Question

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

Answer 1

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

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

Proof
In $\Delta AODand\Delta BOC$,
$\angle OAD=\angle OBC$ [since, same segments subtends equal angle to the circle]
$\angle ODA=\angle OCB$ [since, same segments subtends equal angle to the circle]
AD = BC (Given)
$\Delta AOD\cong \Delta BOC$ [ASA congruence rule]

Adding $\Delta DOC$ on both sides, we get
$\Delta AOD+\Delta DOC\cong \Delta BOC+\Delta DOC$
$\Rightarrow \Delta ADC\cong \Delta BCD$
$\therefore AC=BD$ (By CPCT)