Question

prove that equal chords of a circle subtend equal angles at the centre

Answer 1

Hi there,

Given:  AB and CD are equal chords of the same circle with center as O.
To prove: angle AOB = angle COD

Proof:   In triangle AOB and triangle COD.
             AO = CO (radii of the same circle)
             AB = CD (given)
             OB = OC (radii of the same circle)

Therefore triangle AOB is congruent to triangle COD by SAS congruence rule..

This implies  angle AOB is equal to angle COD....
                                        Hence proved...

Hope this helps you.....
Thank you

Answer 2

Statement : Equal chords of a circle subtend equal angles at the centre.

Given : AB and CD are chords of a circle with centre O, such that AB = CD.

To prove : ∠AOC = ∠COD

Proof :

In ΔAOB and ΔCOD,

AO = CO [radii of same circle]
BO = DO [radii of same circle]
AB = CD [given]

ΔAOB ≅ ΔCOD [SSS]
∠AOB = ∠COD [C. P. C. T]

Hence, Equal chords of a circle subtend equal angles at the centre.