Exercise 51 Page 467

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

1. Angle-Angle Similarity Postulate. If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
2. Side-Side-Side Similarity Theorem. If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
3. 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.

We are asked to determine whether the given triangles are similar.

Notice that there is a pair of vertical angles. These angles have equal measures. Therefore, ∠ MNR = ∠ PNQ.

As we can see, two angles of one triangle are congruent to two angles of another triangle. Therefore, by the Angle Angle Similarity Postulate the triangles are similar. △ NRM ~ △ NPQ