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:

Two congruent circles with centres O and O′. AB and CD are equal chords of the circles with centres O and O′  respectively.

To Prove : ∠AOB = ∠CO’D

Proof :

In triangles AOB and CO’D,

OA = O’C [since radii of congruent circles)

OB = O’D [Since radii of congruent circles]

AB = CD [Given]

⇒ ∆AOB ≅ ∆CO′D [SSS axiom]

⇒ ∠AOB ≅ ∠CO′D (CPCT)

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

Solution:

Two congruent circles with centres O and O′. AB and CD are chords of circles with centre O and O′ respectively such that ∠AOB = ∠CO′D

To Prove : AB = CD

Proof : In triangles AOB and CO′D,

AO = CO[Radii of congruent circle]

BO = DO [Radii of congruent circle]

∠AOB = ∠CO′D [Given]

⇒ ∆AOB ≅ ∆CO′D [SAS axiom]

⇒ AB = CD [CPCT]

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