Question

Recall the two circles are congruent if they have the same radii .Prove that equal chords of congruent circles subtend equal angles at their centres​

Answer 1

Answer:

As per the theorem; equal chords (of a circle) subtend equal angles at the center. Hence, it is clear that equal chords of congruent circles would subtend equal angles at their centers. The theorem says that if two chords subtend equal angles at center then the chords are equal. Hence, if two chords in two congruent circles subtend equal angles at their centers then the chords are equal.

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Answer 2

Answer:

To recall, a circle is a collection of points whose every point is equidistant from its centre. So, two circles can be congruent only when the distance of every point of both the circles are equal from the centre.

For the second part of the question, it is given that AB = CD i.e. two equal chords.

Now, it is to be proven that angle AOB is equal to angle COD.

Proof:

Consider the triangles ΔAOB and ΔCOD,

OA = OC and OB = OD (Since they are the radii of the circle)

AB = CD (As given in the question)

So, by SSS congruency, ΔAOB ΔCOD

∴ By CPCT we have,

AOB = COD. (Hence proved).