Question

The quadrilateral(PQRS) formed by the bisectors of the angles of a parallelogram(ABCD) is a rectangle.

• A. True
• B. False

Answer 1

The correct option is A

True

Given:
BCD is a parallelogram and AP, BR, CR, be are the bisectors of $\angle A,\angle B,\angle Cand\angle D$, respectively.
We know that:
DC||AB and DA is a transversal.
$\therefore \angle A+\angle D={180}^{\circ }$
[sum of co-interior angles of a parallelogram is ${180}^{\circ }$]

$\Rightarrow \frac{1}{2}\angle A+\frac{1}{2}\angle ={90}^{\circ }$

$\Rightarrow \angle PAD+\angle PDA={90}^{\circ }$

$\Rightarrow \angle APD={90}^{\circ }$ [Sum of all angles of a triangle is ${180}^{\circ }$]

$\therefore \angle SPQ={90}^{\circ }$ [vertically opposite angles]

Similarly, $\angle PQR={90}^{\circ }$

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

$PSR={90}^{\circ }$

Thus, PQRS is a quadrilateral whose each angle is ${90}^{\circ }$.

Hence, PQRS is a rectangle.