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**Q.1Â Â Â Â RecallÂ that two circles are congruentÂ if they have the same radii. Prove that equalÂ chords ofÂ congruent circles subtend equalÂ angles at theirÂ centers. **

**Sol. **

Given : AB andÂ CD are two equalÂ chords of a circleÂ with center at O.

To prove :

Proof : In AOB and COD,Â we have

AO = CO Â Â Â Â Â Â Â Â Â Â Â Â Â [Radii of the same circle]

BO = DO Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â Â [Radii of the same circle]

and Â Â Â AB = CDÂ Â Â Â Â Â Â Â [Given]

ThereforeÂ byÂ SSS criterionÂ of congruence, we have

Â Â Â Â Â Â Â Â Â Â Â Â Â Â [C.P.C.T.]

**Q.2Â Â Â Â Â Prove that if chords of congruent circles subtendÂ equalÂ angles at theirÂ centres, then the chords are equal. **

**Sol.Â **

Given : AB and CD are two chords such that angles subtended by these chords at the centre of theÂ circle are equal.Â

i.e. Â Â Â Â Â Â Â

To prove : Â Â Â Â Â Â AB = CD

ProofÂ : In AOB andÂ COD we have

AO = CO Â Â Â Â Â Â [Radii of the sameÂ circle]

BO = DO Â Â Â Â Â Â [Radii of the same circle]

and

ThereforeÂ Â By SAS criterionÂ of congruence,Â we have

Â Â Â Â Â Â AB = CD Â Â Â Â Â Â Â Â Â Â Â Â Â Â [C.P.C.T.]

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