Angles and Arcs

Overview

Every angle formed with a circle is tied to an arc. An angle whose vertex is the center measures exactly its intercepted arc, while an angle whose vertex is on the circle measures half of its intercepted arc. This lesson names the arcs (minor, semicircle, major), adds adjacent arcs, and uses the fact that all the arcs of a circle total 360°.

Vocabulary

Central Angle: An angle whose vertex is the center of a circle and whose sides contain radii of the circle.
Arc: An unbroken part of a circle, named by its endpoints. The measure of an arc is the measure of its central angle.
Minor Arc: An arc whose measure is less than 180°. A minor arc is named by its two endpoints.
Semicircle: An arc whose measure is exactly 180°. A semicircle is named by three points: its two endpoints and one point on the arc.
Major Arc: An arc whose measure is greater than 180°. A major arc is named by three points: its two endpoints and one point on the arc.
Inscribed Angle: An angle whose vertex is on the circle and whose sides contain chords of the circle.
Intercepted Arc: The arc that lies in the interior of an angle and has endpoints on the sides of the angle.

Main Ideas

The measure of a central angle is equal to the measure of its intercepted arc.
The measure of an inscribed angle is equal to half the measure of its intercepted arc.
Always focus on where the vertex is: at the center, the angle equals its arc; on the circle, the angle is half its arc.
Arc Addition: the measures of adjacent arcs add to the measure of the arc they form together.
The arcs that make up a whole circle total 360°, so a major arc is 360° minus its matching minor arc.