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All courses Geometry · G-C.2 15 of 57
Analyze relationships among inscribed angles, central angles, circumscribed angles, radii, chords, diameters, and tangents.

Find an inscribed angle measure from its intercepted arc

Problem
Find the inscribed angle measure when the intercepted arc is \(80^{\circ}\).
Unworked circle diagram showing only the supplied geometric configuration and measures. Open full size
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Hint

Use the inscribed-angle theorem: an inscribed angle measures half its intercepted arc.

Inscribed angle = \(1/2\) × intercepted arc.

Solution walkthrough

01

Verify theorem applicability

\[\text{vertex}~C~\text{lies}~\text{on}~\text{the}~\text{circle};~\text{rays}~CA~\text{and}~CB~\text{intercept}~\text{arc}~AB\]

An angle with vertex on the circle and sides through arc endpoints is an inscribed angle, so the inscribed-angle theorem applies.

02

Apply the theorem

\[m~\angle~ACB=(1/2)(m~\text{arc}~AB)=(1/2)(80^{\circ})\]

An inscribed angle measures half its intercepted arc.

03

Calculate and interpret

\[m~\angle~ACB=40^{\circ}\]

The angle at C is 40 degrees, matching the inspected diagram's highlighted arc and vertex.

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Another way

  1. Use arc measure=2 times inscribed angle and solve 80=2x.

!

Common mistake

Do not set the angle equal to 80 degrees. Equality with arc measure applies to a central angle, not an inscribed angle.

Answer circle diagram highlighting the defining configuration, applicable theorem, and exact calculation.
angle measure: 40 degrees

Solution walkthrough video