Question

in which figure is $\angle 1$ congruent to $\angle 2$?

(the interior angles are on the same side of the transversal and the lines intersected by the transversal are not parallel.)

Explanation:

Step1: Recall angle congruence rules

For two angles formed by a transversal to be congruent, they can be corresponding angles (if lines are parallel), alternate interior/exterior angles, or vertical angles. Also, note the hint: same-side interior angles are non-congruent when lines are not parallel, so we eliminate that case.

Step2: Analyze each figure

1. Top-left: ∠1 and ∠2 are same-side interior angles with non-parallel lines (per hint), so they are supplementary, not congruent.
2. Top-right: ∠1 and ∠2 are vertical angles, which are always congruent regardless of line parallelism.
3. Bottom-left: ∠1 and ∠2 are corresponding angles with non-parallel lines, so they are not congruent.
4. Bottom-right: ∠1 and ∠2 are neither vertical, alternate, nor corresponding angles with parallel lines, so they are not congruent.

Answer:

The top-right figure (with two horizontal parallel lines and a transversal, where ∠1 and ∠2 are vertical angles) is the one where ∠1 is congruent to ∠2.