Circle Theorems
Rule

Inscribed Angles of a Circle Theorem

If two inscribed angles of a circle intercept the same arc, then they are congruent.
A circle with two inscribed angles that intercept the same arc

By this theorem, and in the above diagram are congruent angles.

Proof

Consider two inscribed angles and that intercept the same arc in a circle.

Two inscribed angles intercept the same arc
By the Inscribed Angle Theorem, the measures of and are half the measure of
Therefore, by the Transitive Property of Equality, it can be said that these two angles have the same measure.
Consequently, by the definition of congruent angles, and are congruent.

Exercises