Exercise 18 Page 484

Let's review the theorems that can help us prove that two triangles are similar.

Angle-Angle Similarity Theorem. If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
Side-Side-Side Similarity Theorem. If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
Side-Angle-Side Similarity Theorem. If the lengths of two sides of one triangle are proportional to the lengths of two corresponding sides of another triangle and the included angles are congruent, then the triangles are similar.

With this in mind, we will identify the similar triangles and find the measures, one at a time.

We want to identify similar triangles in the given diagram. We will consider two pairs of triangles at a time.

Let's separate these triangles. Be aware that by the Reflexive Property of Congruence, $∠ Z$ is congruent to itself.

In the diagram above we can see that

∠ U W Z

and

∠ W Y Z

are congruent angles, and that

∠ W Z U

and

∠ Y Z W

are also congruent angles. This means that

△ Z U W

and

△ Z W Y

have two pairs of congruent angles. Therefore, by the Angle-Angle Similarity Theorem, these two triangles are similar.