3.1.2 Angles, Parallel Lines, and Transversals Key Objectives • Prove and use theorems about the angles formed by parallel lines and a transversal. Theorems, Postulates, Corollaries, and Properties • Corresponding Angles Postulate If two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. • Alternate Interior Angles Theorem If two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. • Alternate Exterior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of alternate exterior angles are congruent. • Same-Side Interior Angles Theorem If two parallel lines are cut by a transversal, then the two pairs of same-side interior angles are supplementary. Example 1 Using Congruent Angle Theorems and Postulates
By the Corresponding Angles Postulate, if two parallel lines are cut by a transversal, then the pairs of corresponding angles are congruent. Corresponding angles are on the same side of the transversal and on the same sides of the other lines. By the Alternate Interior Angles Theorem, if two parallel lines are cut by a transversal, then the pairs of alternate interior angles are congruent. Alternate interior angles are on opposite sides of the transversal and on the inside of the other two lines. There are two pairs of alternate interior angles in the figure given here. ∠ 1 and ∠ 3 are alternate interior angles, as well as ∠ 2 and ∠ 4. By the Alternate Exterior Angles Theorem, if two parallel lines are cut by a transversal, then the pairs of alternate exterior angles are congruent. Alternate exterior angles are on opposite sides of the transversal and on the outside of the other two lines.
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