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Arcs of a Circle
Lesson Demo

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.