Exercise 18 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.

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

Similar Triangles

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

△ ZUW and △ ZWY

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 ∠ UWZ and ∠ WYZ are congruent angles, and that ∠ WZU and ∠ YZW are also congruent angles. This means that △ ZUW and △ ZWY have two pairs of congruent angles. Therefore, by the Angle-Angle Similarity Theorem, these two triangles are similar. △ ZUW ~ △ ZWY

△ ZUW and △ WUY

To get these two triangles, we will divide the main triangle along the height. Notice that ∠ ZUW and ∠ WUY form a linear pair. Therefore, they are supplementary angles. Since ∠ ZUW is a right angle, so is ∠ WUY. This means that they are congruent angles.

Also, ∠ UWZ and ∠ UYW are marked as congruent angles. Again, by the Angle-Angle Similarity Theorem, these triangles are similar. △ ZUW ~ △ WUY

△ ZWY and △ WUY

We found that △ ZUW and △ ZWY are similar triangles. We also found that △ ZUW and △ WUY are similar triangles as well. Therefore, by the Transitive Property of Similarity, △ ZWY and △ WUY are similar triangles. △ ZUW ~ △ ZWY △ ZUW ~ △ WUY ⇓ △ ZWY ~ △ WUY This means that there are three similar triangles in the given diagram. △ ZUW ~ △ ZWY ~ △ WUY

Finding the Measures

Knowing that △ ZUW and △ ZWY are similar triangles, we can identify corresponding sides in these triangles.

Let's state the pairs of corresponding sides. ccc UZ & WZ & UW and & and & and WZ & YZ & WY Recall that corresponding sides of similar figures have proportional lengths. We are given expressions for the lengths of these sides which we can use to write a proportion. UZ/WZ = WZ/YZ = UW/WY ⇓ x+6/3x-6 = 3x-6/( x+6)+ 32 = UW/40 First, we can find UW using the two already known side lengths of △ WUY and the Pythagorean Theorem.

Note that, when solving the equation, we only kept the principal root. This is because a side length can never be negative! We can now substitute UW= 24 in the equation 3x-6( x+6)+ 32 = UW 40, and solve for x.

Now that we know the value of x, we can find WZ and UZ. To do so, we will substitute x= 12 in the expressions for the lengths.

We found that WZ=30 and UZ=18.