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Question
If a pair of opposite sides of a cyclic quadrilateral are equal, prove that its diagonals are also equal.
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Solution
Given: Let ABCD be a cyclic quadrilateral and AD = BC.
Join AC and BD.
To prove: AC = BD
Proof: In ΔAOD and ΔBOC,
∠OAD = ∠OBC and ∠ODA = ∠OCB ...[Since, same segments subtends equal angle to the circle]
AB = BC ...[Given]
ΔAOD = ΔBOC ...[By ASA congruence rule]
Adding is DOC on both sides, we get
ΔAOD + ΔDOC ≅ ΔBOC + ΔDOC
⇒ ΔADC ≅ ΔBCD
AC = BD ...[By CPCT]
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