Inscribed Angles & the Inscribed-Angle Theorem

An inscribed angle has its vertex on the circle, with two chords as its sides. The arc that sits inside the angle — between where the two sides meet the circle — is called the intercepted arc.

The inscribed-angle theorem says the angle is half the arc: $m\angle = \dfrac{1}{2}\,m\overset{\frown}{\text{arc}}$ . Compare that with a central angle, which equals its arc exactly. Moving the vertex from the center out to the circle cuts the angle in half.

The theorem, both directions

Given the arc, halve it to get the inscribed angle. A $96^\circ$ arc gives a $48^\circ$ inscribed angle.

Given the angle, double it to get the arc. A $40^\circ$ inscribed angle intercepts an $80^\circ$ arc.

Before you apply either direction, check where the vertex is. Vertex at the center: angle equals arc. Vertex on the circle: angle is half the arc. Mixing these up is the single most common error in this unit.

Two consequences worth memorizing

Inscribed angles that intercept the same arc are congruent. Two different vertices on the circle looking at the same arc each measure half of it, so they measure the same.

An angle inscribed in a semicircle is a right angle. If the intercepted arc is a semicircle, it measures $180^\circ$ , and half of $180^\circ$ is $90^\circ$ . Any triangle drawn with one side as a diameter is automatically a right triangle — as the figure below shows.

Worked examples

Example 1: angle from the arc

Inscribed angle $ABC$ intercepts arc $AC$ , which measures $124^\circ$ . Find $m\angle ABC$ .

Inscribed-angle theorem

m\angle ABC = \dfrac{1}{2}\,m\overset{\frown}{AC}

Substitute

m\angle ABC = \dfrac{124^\circ}{2}

Simplify

m\angle ABC = 62^\circ

Answer: $m\angle ABC = 62^\circ$

Example 2: arc from the angle

An inscribed angle measures $35^\circ$ . Find the measure of its intercepted arc.

The arc is double the inscribed angle

m\overset{\frown}{\text{arc}} = 2 \cdot 35^\circ

Simplify

m\overset{\frown}{\text{arc}} = 70^\circ

Answer: $70^\circ$

Example 3: inscribed in a semicircle

In circle $O$ , $\overline{AC}$ is a diameter and $B$ is another point on the circle. Find $m\angle ABC$ .

The intercepted arc is a semicircle

m\overset{\frown}{AC} = 180^\circ

Inscribed-angle theorem

m\angle ABC = \dfrac{1}{2}(180^\circ)

Simplify

m\angle ABC = 90^\circ

Answer: $m\angle ABC = 90^\circ$ — a right angle

Try one yourself

Common questions

How is an inscribed angle different from a central angle?

The vertex location. A central angle's vertex is at the center and equals its arc. An inscribed angle's vertex is on the circle and equals half its arc. Same arc, and the central angle is exactly twice the inscribed angle.

Which arc is the intercepted arc?

The arc that lies in the interior of the angle — the piece of the circle between the two points where the angle's sides meet it, on the far side from the vertex. It never contains the vertex itself.

Do two inscribed angles on the same arc have to be equal?

Yes. Each one is half of the same intercepted arc, so they're congruent no matter where their vertices sit on the rest of the circle.

Why does a diameter always create a right angle?

A diameter splits the circle into two $180^\circ$ arcs. An inscribed angle whose sides reach the ends of the diameter intercepts one of those semicircles, and half of $180^\circ$ is $90^\circ$ .

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Inscribed Angles & the Inscribed-Angle Theorem

The theorem, both directions

Two consequences worth memorizing

Worked examples

Example 1: angle from the arc

Example 2: arc from the angle

Example 3: inscribed in a semicircle

Try one yourself

Common questions

Keep learning

Central Angles & Arc Measure

Tangents, Secants & Chords: Angle Relationships

Arcs & Chords

Circle Vocabulary: Center, Radius, Chord, Secant, Tangent