Question

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

Explanation:

Step1: Recall the concept of vertical angles

Vertical angles are formed when two lines intersect, and they are congruent (equal in measure).

Step2: Analyze each figure

• Top - left figure: $\angle1$ and $\angle2$ are same - side interior angles (assuming the two horizontal lines are parallel and the transversal cuts them). Same - side interior angles are supplementary, not necessarily congruent.
• Top - right figure: $\angle1$ and $\angle2$ are not vertical angles or angles with a known congruence relationship from basic angle - related theorems (like corresponding angles, alternate interior angles etc. don't apply here in a way that makes them congruent).
• Bottom - left figure: $\angle1$ and $\angle2$ are vertical angles (formed by the intersection of two pairs of lines). By the vertical angles theorem, vertical angles are congruent.
• Bottom - right figure: $\angle1$ and $\angle2$ are corresponding angles only if the two slant lines are parallel, but there's no indication of parallelism, and visually they don't seem to be vertical angles or have a congruence - guaranteeing relationship.

Answer:

The bottom - left figure (the third figure)