### Question 1. 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.

**Solution:**

Given:

Two Congruent Circles

C1andC2

ABis the chord of C1

and

PQis the chord of C2AB = PQ

To Prove: Angle subtended by the Chords AB and PQ are equal i.e. ∠AOB = ∠PXQ

Proof:In △AOB & △PXQ

AO = PX (Raduis of congruent circles are equal)

BO = QX (Raduis of congruent circles are equal)

AB = PQ (Given)

△AOB ⩭ △PXQ (SSS congruence rule)

Therefore, ∠AOB = ∠PXQ (CPCT)

### Question 2. Prove that if chords of congruent circles subtend equal angles at their centers, then the chords are equal.

**Solution: **

Given:

Two Congruent circles C1 and C2

AB is the chord of C1 and PQ is chord of C2

& ∠AOB = ∠PXQ

To prove :

In △AOB and △PXQ ,

AO = PX (Raduis of congruent circles are equal)

∠AOB = ∠PXQ (Given)

BO = QX (Raduis of congruent circles are equal)

△AOB ⩭ △PXQ (SAS congruence rule)

Therefore, AB = PQ (CPCT)