Exercise 6 Page 565

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

Since m∠ Y=51° and m∠ K=51°, this means ∠ Y and ∠ K are congruent. We can also see that ∠ X is congruent to ∠ J. This means that two angles of △ XYZ are congruent to two angles of △ JKL. Therefore, by the Angle-Angle Similarity Theorem, △ XYZ and △ JKL are similar. △ XYZ ~ △ JKL

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. XY corresponds with JK YZ corresponds with KL 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. XY/JK = YZ/KL ⇕ 5/4 = 15/x Let's solve this equation to find x.

5/4 = 15/x

▼

Solve for x

CrossMult

Cross multiply

5x=4 * 15

Multiply

Multiply

5x=60

DivEqn

.LHS /5.=.RHS /5.

x=12

Since KL=x, we found that KL=12.