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Question 12
If a pair of opposite sides of a cyclic quadrilateral are equal, then prove that its diagonals are also equal.
 


Solution

Given, ABCD be a cyclic quadrilateral and AD = BC.

Construction : Join AC and BD
To prove that  AC = BD.

Proof
 In ΔAOD and ΔBOC,
OAD=OBC [since, same segments subtends equal angle to the circle]
ODA=OCB [since, same segments subtends equal angle to the circle]
AD = BC (Given)
ΔAODΔBOC [ASA congruence rule]

Adding ΔDOC on both sides, we get
ΔAOD+ΔDOCΔBOC+ΔDOC
 ΔADCΔBCD
 AC=BD (By CPCT)

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