Question

In the given figure, P is the centre of the circle. chord AB and chord CD intersect on the diameter at the point E

If $\angle$AEP $\cong$ $\angle$DEP then prove that AB = CD.

Answer 1

Construction: Draw perpendicular from point P to the chords CD and AB.
To prove: AB = CD
Proof: In $△$MEP and $△$NEP,
$\angle$EMP = $\angle$ENP = 90$°$
$\angle$AEP = $\angle$DEP (Given)
Thus, by AAS congruence rule, $△$MEP$\cong$$△$NEP.
So, MP = NP (CPCT)
We know that chords equidistant from the centre are equal to each other.
Hence, AB = CD.