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.
Notice that ∠ SWR and ∠ RWT form a linear pair. Thus, they are supplementary angles. Since ∠ RWT is a right angle so is ∠ SWR, and hence they are congruent. Let's check whether the corresponding sides that include ∠ SWR and ∠ RWT are proportional.
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SW/WR & = & 16/12 & = & 4/3 [0.8em]
WR/WT & = & 12/9 & = & 4/3
As we can see, with the ratios being equal this means that the corresponding sides are proportional. According to the Side-Angle-Side Similarity Theorem, the given triangles are similar.
△ SWR ~ △ RWT
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 RT
SW corresponds with WR
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.
RS/RT = SW/WR
⇕
6x+2/4x+3 = 16/12
Let's solve this equation to find x.
