Chapter 7.2, Problem 14PSB

Solution

To Prove: The congruency of $\angle X$ with respect to $\angle Z$

Given information: $\overleftrightarrow{AC}\parallel \overline{XY}$ , $\overleftrightarrow{AB}\parallel \overleftrightarrow{CY}$ and $\angle ZAC\cong \angle XAB$ are provided in the question which will help to find the solution.

Proof: As, $\overleftrightarrow{AC}\parallel \overline{XY}$

So, $\angle ZAC=\angle AXB$ .......... $\left(1\right)$

As, $\overleftrightarrow{AB}\parallel \overleftrightarrow{CY}$

So, $\angle XAB=\angle AZC$ .......... $\left(2\right)$

  $\angle ZAC\cong \angle XAB$ (given)......... $\left(3\right)$

From $\left(1\right)$ and $\left(3\right)$

  $\angle AXB\cong \angle XAB$ .......... $\left(4\right)$

Now, from $\left(2\right)$ and $\left(4\right)$

  $\angle AXB\cong \angle AZC$

Hence, $\angle X\cong \angle Z$