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

Solution:

Consider a parallelogram ABCD

AP, BR and CR be the bisectors of ∠A, ∠B, ∠C and ∠D

We have to prove that PQRS is a rectangle.

We know that the opposite sides of a parallelogram are parallel and congruent.

So, DC || AB

Now, DC || AB and DA is a transversal

We know that if two parallel lines are cut by a transversal, the sum of interior angles lying on the same side of the transversal is always supplementary.

∠A + ∠D = 180° ------------------ (1)

Dividing by 2 on both sides,

1/2 ∠A + 1/2 ∠D = 180°/2

=> 1/2 ∠A+ 1/2 ∠D = 90°

∠PAD + ∠PDA = 90° ----------- (2)

Considering triangle PDA,

By angle sum property,

∠APD + ∠PAD + ∠PDA = 180°

From (2),

∠APD + 90° = 180°

∠APD = 180° - 90°

∠APD = 90°

We know that the vertically opposite angles are equal.

So, ∠APD = ∠SPQ

∠SPQ = 90°

Similarly, ∠PQR = 90°

∠QRS = 90°

∠PSR = 90°

PQRS is a quadrilateral with each angle equal to 90°.

Therefore, PQRS is a rectangle.

✦ Try This: Prove that the perpendicular bisectors of the sides of a cyclic quadrilateral are concurrent.

☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 8

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NCERT Exemplar Class 9 Maths Exercise 8.4 Problem 13

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

Summary:

The opposite sides of a parallelogram are equal in length, and the opposite angles are equal in measure. It is proven that the quadrilateral formed by the bisectors of the angles of a parallelogram is a rectangle

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