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Recall that two circles are congruent if they have the same radii. Prove that equalchords of congruent circles subtend equal angles at their centres.
A circle is a collection of points which are equidistant from a fixed point. This fixed point is called as the centre of the circle and this equal distance is called as radius of the circle. And thus, the shape of a circle depends on its radius. Therefore, it can be observed that if we try to superimpose two circles of equal radius, then both circles will cover each other. Therefore, two circles are congruent if they have equal radius.
Consider two congruent circles having centre O and O' and two chords AB and CD of equal lengths.
In ΔAOB and ΔCO'D,
AB = CD (Chords of same length)
OA = O'C (Radii of congruent circles)
OB = O'D (Radii of congruent circles)
∴ ΔAOB ≅ ΔCO'D (SSS congruence rule)
⇒ ∠AOB = ∠CO'D (By CPCT)
Hence, equal chords of congruent circles subtend equal angles at their centres.
Prove that if chords of congruent circles subtend equal angles at their centres, thenthe chords are equal.
Let us consider two congruent circles (circles of same radius) with centres as O and O'.
In ΔAOB and ΔCO'D,
∠AOB = ∠CO'D (Given)
OA = O'C (Radii of congruent circles)
OB = O'D (Radii of congruent circles)
∴ ΔAOB ≅ ΔCO'D (SAS congruence rule)
⇒ AB = CD (By CPCT)
Hence, if chords of congruent circles subtend equal angles at their centres, then the chords are equal.
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