34. Prove that quadrilateral formed by theintersection of angle bisect

1.	34. Prove that quadrilateral formed by theintersection of angle bisectors of allangles of a parallelogram is a rectangle(Figure 5.24)Fie. 5.24
Answer» Given,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.

34. Prove that quadrilateral formed by theintersection of angle bisectors of allangles of a parallelogram is a rectangle(Figure 5.24)Fie. 5.24

Answer»

Given,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.