Question

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

Answer 1

Let ABCD be a parallelogram. To show that ABCD is a rectangle, we must prove that one of its interior angles is 90°.

In ΔABC and ΔDCB,

AB = DC       ...(Opposite sides of a parallelogram are equal)

BC = BC       ...(Common)

AC = DB       ...(Given)

∴ ΔABC ≅ ΔDCB      ...(By SSS Congruence rule)

⇒ ∠ABC = ∠DCB

It is known that the sum of the measures of angles on the same side of transversal is 180°.

∠ABC + ∠DCB = 180°      ...(AB || CD)

⇒ ∠ABC + ∠ABC = 180°

⇒ 2∠ABC = 180°

⇒ ∠ABC = 90°

Since ABCD is a parallelogram and one of its interior angles is 90°, ABCD is a rectangle.