Exercise 16 Page 484

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

1. Angle-Angle Similarity Theorem: 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.

Now we will identify the similar triangles and find the measures, one at a time.

Similar Triangles

We want to identify the similar triangles in the given diagram.

Notice that ∠ J is congruent to ∠ P. We can also see that ∠ KLJ and ∠ MLP are vertical angles, thus they are congruent. This means that two angles of △ JKL are congruent to two angles of △ PML. Therefore, by the Angle-Angle Similarity Theorem, △ JKL and △ PML are similar. △ JKL ~ △ PML

Finding the Measures

Using our similarity statement from above, we can identify two pairs of corresponding sides that will help us find the requested lengths. JK corresponds with PM LJ corresponds with LP Recall that corresponding segments of similar figures will have proportional lengths. We are given expressions for the lengths of these sides which we can use to write a proportion. JK/PM = LJ/LP ⇕ x/12 = 4/6 Let's solve this equation to find x.

Since JK=x, we found that JK=8.