# Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

**Solution:**

Using equal angles at the centers and the fact that circles are congruent, we prove the statement using Side-Angle-Side (SAS criteria) and corresponding parts of congruent triangles (CPCT).

Draw chords QR and YZ in two congruent circles as shown above. Join the radii PR, PQ, and XY, XZ respectively.

Given that chords subtend equal angles at the center. So, ∠QPR = ∠YXZ.

We need to prove that chords are equal, that is, QR = YZ

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

PR = PQ = XZ = XY

Consider the two triangles ∆PQR and ∆XYZ.

PQ = XY (Radii are equal)

∠QPR = ∠YXZ (Chords subtend equal angles at center)

PR = XZ (Radii are equal)

By SAS criteria, ∆PQR is congruent to ∆XYZ.

So, QR = YZ (Corresponding parts of congruent triangles)

Hence proved if chords of congruent circles subtend equal angles at their center then the chords are equal.

**Video Solution:**

## Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal.

### Maths NCERT Solutions Class 9 - Chapter 10 Exercise 10.2 Question 2:

**Summary:**

If chords of congruent circles subtend equal angles at their centers, then the chords are equal.