Question

Which of the following statement(s) is/are true , if a quadrilateral PQRS is formed by the angle bisectors of a parallelogram ABCD and PQ is not equal to QR?

• A. Each interior angle of the quadrilateral PQRS is 60°.
• B. PQRS is a rectangle.
• C. PQRS is a square.
• D. Each interior angle of the quadrilateral PQRS is 90°.

Answer 1

Answer: (B), (D)

The correct options are
B PQRS is a rectangle.
D Each interior angle of the quadrilateral PQRS is 90°.

Given, ABCD is a parallelogram,
∠A = ∠C and ∠B = D

AE, BF, CG and DH are angle bisectors.
So, ∠DAE = ∠EAB,
∠ABF = ∠FBC,
∠BCG = ∠GCD and
∠CDH = ∠ HDA

$\because$∠A + ∠B = 180°
$\Rightarrow$∠EAB + ∠ABF = 90°
∠APB = 180° - (∠EAB + ∠ABF) = 90° = ∠QPS (vertically opposite angles)

Similarily,
∠PSR = ∠SRQ = ∠RQP = 90°
Also, PQ $\ne$ QR
Hence, the quadrilateral PQRS is a rectangle.