The diagonals of a rectangle intersect each other at right angles. Prove that the rectangle is a square.

Let ABCD be the rectangle.

Since, opposite sides of rectangle are equal.

∴ AB = DC ……..(1)

∴ AD = BC ………(2)

Given,

Diagonals intersect at right angle.

∴ ∠AOB = 90°, ∠AOD = 90°.

In △ AOB and △ AOD,

⇒ AO = AO (Common side)

⇒ ∠AOB = ∠AOD (Both equal to 90°)

⇒ OB = OD (Diagonals of rectangle bisect each other)

∴ △ AOB ≅ △ AOD (By S.A.S. axiom)

We know that,

Corresponding parts of congruent triangle are equal.

∴ AD = AB …….(3)

From equations (1), (2) and (3), we get :

⇒ AB = BC = CD = AD.

Since, all sides are equal and diagonals intersect at right angle.

Hence, proved that the rectangle is a square.