Question

Bisectors of angles of a parallelogram form a __________ .

• A. square
• B. parallelogram
• C. rectangle
• D. trapezium

Answer 1

The correct option is C rectangle

Let $ABCD$ be a parallelogram as given below:

Since, sum of adjacent angles in a parallelogram is supplementary,

So, $\angle A+\angle B={180}^{\circ }$

$\Rightarrow \frac{\angle A}{2}+\frac{\angle B}{2}=\frac{{180}^{\circ }}{2}$

Since $AR$ and $BR$ are bisectors of $\angle A$ and $\angle B$ respectively.

So, $\angle RAB+\angle RBA={90}^{\circ }$....(i)

Since, In a triangle $\Delta ARB$,

We have $\angle ARB+\angle RAB+\angle RBA={180}^{\circ }$

$\Rightarrow \angle ARB+90=180$ [from equation (i)]

$\therefore \angle ARB=\angle QRS={90}^{\circ }$

Similar way, we can prove $\angle QPS={90}^{\circ }$

Since, $\angle A+\angle D={180}^{\circ }$

$\Rightarrow \frac{\angle A}{2}+\frac{\angle D}{2}=\frac{{180}^{\circ }}{2}$

$\Rightarrow \angle DAQ+\angle ADQ={90}^{\circ }$

In a triangle $AQD$, we have $\angle AQD+\angle DAQ+\angle ADQ=180$

$\Rightarrow \angle AQD+90=180$

$\therefore \angle AQD=\angle PQR={90}^{\circ }$ [As vertically opposite angles]

Similarly, we can prove $\angle PSR={90}^{\circ }$

Therefore, $PQRS$ is a rectangle.

Hence, the correct option is C (Rectangle).