Equal Arcs Theorem

1. What is the Equal Arcs Theorem?

The Equal Arcs Theorem states that:

In the same circle or in congruent circles, equal arcs subtend equal central angles and correspond to equal chords.

This theorem links arc length, chord length, and angle measure.

2. Equal Arcs → Equal Central Angles

If two arcs have the same length (or the same degree measure), then the central angles that subtend them are equal.

\overset{\frown}{AB} = \overset{\frown}{CD} \Rightarrow \angle AOB = \angle COD

2.1. Why?

The measure of an arc is directly defined by its central angle. If the arcs match in size, the central angles must match as well.

3. Equal Arcs → Equal Chords

Equal arcs always correspond to equal chords.

If arc \(AB\) equals arc \(CD\), then:

AB = CD

3.1. Reason

Chords that subtend equal angles at the centre are equal in length, so equal arcs must have equal chords.

4. Converse of the Equal Arcs Theorem

The converse is also true:

If chords are equal or if central angles are equal, then the arcs they subtend are equal.

4.1. Equal Central Angles → Equal Arcs

\angle AOB = \angle COD \Rightarrow \overset{\frown}{AB} = \overset{\frown}{CD}

4.2. Equal Chords → Equal Arcs

AB = CD \Rightarrow \overset{\frown}{AB} = \overset{\frown}{CD}

5. Visual Understanding

Imagine two equal curves on the boundary of a circle. If you connect the endpoints of each curve, the resulting chords look identical and the angles at the centre also match. Equal arcs create completely symmetric structures inside the circle.

6. Example

In a circle, suppose arc \(AB\) and arc \(CD\) both measure \(60^\circ\).

Then:

\angle AOB = \angle COD = 60^\circ