Exercise 17 Page 566

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 RS and PT are parallel, we can state that ∠ QRS is congruent to ∠ QPT, and that ∠ RSQ is congruent to ∠ PTQ. This means that two angles of △ QRS are congruent to two angles of △ QPT. Therefore, by the Angle-Angle Similarity Theorem, △ QRS and △ QPT are similar. △ QRS ~ △ QPT

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. RS corresponds with PT SQ corresponds with TQ Recall that corresponding sides of similar figures will have proportional lengths. We are given expressions for the lengths of these sides, which can be used to write a proportion. RS/PT = QS/QT ⇕ 12/16 = x/20 Let's solve this equation to find x.

12/16 = x/20

▼

Solve for x

CrossMult

Cross multiply

12 * 20=16x

Multiply

Multiply

240=16x

DivEqn

.LHS /16.=.RHS /16.

15=x

RearrangeEqn

Rearrange equation

x=15

Now we know the value of x. By the Segment Addition Postulate, we know that ST+QS=TQ. Let's find ST.

ST+QS=TQ

SubstituteII

QS= 15, TQ= 20

ST+ 15= 20

SubEqn

LHS-15=RHS-15

ST=5

Finally, we found that ST=5.