Properties of Parallel Lines Skill 15 Objective HSGCO

Properties of Parallel Lines Skill 15

Objective HSG-CO. 9: Students are responsible for proving theorems about parallel lines, and using properties of parallel lines to find angle measures.

Definition Parallel lines are coplanar lines that do not intersect the symbol || means “is parallel to”.

Postulate 10 Same-Side Interior Angles Postulate If a transversal intersects two parallel lines, then same-side interior angles are supplementary. t l m 4 5 6 3

Theorem 6 Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then alternate interior angles are congruent. t l m 4 6 3

Theorem 6; AIA Theorem Statement Reason 1) Given 3) S. S. I. A Postulate 4) Alg. (Transitive) 5) Alg. (Subtraction) Q. E. D.

Theorem 7 Corresponding Angles Theorem If a transversal intersects two parallel lines, then corresponding angles are congruent. l 5 6 m 8 7 t 1 2 4 3

Theorem 8 Alternate Exterior Angles Theorem If a transversal intersects two parallel lines, then alternate exterior angles are congruent. t 1 2 l m 8 7

Example 1; Identify Supplementary Angles The measure of ∠ 3 is 55. Which angles are supplementary to ∠ 3? For each, tell how you know. 1 2 4 3 5 6 8 7 Definition of Supp. ∠’s Same-Side Interior Angle Postulate

Example 2: Prove Angle Relationships Statement Reason 1) Given 2) Corr. ∠’s Thm 1 a b 5 3) Def. ≅ ∠’s 4) Linear Pair Post. 5) Def. Supp. ∠’s 6) Substitution 7) Def. of Supp. ∠’s 8 7 Q. E. D.

Example 3; Finding Measures of Angles The measure of ∠ 3 is 55. Which angles are supplementary to ∠ 3? For each, tell how you know. l p q 5 2 1 8 3 4 105ᵒ 6 7 m By Alternate Interior Angle Theorem By Same-Side Interior Angles Postulate

Example 4: Prove Angle Measures Statement Reason 1) Given 2) SSIA Postulate 3) Substitution 4) Combine Terms 5) Subtraction Q. E. D. yᵒ 40ᵒ 80ᵒ

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