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

 Solution:

Consider the following diagram-

Here, it is given that ∠AOB = ∠COD i.e. they are equal angles.

Now, we will have to prove that the line segments AB and CD are equal i.e. AB = CD.

Proof:

In triangles AOB and COD,

∠AOB = ∠COD (as given in the question)

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

So, by SAS congruency, ΔAOB ≅ ΔCOD.

∴ By the rule of CPCT, we have

AB = CD. (Hence proved).