Question

prove that the quadrilateral formed by the bisector of angle of 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.