Question

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

Answer 1

Let $ABCD$ be a cyclic quadrilateral and $AD=BC$

$AC$ and $BD$ are diagonals of cyclic quadrilateral.

In $△AOD$ and $△BOC$

$\Rightarrow$ $\angle OAD=\angle OBC$ [ Same segment subtends equal angle to the circle ]

$\Rightarrow$ $AD=BC$ [ Given ]

$\Rightarrow$ $\angle ODA=\angle OCB$ [ Same segment subtends equal angle to the circle ]

$\therefore$ $△AOD\cong △BOC$ [ By $ASA$ congruence rule ]

Adding $△DOC$ on both sides, we get

$\Rightarrow$ $△AOD+△DOC\cong △BOC+△DOC$

$\Rightarrow$ $△ADC\cong △BCD$

$\Rightarrow$ $AC=BD$ [ By CPCT ]

Hence, we have proved that, a pair of opposite sides of a cyclic quadrilateral are equal prove that its diagonals are also equal