Side-Side-Side Similarity Theorem: If the corresponding side lengths of two triangles are proportional, then the triangles are similar.
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 ∠ YWU is congruent to ∠ YZW. We can also see that △ WUY and △ ZWY share ∠ Y. This means that two angles of △ WUY are congruent to two angles of △ ZWY. Therefore, by the Angle-Angle Similarity Theorem, △ WUY and △ ZWY are similar.
△ WUY ~ △ ZWY
Notice that ∠ ZUW and ∠ WUY form a linear pair and so are supplementary angles. Since ∠ ZUW is a right angle, so is ∠ WUY, which means they are congruent. Also, ∠ WZU is a part of △ ZUW and ∠ YWU is a part of △ WUY. Again, by the Angle-Angle Similarity Theorem, these triangles are similar.
△ ZUW ~ △ WUY
We found that △ WUY ~ △ ZWY and △ ZUW ~ △ WUY. By the Transitive Property of Similarity, △ ZWY ~ △ ZUW, which means that there are three similar triangles in the given diagram.
△ ZUW ~ △ ZWY ~ △ WUY
Finding the Measures
Using our first similarity statement, we can identify three pairs of corresponding sides that will help us find the requested lengths.
UZ corresponds with WZ
WZ corresponds with YZ
UW corresponds with WY
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.
UZ/WZ = WZ/YZ = UW/WY
⇕
x-0.8/x+4 = x+4/x+12 = UW/16
First we have to find UW using the two already known side lengths of the △ WUY and the Pythagorean Theorem.
