Q. 84.1( 435 Votes )
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
Answer :
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,
∠CBD = ∠ADB (Alternate interior angles for BC || AD)
∠CDB = ∠ABD
BD bisects D
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