Question

Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle.

Answer 1

Given: Let ABCD be a parallelogram and AP, BR, CR, be are the bisectors of ∠A, ∠B, ∠C and ∠D, respectively.

To prove: Quadrilateral PQRS is a rectangle.

Proof: Since, ABCD is a parallelogram, then DC || AB and DA is a transversal.

∠A + ∠D = 180°  ...[Sum of cointerior angles of a parallelogram is 180°]

⇒ `1/2` ∠A + `1/2` ∠D = 90°  ...[Dividing both sides by 2]

∠PAD + ∠PDA = 90°

∠APD = 90°  ...[Since, sum of all angles of a triangle is 180°]

∴ ∠SPQ = 90°  ...[Vertically opposite angles]

∠PQR = 90°

∠QRS = 90°

And ∠PSR = 90°

Thus, PQRS is a quadrilateral whose each angle is 90°.

Hence, PQRS is a rectangle.