If the diagonals of a parallelogram are equal, then show that it is a rectangle.

To Prove: If diagonals of a parallelogram are equal, the quadrilateral is a rectangle.

Given: Diagonals area equal

Concept Used:

Rectangle: Opposite sides are equal, and all angles are right angles.

SSS Theorem: If the three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent.

Opposite angles of a parallelogram are supplementary.

Diagram:

Explanation:

In ΔABD and Δ BCD,

AB = CD [Opposite sides of a parallelogram are equal]

AD = BC [Opposite sides of a parallelogram are equal]

BD = Common

Therefore, ΔABD and ΔBCD are congruent.

As the triangles are congruent,

∠ BAD = ∠ BCD [By C.P.C.T]

Now, we also know that,

Opposite angles of a parallelogram are supplementary.

Therefore,

∠ BAD + ∠ BCD = 180˚

2∠ BAD = 180˚

∠ BAD = 90˚ = ∠ BCD

And similarly

∠ ADC = ∠ ABC = 90˚

Now, all angles of the parallelogram are right angles and opposite sides are equal.

The parallelogram is a rectangle.

Hence, Proved.