Chapter 12, Problem 11CUR

To determine

To prove: $\overline{WZ}\parallel \overline{XY}$ .

Explanation of Solution

Given information:

It is given that $\overline{WZ}\perp \overline{ZY}$ , $\overline{WX}\perp \overline{XY}$ , and $\overline{WX}\cong \overline{YZ}$ .

  

Calculation:

Two triangles are congruent if they have exactly the same three sides and exactly the same three angles.

From the above figure, $\Delta WXY$ and $\Delta WYZ$ have exactly the same three sides and exactly the same three angles.

In $\Delta WXY$ and $\Delta WYZ$ , $\overline{WY}$ is common side, $\angle Z=90$ , $\angle X=90$ ,and $\overline{WX}=\overline{YZ}$ .

  $\begin{array}{c}\overline{WX}=\overline{YZ}\\ \angle Z=\angle X\end{array}$

Hence, both triangles are congruent triangle.

Therefore, $\overline{WZ}\parallel \overline{XY}$.