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

2 3-1 Definition of transversal A transversal is a line that intersects two coplanar lines at two distinct points b c Nonparallel Lines r is a transversal for b and c. r l m Parallel Lines t is a transversal for l and m. t

3 3-1 Angles associated with transversals b c Lines b and c r is a transversal for b and c. r Interior angles lie between the two lines. Examples: ∠4, ∠6 Exterior angles lie outside the two lines. Examples: ∠1, ∠8 Alternate Interior angles are on the opposite sides of the transversal. Example: ∠4 and ∠6 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 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

4 3-1 Theorems/postulates for parallel lines If a transversal intersects two parallel lines, then –Corresponding angles are congruent (Corr. Angles Postulate) –Alternate interior angles are congruent (Alt. Int. Angles Thm) –Alternate exterior angles are congruent (Alt. Ext. Angles Thm) –Same side interior angles are supplementary (Same Side Int. Angles Thm) –Same side exterior angles are supplementary (Same Side Ext. Angles Thm) l m Parallel Lines t is a transversal for l and m. t

5 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

6 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.