Prove that quadrilateral formed by the intersection of angle bisectors of all angles of a parallelogram is a rectangle. (Figure 5.24)
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Let ABCD be a parallelogram
To prove Quadrilateral PQRS is a rectangle.
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]
∠SAD + ∠SDA = 90°
∠ASD = 90° [since,sum of all angles of a triangle is 180°]
∴ ∠PSR = 90° and ∠PQR = 90° [vertically opposite angles]
∠QRS = 90°and ∠QPS = 90° [vertically opposite angles]
So, PQRS is a quadrilateral whose each angle is 90°.
Hence, PQRS is a rectangle.
Let ABCD be a parallelogram
To prove Quadrilateral PQRS is a rectangle.
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]
∠SAD + ∠SDA = 90°
∠ASD = 90° [since,sum of all angles of a triangle is 180°]
∴ ∠PSR = 90° and ∠PQR = 90° [vertically opposite angles]
∠QRS = 90°and ∠QPS = 90° [vertically opposite angles]
So, PQRS is a quadrilateral whose each angle is 90°.
Hence, PQRS is a rectangle.
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