Chapter 5.4, Problem 36WE

a.

To determine

To Find:nature of quadrilateral $WXYZ$ formed by bisector of the angles of $▱ABCD$ shown in figure.

Answer to Problem 36WE

Rectangle.

Explanation of Solution

Given:

Figure is shown below.

  

quadrilateral $WXYZ$ formed by bisector of the angles of $▱ABCD$

Calculation:

  $WXYZ$ is a rectangle because intersection angle will be ${90}^{\circ }$ .

Therefore, the answer is rectangle.

b.

To determine

To Prove: quadrilateral $WXYZ$ formed by bisector of the angles of $▱ABCD$ is a rectangle.

Explanation of Solution

Given:

Figure is shown below.

  

quadrilateral $WXYZ$ formed by bisector of the angles of $▱ABCD$

Proof:

AnglesA andDare bisected, therefore.

  $\angle DAW=\frac{1}{2}\angle A\text{ and }\angle WDA=\frac{1}{2}\angle D$

Since, A and B is supplementary angle of parallelogram, therefore.

  $\begin{array}{l}\angle A+\angle D=180\\ 2\angle DAW+2\angle WDA=180\\ \angle DAW+\angle WDA=90\mathrm{......}\left(1\right)\end{array}$

In $\Delta AWD$ .

  $\begin{array}{l}\angle DAW+\angle WDA+\angle DWA=180\\ \text{From}\left(1\right)\\ 90+\angle DWA=180\\ \angle DWA=90\\ \Rightarrow \boxed{\angle XWZ=90}(\angle DWA\text{ and }\angle XWZ\text{ are vertically opposite angles})\end{array}$

Similarly,

  $\angle WZY=\angle ZYX=YXW=90$

Since all angles in $WXYZ$ are $90°$ therefore it is a rectangle.