Q12 of 40 Page 103

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

Let ABCD be a cyclic quadrilateral with AD = BC.


We need to prove that AC = DB.


In triangles AOD and BOC:


AD = DB (given)


∠OAD = ∠OBC and ∠ODA = ∠OCB (angle subtended by the same segment are equal)


∴ ΔAOD ≅ ΔBOC (By ASA congruence rule)


⇒ ΔAOD + ΔDOC ≅ ΔBOC + ΔDOC (adding a similar quantity on both sides)


⇒ ΔADC ≅ ΔBCD


∴ AC = BD (by CPCT).


Hence, proved.


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