Question

1. what is mn?

Explanation:

Step1: Identify similar - triangles

By comparing the angles and side - lengths of the two polygons, we can see that the two polygons are similar. The corresponding angles are equal: $\angle L = \angle C=109^{\circ}$, $\angle M=\angle D = 25^{\circ}$, and the side - length ratios are consistent.

Step2: Set up proportion

If the two polygons are similar, then the ratios of corresponding sides are equal. We can set up the proportion using the known corresponding sides. Let's assume the two polygons are similar. We know that if we consider the side - length pairs, for example, if we take the pair of sides with lengths $12$ cm and $12$ cm as one corresponding pair, and the pair of sides with lengths $8$ cm and $8$ cm as another corresponding pair. The side $MN$ corresponds to the side with length $10$ cm in the other polygon.
Since the polygons are similar, the ratio of corresponding sides is the same. We can see that the side corresponding to $MN$ in the other polygon has length $10$ cm.

Answer:

$10$ cm