Bisectors of the Angles of a Parallelogram form a Rectangle

This article shows that the Bisectors of the Angles of a Parallelogram form a Rectangle. ABCD is a parallelogram in which bisectors of the angles A, B, C, and D intersect at P, Q, R, S to form the quadrilateral PQRS. The angle bisectors of a parallelogram form a rectangle as all the angles are right angles. So, by seeing the figure we can say it is a rectangle. Go through the below theorem and know how to show that the angle bisectors of all four angles of a parallelogram form a rectangle.

Bisectors of the Angles of a Parallelogram form a Rectangle

Statement: Prove that the bisectors of angles of a parallelogram form a rectangle.

Angle bisectors of a parallelogram form a rectangle

Given:
ABCD is a parallelogram AP, BP, CR, DR are bisectors of ∠A, ∠B, ∠C, ∠D respectively.
To prove that:
PQRS is a rectangle
Proof:
A rectangle is a parallelogram with one angle of 90 degrees.
We will prove PQRS is a parallelogram.
Now,
AB || DC
AD is transversal
∠A + ∠D = 180
1/2 ∠A + 1/2 ∠D = 1/2 × 180 degrees
1/2 ∠A + 1/2 ∠D = 90 degrees (DR bisects ∠D and AS bisects ∠A) …..(1)
Now,
In ΔADC
∠DAS + ∠ADS + ∠DSA = 180°
90° + ∠DSA = 180°
∠DSA = 180° – 90°
∠DSA = 90°
Also lines AP and DR intersect
So, ∠PSR = ∠DSA
Therefore, ∠PSR = 90°
Similarly we can prove that,
∠SPQ = 90°, ∠PSR = 90° and ∠SRQ = 90°
So, ∠PSR = ∠PQR = ∠SPQ = ∠SRQ = 90°
Therefore is a parallelogram in which one angle 90°
PQRS is a rectangle.
Hence proved.

Also, Check:

FAQs on What do the Bisectors of the Angles of a Parallelogram Enclose?

1. What makes a parallelogram a rectangle?

A parallelogram with one right angle is a rectangle. A quadrilateral whose diagonals are equal and bisect each other is a rectangle.

2. What are the diagonals of a parallelogram?

A parallelogram is a quadrilateral whose opposite sides are parallel and equal. Diagonals of a parallelogram are the segments that connect the opposite corners of the figure.

3. Do diagonals of parallelogram perpendicularly bisect?

If the diagonals of a parallelogram are perpendicular, then it is a rhombus.

4. Which shape is formed by the bisectors of the angles of a parallelogram?

Rectangle is formed by the bisectors of the angles of a parallelogram.

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