Given Let ABCD be a parallelogram and AP, BR, CR, be are the bisectors of ∠A,∠B,∠C and ∠D, respectively.
To prove Quadrlateral PQRS is a rectangle.
Proof Since, ABCD is a parallelogram, then DC||AB and DA is a transversal.
We have, ∠A+∠D=180∘
[sum of cointerior angles of a parallelogram is 180∘]
⇒12∠A+12∠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]
Similarly, ∠PQR=90∘
∠QRS=90∘
PSR=90∘
Thus, PQRS is a quadrilateral whose each angle is 90∘.
Hence, PQRS is a rectangle.