The quadrilateral(PQRS) formed by the bisectors of the angles of a parallelogram(ABCD) is a square.
False
Given:
BCD is a parallelogram and AP, BR, CR, be are the bisectors of ∠A,∠B,∠C and ∠D, respectively.
We know that:
DC||AB and DA is a transversal.
∴∠A+∠D=180∘
[sum of cointerior angles of a parallelogram is 180∘]
⇒12∠A+12∠=90∘
⇒∠PAD+∠PDA=90∘
⇒∠APD=90∘ [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.