Q. 24.2( 677 Votes )

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

Answer :

Let ABCD be a parallelogram. To show that ABCD is a rectangle,

We have to prove that: One of its interior angles is 90^{0}

In ΔABC and ΔDCB,

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

BC = BC (Common)

AC = DB (Given that the diagonals are equal)

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^{0}

∠ABC + ∠DCB = 180^{0} (AB || CD)

∠ABC + ∠ABC = 180^{0}

∠ABC = 90^{0}

**Since ABCD is a parallelogram and one of its interior angles is 90 ^{0}, ABCD is a rectangle.**

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