# Honors Geometry Companion Book, Volume 1

3.1.3 Proving that Lines are Parallel Key Objectives • Use the angles formed by a transversal to prove two lines are parallel. Theorems, Postulates, Corollaries, and Properties • Converse of the Corresponding Angles Postulate If two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel. • Parallel Postulate Through a point P not on line  , there is exactly one line parallel to  . • Converse of the Alternate Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel. • Converse of the Alternate Exterior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel. • Converse of the Same-Side Interior Angles Theorem If two coplanar lines are cut by a transversal so that a pair of same-side interior angles are supplementary, then the two lines are parallel. Example 1 Showing that Lines are Parallel

By the Converse of the Corresponding Angles Postulate, if two coplanar lines are cut by a transversal so that a pair of corresponding angles are congruent, then the two lines are parallel.

By the converse of the Alternate Exterior Angles Theorem, if two coplanar lines are cut by a transversal so that a pair of alternate exterior angles are congruent, then the two lines are parallel.

By the converse of the Alternate Interior Angles Theorem, if two coplanar lines are cut by a transversal so that a pair of alternate interior angles are congruent, then the two lines are parallel.

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