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← 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, $∠ A D B$ and $∠ A C B$ in the above diagram are congruent angles.

$∠ A D B ≅ ∠ A C B$

Proof

Consider two inscribed angles $∠ A D B$ and $∠ A C B$ that intercept the same arc $A B$ in a circle.

By the Inscribed Angle Theorem, the measures of

∠ A D B

and

∠ A C B

are half the measure of

A B .

m ∠ A D B = \frac{1}{2} m A B m ∠ A C B = \frac{1}{2} m A B

Therefore, by the Transitive Property of Equality, it can be said that these two angles have the same measure.

m ∠ A D B = m ∠ A C B

Consequently, by the definition of congruent angles,

∠ A D B

and

∠ A C B

are congruent.

$∠ A D B ≅ ∠ A C B$

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