Question

1. in the figure below, ab is parallel to cd. which statement proves that ∠2 ≅ ∠8? a. if two parallel lines are cut by a transversal, the vertical angles are congruent. b. if two parallel lines are cut by a transversal, the alternate exterior angles are congruent. c. if two parallel lines are cut by a transversal, the alternate interior angles are congruent. d. if two parallel lines are cut by a transversal, the corresponding angles are congruent.

Explanation:

• Option A: Vertical angles are congruent regardless of parallel lines. For example, \(\angle1\cong\angle3\), \(\angle2\cong\angle4\), \(\angle5\cong\angle7\), \(\angle6\cong\angle8\). But this property alone does not relate \(\angle2\) and \(\angle8\) through the parallel - line transversal relationship.
• Option B: Alternate exterior angles: When two parallel lines \(AB\parallel CD\) are cut by a transversal, alternate exterior angles are non - adjacent exterior angles that lie on opposite sides of the transversal. \(\angle2\) and \(\angle8\) are non - adjacent, exterior angles, and since \(AB\parallel CD\), by the alternate exterior angles theorem (if two parallel lines are cut by a transversal, then alternate exterior angles are congruent), \(\angle2\cong\angle8\).
• Option C: Alternate interior angles are non - adjacent interior angles that lie on opposite sides of the transversal. Examples of alternate interior angles in the given figure are \(\angle4\) and \(\angle6\), which is not relevant to \(\angle2\) and \(\angle8\).
• Option D: Corresponding angles are in the same relative position. For example, \(\angle2\) and \(\angle6\) are corresponding angles (not \(\angle2\) and \(\angle8\)).

Answer:

B. If two parallel lines are cut by a transversal, the alternate exterior angles are congruent.