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. We have a circle having its centre at O and two equal chords AB and CD such that they subtend ∠AOB and ∠COD respectively at the centre, i.e. at O. We have to prove that ∠AOB = ∠COD Now, in ΔAOB and ΔCOD, we have AO = CO [Radii of the same circle] BO = DO [Radii of the same circle] AB = CD [Given] ∴ ΔAOB ≌ ΔCOD [SSS criterion] ⇒ Their corresponding parts are equal. ∴ ∠AOB = ∠COD 2. Prove that if chords of congruent circles subtend equal angles at their centres, then the chords are equal. Solution: We have a circle having its centre at O, and its two chords AB and CD such that ∠AOB = ∠COD We have to prove that AB = CD ∵ In ΔAOB and ΔCOD, we have: AO = CO [Radii of the same circle] BO = DO [Radii of the same circle] ∠AOB = ∠COD [Given] ∴ ΔAOB ≌ ΔCOD [SAS criterion] ∴ Their corresponding parts are equal, i.e. AB = CD