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Question:

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:

Let us consider a circle with center O and two equal chords of a circle AB and CD. We need to prove that ∠AOB = ∠COD. In △AOB and △COD, we have AO = CO (Radius of the circle) BO = DO (Radius of the circle) AB = CD (Equal chords). By SAS criterion of congruence, we have △AOB ≅ △COD ⇒ ∠AOB = ∠COD