice On Worksheet 2.7 – problems 1+2 When two lines intersect, pairs of vertical angles and linear pairs are formed (lesson 1.5). Linear Pair Postulate If two angles form a linear pair, then they are supplementary. 1 2 ∠1 and ∠2 form a linear pair, so ∠1 and ∠2 are supplementary and m∠1 + m∠2 =180o Theorem 2.6 Vertical Angles Congruence Theorem Vertical angles are congruent. ∠1 ≅ ∠3 ∠2 ≅ ∠4 Example 3 Prove the Vertical Angles Congruence Theorem Given: ∠5 and ∠7 are vertical angles Prove: ∠5 ≅ ∠7 Statements 1. ∠5 and ∠7 are vertical angles 2. ∠5 and ∠6 are a linear pair ∠6 and ∠7 are a linear pair 3. ∠5 and ∠6 are supplementary ∠6 and ∠7 are supplementary 4. ∠5 ≅ ∠7 2 1 4 3 7 5 6 Reasons 1. Given 2. Definition of a linear pair, as shown in the diagram. 3. Linear Pair Postulate 4. Congruent Supplements Theorem Guided Practice on Worksheet 2.7 – problems 3+4