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

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Solution

In $△ A O B$ and $△ P X Q$

$A O = P X$ [Radius of congruent circles are equal]

$∠ A O B = ∠ P X Q$ [Given]

$B O = Q X$ [Radius of congruent circles are equal]

$△ A O B ≅ △ P X Q$ [SAS congruence rule]

$∴ A B = P Q$ [CPCT]

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Q. Recall that two circles are congruent if they have the same radius then equal chords of congruent circles subtend equal angles at their centres.

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

In △AOB and △PXQAO=PX [Radius of congruent circles are equal]∠AOB=∠PXQ [Given]BO=QX [Radius of congruent circles are equal]△AOB≅△PXQ [SAS congruence rule]∴AB=PQ [CPCT]

In $△ A O B$ and $△ P X Q$

$A O = P X$ [Radius of congruent circles are equal]

$∠ A O B = ∠ P X Q$ [Given]

$B O = Q X$ [Radius of congruent circles are equal]

$△ A O B ≅ △ P X Q$ [SAS congruence rule]

$∴ A B = P Q$ [CPCT]