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