The 2 triangles intersect at vertex upper Q. The measure of angle upper X upper W upper Q is a/li

Question

In the figure shown, $\bar{W Z}$ and $\bar{X Y}$ intersect at point $Q$ . $upper Y upper Q equals 63$ , $upper W upper Q equals 70$ , $upper W upper X equals 60$ , and $upper X upper Q equals 120$ . What is the length of $\bar{Y Z}$ ?

Accepted Answer

The correct answer is 54 . The figure shown includes two triangles, triangle W Q X and triangle Y Q Z , such that angle W Q X and angle Y Q Z are vertical angles. It follows that angle W Q X is congruent to angle Y Q Z . It&rsquo;s also given in the figure that the measures of angle W and angle Y are a &#176; . Therefore angle W is congruent to angle Y . Since triangle W Q X and triangle Y Q Z have two pairs of congruent angles, triangle W Q X is similar to triangle Y Q Z by the angle-angle similarity postulate, where Y Z &#175; corresponds to W X &#175; , and Y Q &#175; corresponds to W Q &#175; . Since the lengths of corresponding sides in similar triangles are proportional, it follows that Y Z W X = Y Q W Q . It&rsquo;s given that Y Q = 63 , W Q = 70 , and W X = 60 . Substituting 63 for Y Q , 70 for W Q , and 60 for W X in the equation Y Z W X = Y Q W Q yields Y Z 60 = 63 70 . Multiplying each side of this equation by 60 yields Y Z = 63 70 60 , or Y Z = 54 . Therefore, the length of Y Z &#175; is 54 .

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