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 centers.

Answer 1

Circle property:

Let we are having two circles having centers $O$ and $O\text{'}$ and having an equal radius $\left(r\right)$

Let we are having chords $\mathrm{AB}\mathrm{and}\mathrm{CD}$ of equal length.

$AB=CD$ i.e. two equal chords.

From $\Delta AOBand\Delta COD$,

$OA=O\text{'}C$ ( Radii of congruent circles )

$OB=O\text{'}D$ (Radii of congruent circles)

$AB=CD$ (Given)

So, by $\text{SSS}$ congruency,

$\Delta AOB\cong \Delta COD$

∴ By CPCT we get

$\angle AOB=\angle COD.$

Hence it is proved that equal chords of congruent circles subtend equal angles at their centers.