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.

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:

- If Each Diagonal of a Quadrilateral Divides it in Two Triangles of Equal Area then Prove that the Quadrilateral is a Parallelogram
- A Parallelogram, whose Diagonals are of Equal Length, is a Rectangle
- Opposite Angles of a Parallelogram are Equal

### 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?**