We know that,
The circles are congruent, their radii will be equal.
Let, there be two circles with center P and X with equal radius.
Since,
Chords of congruent circles subtend equal angles at their centres
∴ ∠QPR = ∠YXZ.
⇒ PR = PQ = XZ = XY
In ∆ PQR and ∆ XYZ,
⇒ PQ = XY (Radius of congruent circles are equal)
⇒ ∠QPR = ∠YXZ (Chords subtend equal angles at center)
⇒ PR = XZ (Radius of congruent circles are equal)
∴ ∆ PQR ≅ ∆ XYZ. (By S.A.S. congruence rule)
We know that,
Corresponding parts of congruent triangles are equal.
∴ QR = YZ (By C.P.C.T.)
Hence, proved that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.