Question

1. in the figure below, ab is parallel to cd. which statement proves that ∠3 ≅ ∠7? a. if two parallel lines are cut by a transversal, the alternate exterior angles are congruent. b. if two parallel lines are cut by a transversal, the vertical angles are congruent. c. if two parallel lines are cut by a transversal, the corresponding angles are congruent. d. if two parallel lines are cut by a transversal, the alternate interior angles are congruent.

Explanation:

• Option A: Alternate exterior angles are angles outside the two parallel lines and on opposite sides of the transversal. \(\angle3\) and \(\angle7\) are not alternate exterior angles.
• Option B: Vertical angles are pairs of opposite angles formed by two intersecting lines. The property of vertical - angle congruence is not related to the parallel - line transversal relationship for proving \(\angle3\cong\angle7\).
• Option C: Corresponding angles are in the same relative position with respect to the parallel lines and the transversal. \(\angle3\) and \(\angle7\) are not corresponding angles.
• Option D: Alternate interior angles are angles between the two parallel lines and on opposite sides of the transversal. When two parallel lines \(AB\) and \(CD\) are cut by a transversal, \(\angle3\) and \(\angle7\) are alternate interior angles. By the alternate - interior - angles theorem (if two parallel lines are cut by a transversal, then the alternate interior angles are congruent), \(\angle3\cong\angle7\).

Answer:

D. If two parallel lines are cut by a transversal, the alternate interior angles are congruent.