**Circles**

**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 centres**

**Solution:**

Given: Two congruent circles with centres O and O’ and radii r, which have chords AB and CD respectively such that AB = CD.

To Prove: ∠AOB = ∠CO’D

Proof: In ∆AOB and ∆CO’D, we have

AB = CD [Given]

OA = O’C [Each equal to r]

OB = O’D [Each equal to r]

∴ ∆AOB ≅ ∆CO’D [By SSS congruence criteria]

⇒ ∠AOB = ∠CO’D [C.P.C.T.]

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

**Solution:**

Given: Two congruent circles with centres O & O’ and radii r which have chords AB and CD respectively such that ∠AOB = ∠CO’D.

To Prove: AB = CD

Proof: In ∆AOB and ∆CO’D, we have

OA = O’C [Each equal to r]

OB = O’D [Each equal to r]

∠AOB = ∠CO’D [Given]

∴ ∆AOB ≅ ∆CO’D [By SAS congruence criteria]

Hence, AB = CD [C.P.C.T.]