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Geometry 1A Name _____________________ 3.3 Parallel Lines and Angle Relationships A transversal is a line that intersects two coplanar lines at two distinct points. t a 1 2 4 3 5 6 b 8 7 Alternate interior angles Alternate interior angles are angles that lie on opposite sides of the transversal in the interior. Angle Pairs: Same-side interior angles Same-side interior angles are angles that lie on the same side of the transversal in the interior. Angle Pairs: Corresponding angles Corresponding angles are ones at the same location at each intersection. Angle Pairs: Example 1: Classify each pair of angles as alternate interior angles, same-side interior angles, or corresponding angles. 1. 2. 1 3. 1 1 2 2 2 4. 1 5. 6. 1 1 2 2 2 Corresponding Angles Postulate If a transversal intersects two parallel lines, then corresponding angles are congruent. Alternate Interior Angles Theorem If a transversal intersects two parallel lines, then alternate interior angles are congruent. Same-Side Interior Angles Theorem If a transversal intersects two parallel lines, then same-side interior angles are supplementary. Example 2: Find the measure of ∠1 and ∠2 . Justify each answer. a. b. c. 2 o 100 1 o 1 135 o 75 1 2 2 Example 3: Find x and the measure of each angle. a. b. x c. o (7x) (x – 26) o (x + 55) (3x – 5) o o o Example 4: In the figure to the right, m || n. Find x. a. m∠2 = ( 9 x − 2 ) , m∠4 = (10 x − 8 ) 1 5 2 m 6 7 3 4 b. m∠6 = ( 4 x + 20 ) , m∠7 = ( 2 x + 4 ) 8 Converse of the Corresponding Angles Postulate If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel. Converse of the Alternate Interior Angles Theorem If two lines and a transversal form alternate interior angles that are congruent, then the two lines are parallel. Converse of the Same-Side Interior Angles Theorem If two lines and a transversal form same-side interior angles that are supplementary, then the two lines are parallel. n Example 4: Using the given information, which lines, if any, can you conclude are parallel? Justify each conclusion with a theorem or postulate. a. ∠ 3 is supplementary to ∠ 10 1 2 r 5 6 7 8 4 3 b. ∠3 ≅ ∠1 s 9 10 11 12 13 14 16 15 t v c. ∠6 ≅ ∠14 d. ∠7 ≅ ∠2 e. ∠11 ≅ ∠14 f. ∠4 ≅ ∠10