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Rule

Congruent Central Angles Theorem

In a circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.

Based on the above diagram, the theorem can be written as follows.

Proof

The biconditional statement will be proved in two parts.

If Two Minor Arcs Are Congruent, Then the Corresponding Central Angles Are Congruent

Consider a circle and two congruent minor arcs.

By definition, the measure of a minor arc is equal to the measure of its related central angle. Since these two arcs are congruent, they have the same measure. Then, by the Transitive Property of Equality, the two central angles have the same measure.
Therefore, and are congruent angles.

It has been proved that if two minor arcs are congruent, then their corresponding central angles are congruent.

If Two Central Angles Are Congruent, Then the Minor Arcs Are Congruent

Consider now a circle centered at point Let and be two congruent central angles.

By definition, the measure of a minor arc is equal to the measure of its related central angle. Since the central angles are congruent, they have the same measure. Again, by the Transitive Property of Equality, the two minor arcs have the same measure.
Consequently, since the arcs have the same measure, they are congruent.

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