Theorem – If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal

If the angles subtended by chords of a circle at the center are equal, then the chords are equal.

Given:  A circle with center O, AB and CD are chords of circle that subtend equal angles at center O.

i,e. ∠AOB = ∠COD.

To prove: chord AB = chord CD

Proof: In △AOB and △ COD
OA = OC (Radius of circle)
∠AOB = ∠COD (Given
OB = OD (Radius of circle)
AB = CD  (Given)

Therefore, △AOB ≅ △ COD (SAS rule)

    ∴   AB = CD (Corresponding parts of congruent triangles)

    Hence proved

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