Question

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

Answer 1

$PROOF$

We are given two equal chords AB and CD of a circle with centre O as given in the figure.

We want to prove that angle ABC is equal to angle COD

In triangles ABC and triangle COD,

OA = OC. (Radii of a circle)

OB = OD. (Radii of a circle)

AB = CD. (Given)

Therefore,. ∆AOB ≈ ∆COD. (SSS rule)

This gives Angle AOB = Angle COD

(Corresponding parts of congruent Triangle)

HENCE PROVED

Answer 2

GIVEN:-

AB=CD

TO PROVE:-

$\large\sf{\angle{AOB}=\angle{COD}}$

PROOF:-

$\large\sf{In\:∆AOB\:and\:∆COD,}$

$\large\sf{OA=OC(given)}$

$\large\sf{OB=OD(radii)}$

$\large\sf{AB=CD(radii)}$

$\large\therefore$ $\large\sf{∆AOB\:\cong\:∆COD}$ (By SSS congruence rule)

$\large\sf{\angle{AOB}=\angle{COD}(By\:CPCT)}$