Question

ABCD is a cyclic quadrilateral in which BA and CD when produced meet in E and EA = ED. Prove that :

(i) AD || BC

(ii) EB = EC

Answer 1

Given : ABCD is a cyclic quadrilateral in which sides BA and CD are produced to meet at E and EA = ED.

To Prove: (i) AD || BC

(ii) EB = EC

Proof: $\because$ EA = ED

$\therefore In\Delta EAD$, $\angle EAD=\angle EDA$ (Angles opposite to equal sides)

In a cyclic quadrilateral ABCD, Ext. $\angle EAD=\angle C$

Similarly, Ext. $\angle EDA=\angle B$

$\because \angle EAD=\angle EDA$

$\therefore \angle B=\angle C$

$\therefore EC=EB$ (Sides opposite to equal angles)

and $\angle EAD=\angle B$

But these are corresponding angles.

$\therefore AD||BC$