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3-1 3-2 3-3 Parallel Lines and Angles • Objectives • Define transversal and the angles associated with a transversal • State and apply the properties of angles associated with a transversal that cuts a pair of parallel lines 1 3-1 Definition of transversal • A transversal is a line that intersects two coplanar lines at two distinct points l 1 2 4 3 is a transversal for l and m. c 5 6 8 7 t t b 1 2 4 3 m 5 6 8 7 b || c Nonparallel Lines lm Parallel Lines r r is a transversal for b and c. 2 3-1 Angles associated with transversals Interior angles lie between the two lines. Examples: ∠4, ∠6 Lines b and c Exterior angles lie outside the two lines. Examples: ∠1, ∠8 1 2 4 3 Alternate Interior angles are on the opposite sides of the transversal. Example: ∠4 and ∠6 Corresponding angles are on the same side of the transversal and on the same side of the lines cut by the transversal. Example: ∠2 and ∠6 c 5 6 8 7 Alternate Exterior angles are on the opposite sides of the transversal. Example: ∠2 and ∠8 Same Side Interior angles are on the same side of the transversal. Example: ∠4 and ∠5 b r r is a transversal for b and c. 3 3-1 Theorems/postulates for parallel lines • If a transversal intersects two parallel lines, then Parallel Lines – Corresponding angles are 1 2 congruent (Corr. Angles Postulate) 4 3 – Alternate interior angles are 5 6 congruent (Alt. Int. Angles Thm) 8 7 – Alternate exterior angles are t congruent (Alt. Ext. Angles Thm) – Same side interior angles are supplementary (Same Side Int. t is a transversal for Angles Thm) – Same side exterior angles are supplementary (Same Side Ext. Angles Thm) lm l m l and m. 4 3-2 Converses of previous slide • All converses of the previous slide are true • Example: Converse of the Corresponding Angle Postulate: – If two lines and a transversal form congruent corresponding angles, then the two lines are parallel • Example: Converse of the Alternate Interior Angles Theorem: – If two lines and a transversal form congruent alternate interior angles, then the two lines are parallel 5 3-3 Parallel and perpendicular lines • If two lines are parallel to the same line, then they are parallel to each other. • In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. • In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other. 6