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Question
ABCD is a rectangle in which diagonal BD bisects ∠B. Show that ABCD is a square.
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Solution
Given: In a rectangle ABCD, diagonal BD bisects ∠B.
Construct: Join AC.
To show: ABCD is a square.
Proof: In ΔBAD and ΔBCD,
∠ABD = ∠CBD ...[Given]
∠A = ∠C ...[Each 90°]
And BD = BD ...[Common side]
∴ ΔBAD ≅ ΔBCD ...[By AAS congruence rule]
∴ AB = BC
And AD = CD [By CPCT rule] ...(i)
But in rectangle ABCD, opposite sides are equal.
∴ AB = CD
And BC = AD ...(ii)
From equations (i) and (ii),
AB = BC = CD = DA.
So, ABCD is a square.
Hence proved.
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