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Prove that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle.
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.
[sum of cointerior angles of a parallelogram is 180°]
=> ½ ∠A+ ½ ∠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.