Chapter 10.2, Problem 24E

To determine

To calculate $\angle WXY$ and $\angle XYZ$ .

Answer to Problem 24E

  $\begin{array}{l}\angle WXY={135}^{\circ }\\ \angle XYZ={120}^{\circ }\end{array}$

Explanation of Solution

Given information: WX is parallel to XY,

  $\begin{array}{l}\angle XWZ={45}^{\circ }\\ \angle YZW={60}^{\circ }\end{array}$

Formula used:

Angle sum property of triangle,

  $a1+a2+a3={180}^{\circ }$ , and

Alternate angle property.

Calculation:

Let’s name the points of the triangles as A and B.

In

  $△WAX$ :

To find angle $\angle WXA$ , substituting the value of angles,

  $\begin{array}{l}45+\angle WXA+90=180\\ \angle WXA+135=180\end{array}$

Shifting left-hand side value to right-hand side:

  $\begin{array}{l}\angle WXA=180-135\\ \angle WXA=45\end{array}$

In

  $△YBZ$ :

To find angle $\angle BYZ$ , substituting the value of angles,

  $\begin{array}{l}60+\angle BYZ+90=180\\ \angle BYZ+150=180\end{array}$

Shifting left-hand side value to right-hand side:

  $\begin{array}{l}\angle BYZ=180-150\\ \angle BYZ=30\end{array}$

Considering the alternate angle property, which states that when the lines are parallel, that the alternate interior angles are equal. Hence,

  $\angle XAW=\angle YXA$ ,

  $\angle YXA=90$ , and

  $\angle ZBY=\angle XYB$

  $\angle XYB=90$

For $\angle WXY$ ,

  $\begin{array}{l}\angle WXY=\angle YXA+\angle WXA\\ \angle WXY=45+90\\ \angle WXY=135\end{array}$

For $\angle XYZ$ ,

  $\begin{array}{l}\angle XYZ=\angle XYB+\angle ZYB\\ \angle XYZ=30+90\\ \angle WXY=120\end{array}$

Hence,

  $\begin{array}{l}\angle WXY={135}^{\circ }\\ \angle XYZ={120}^{\circ }\end{array}$