# ABCD is a rectangle in which diagonal AC bisects ∠A as well as ∠C Show that:(i) ABCD is a square(ii) Diagonal BD bisects ∠B as well as ∠D

Given: ABCD is a rectangle,

A = C

A = C
To Prove: ABCD is a square
Proof:

DAC = DCA (AC bisects A and C)

CD = DA (Sides opposite to equal angles are also equal)

However,

DA = BC and AB = CD (Opposite sides of a rectangle are equal)

AB = BC = CD = DA

ABCD is a rectangle and all of its sides are equal.

Hence, ABCD is a square

Hence, Proved.

(ii) Let us join BD

In ΔBCD,

BC = CD (Sides of a square are equal to each other)

CDB = CBD (Angles opposite to equal sides are equal)

However,

CDB = ABD (Alternate interior angles for AB || CD)

CBD = ABD

BD bisects B

Also,

CDB = ABD

BD bisects D

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