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.
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.
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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.
