Question

Recall that two circles are congruent if they have the same radii. Prove that equal chords of congruent circles subtend equal angles at their centres.

Answer 1

Let us consider a circle with center $O$ and two equal chords of a circle $AB$ and $CD.$

We need to prove that $\angle AOB=\angle COD$

In $△$ $AOB$ and $COD$, we have

$AO=CO$ (Radius of the circle)

$BO=DO$ (Radius of the circle)

$AB=CD$ (Equal chords)

By $SAS$ criterion of congruence, we have

$△AOB\cong △COD$

$\Rightarrow \angle AOB=\angle COD$