Question

1. in the figure below, ( ab ) is parallel to ( cd ).

which statement proves that ( angle 4 cong angle 6 )?
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 alternate interior 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 vertical angles are congruent.

Explanation:

• Alternate interior angles are formed when a transversal intersects two parallel lines. The angles lie between the two parallel lines and on opposite sides of the transversal.
• In the given figure, \(\angle4\) and \(\angle6\) are between the parallel lines \(AB\) and \(CD\) and on opposite sides of the transversal.
• Option A: Alternate exterior angles are outside the parallel lines, so this is incorrect for \(\angle4\) and \(\angle6\).
• Option C: Corresponding angles are in the same relative position (e.g., top - left, top - right etc. with respect to the parallel lines and transversal), which \(\angle4\) and \(\angle6\) are not.
• Option D: Vertical angles are pairs of opposite angles formed by two intersecting lines. Here, we are dealing with parallel lines and a transversal, not just two intersecting lines for the relationship of \(\angle4\) and \(\angle6\).

Answer:

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