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Theorem 1 : Equal chords of a circle subtend equal angles at the centre.

Proof :Consider a circle and draw two equal chords $AB$ and $CD$ of a circle with center $O$ as shown in the figure 1.

Fig.1

We want to prove that : $\angle AOB=\angle COD$

From the triangles, $AOB$ and $COD$, we get

$OA=OC$ (Radii of a circle)

$OB=OD$ (Radii of a circle)

$AB=CD$ (Given)

By, using Side-Side-Side (SSS Rule), we can write:

$\triangle AOB\cong \triangle COD$

As the triangles are congruent, the angles should be of equal measurement.

Therefore, $\angle AOB=\angle COD$ [Using Corresponding parts of the congruent triangle (CPCT)]

Hence, the above theorem is proved.

Theorem 2 : If the angles subtended by the chords of a circle at the centre are equal, then the chords are equal.