A parallelogram with a right angle is known as a rectangle. We know that the opposite sides of a parallelogram and rectangle are equal. The diagonals of the parallelogram are of equal lengths. Thus a parallelogram whose diagonals are of equal length is a rectangle. Opposite sides of a parallelogram and rectangle are always congruent. Hence we can say that a parallelogram with equal lengths of a diagonal is a rectangle.

## A Parallelogram whose Diagonals are of Equal Length is a Rectangle

**Theorem:**

Prove that a parallelogram whose diagonals are of equal length is a rectangle.

Given

ABCD is a parallelogram in which AB ∥ DC, AD ∥ BC, and AC = BD.

To prove that:

ABCD is a parallelogram, i.e., in the parallelogram ABCD, one angle, say ∠BAD = 90°.

**Proof:**

In ∆ABC and ∆BDA,

∠CAB = ∠ACD (Since, DC ∥ AB),

∠BCA = ∠DAC(Since, AD ∥ BC),

AC = AC

Therefore, ∆ABC ≅ ∆CDA, (By AAS criterion of congruence)

Therefore, BC = AD (CPCTC).

In ∆ABC and ∆BAD,

BC = AD

AC = BD (Given),

AC = AC

Therefore, ∆ABC ≅ ∆BAD (By SSS criterion of congruence)

∠ABC = ∠BAD (CPCTC).

But ∠ABC + ∠BAD= 180° (Since, QR ∥ PS)

Therefore, ∠ABC = ∠BAD = 90°

Hence proved.

Also, See:

- Opposite Angles of a Parallelogram are Equal
- Pair of Opposite Sides of a Parallelogram are Equal and Parallel
- Opposite Sides of a Parallelogram are Equal

### FAQs on a Parallelogram Whose Diagonals are of Equal Length is a Rectangle

**1. Is it true that a parallelogram is a rectangle?**

A parallelogram has two sets of parallel sides and two pairs of opposite sides that are congruent. A rectangle is always a parallelogram.

**2. Do diagonals have an equal length?**

Square, Rectangle, Parallelogram have diagonals of equal length.

**3. Are the diagonals of a rectangle are equal Why?**

A rectangle is a parallelogram where each angle is a right angle.

Therefore, the Diagonals of a rectangle are equal.