Every day, you come across many things circular in shape. The collection of all that points in a plane that are at a fixed distance from a fixed point in the plane is called a circle. The fixed point is called the centre of the circle, and the fixed distance is called the radius of the circle.
A part of a circle is called an arc. Arcs of a circle that superimpose each other completely are called congruent arcs. A segment with its endpoints on a circle is called a chord. A diameter is the longest chord. If two arcs of a circle are congruent, then their corresponding chords are equal. Conversely, if two chords of a circle are equal, then their corresponding arcs are congruent.
Corresponding Arcs Of Two Equal Chords Of A Circle Are Congruent
Theorem: Congruent arcs of a circle subtend equal angles at the centre.
Given: Two congruent arcs AB and CD.
To prove: ∠ AOB = ∠ COD
Construction: Draw chords AB and CD.
Proof: The angle subtended by an arc at the centre is equal to the angle subtended by its corresponding chord at the centre.
In the given figure,
AB = CD (Chords corresponding to congruent arcs of a circle are equal)
∠ AOB = ∠ COD (Equal chords subtend equal angles at the centre)
Hence, the theorem is proved.