If chords of congruent circles subtend equal angles at their centres, then

the chords are equal.

Concept: By using equal angles at the centres and the assertion that circles are congruent, we confirm the statement using Side-Angle-Side (SAS) criteria and corresponding parts of congruent triangles (CPCT).

First, draw chords QR and YZ in two congruent, then connect the radii PR, PQ, and XY, XZ respectively.

It is given that chords subtend equal angles at the center.

Then, $\angle QPR=\angle YXZ$.

Since the circles are congruent, their radii will be equal.

$PR=PQ=XZ=XY$

Let’s consider the two triangles$∆PQR and ∆xyz$.

$PQ=XY$

We know chords subtend equal angles at center.

$\angle QPR=\angle YXZ$

$PR=XZ$

By using SAS criteria, 

$∆PQR$is congruent to $∆XYZ$.

By CPCT,

Thus, $QR=YZ$

So, the chords are equal if chords of congruent circles subtend equal angles at their centres.

Hence, the given statement is true.