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Name:_________________________ 2-8 Proving Angle Relationships Notes Objectives: Students will write proofs involving supplementary and complementary angles Students will write proofs involving congruent and right angles. Postulate 2.11: Angle Addition Postulate: If S is in the interior of ∠𝑃𝑄𝑅, then 𝑚∠𝑃𝑄𝑆 + 𝑚∠𝑆𝑄𝑅 = 𝑚∠𝑃𝑄𝑅. Example 1: Using the Angle Addition Postulate: If 𝑚∠𝐴𝐵𝐶 = 103 find the value of x. Then find 𝑚∠𝐴𝐵𝐷 and 𝑚∠𝐷𝐵𝐶 x= 𝑚∠𝐴𝐵𝐷 = 𝑚∠𝐷𝐵𝐶 = Supplementary and Complementary Angles Theorem 2.3: If two angles form a ___________ ____________, then they are ___________________ angles. 1 2 𝑚∠1 + 𝑚∠2 = 180 Theorem 2.4: If the non-common sides of two _____________ angles form a ____________ angle, then the angles are complimentary angles. 1 1 2 𝑚∠1 + 𝑚∠2 = 90 Example 2: Using Supplementary Angles If ∠1 and ∠2 form a linear pair, and 𝑚∠1 = 4𝑥 − 5 and the 𝑚∠2 = 14𝑥 + 5, find x and the measurements of ∠1 and ∠2. x= ∠1 = ∠2 = 1 2 Theorem 2.5 Angle Congruence Reflexive Property ∠𝟏 ≅ ∠𝟏 Symmetric Property If ∠𝟏 ≅ ∠𝟐, ∠𝟐 ≅ ∠𝟏 Transitive Property If ∠𝟏 ≅ ∠𝟐 and ∠𝟐 ≅ ∠𝟑, then ∠𝟏 ≅ ∠𝟑 Theorem 2.6: Angles supplementary to the same angle or to congruent angles are congruent. If 𝑚∠1 + 𝑚∠2 = 180, and 𝑚∠2 + 𝑚∠3 = 180, then ______ ≅ _________. Example 3: Proof of Theorem 2.6 Given: ∠1 and ∠2 are supplementary ∠2 and ∠3 are supplementary Prove: ∠1 ≅ ∠3 Statements Reasons 1. ∠1 and ∠2 are supplementary ∠2 and ∠3 are supplementary 2. 𝑚∠1 + 𝑚∠2 = 180 𝑚∠2 + 𝑚∠3 = 180 1. Given 3. 3. Substitution. 4. 𝑚∠1 = 𝑚∠3 4. 5. ∠1 ≅ ∠3 5. 2. Theorem 2.7: Angles complementary to the same angle or to congruent angles are congruent. 1 If 𝑚∠1 + 𝑚∠2 = 90, and 𝑚∠2 + 𝑚∠3 = 90, then ______ ≅ _________. 2 3 Theorem 2.8: If two angles are vertical angles, then they are congruent ( ≅ ) Example 4: Prove Vertical Angles are ≅ Given: ∠1 and ∠2 form a linear pair. ∠2 and ∠3 form a linear pair. Prove: ∠1 ≅ ∠3 Statements Reasons 1. ∠1 and ∠2 form a linear pair ∠2 and ∠3 form a linear pair 1. Given 2. ∠1 and ∠2 are supplementary ∠2 and ∠3 are supplementary 2. ______________________________ 3. ∠1 ≅ ∠3 3. ______________________________ ∠1 ≅ ∠3 and ∠2 ≅ ∠4 Example 5: Using Vertical Angles Find the value of x using vertical angles. Theorem 2.9: Perpendicular lines intersect to four ______________________. Theorem 2.10: All right angles are congruent. Theorem 2.11: Perpendicular lines form congruent adjacent angles. Theorem 2.12: If two angles are congruent and supplementary, then each angle is a __________________. Theorem 2.13: If two congruent angles form a linear pair, then they are ___________________.