Arcs and Central Angles
Apply the Arc Addition Postulate and the relationship that a central angle and its intercepted arc have equal measures. Solve for arc measures in multi-arc diagrams.
A lesson of Geometry, in Circles, about 10 minutes.
By the end of this lesson, you will be able to find arc measures using central angles, apply the Arc Addition Postulate, use congruent arcs and chords, and prove that congruent chords have equidistant centers.
The Big Idea
The central angle is the key to unlocking arc measure — they are equal by definition. This simple relationship cascades into theorems about chords and their distances from the center, giving us a powerful toolkit for circle problems.
Arc Measure and Central Angles
Central Angle Theorem: The measure of a central angle equals the measure of its intercepted arc.
$$m\angle AOB = m\overset{\frown}{AB}$$
A full circle has 360°. A semicircle = 180°.
Arc Addition Postulate
If $C$ is a point on arc $\overset{\frown}{AB}$ (between $A$ and $B$ along the arc):
$$m\overset{\frown}{AC} + m\overset{\frown}{CB} = m\overset{\frown}{AB}$$
This mirrors the Segment Addition Postulate.
Congruent Chords and Arcs
Theorem: In the same circle (or congruent circles):
- Congruent chords have congruent arcs.
- Congruent arcs have congruent chords.
$$AB = CD \iff \overset{\frown}{AB} \cong \overset{\frown}{CD}$$
Perpendicular Bisector of a Chord
Theorem: The perpendicular bisector of a chord passes through the center of the circle.
Corollary: The perpendicular from the center to a chord bisects the chord.
Converse: The segment from the center to the midpoint of a chord is perpendicular to the chord.
Equidistant Chords
Theorem: In the same circle, chords equidistant from the center are congruent.
Converse: Congruent chords are equidistant from the center.
Example 1 — Arc Measure from Central Angle
Central angle $\angle AOB = 130°$. Find minor arc $\overset{\frown}{AB}$, major arc $\overset{\frown}{AB}$, and the arc if the radius is 10.
Minor arc $= 130°$
Major arc $= 360° - 130° = 230°$
Arc length (in the next lesson — preview): $\dfrac{130}{360} \times 2\pi(10) \approx 22.7$ units
Example 2 — Arc Addition
Circle $O$ with points $A$, $B$, $C$ on the circle. $m\overset{\frown}{AB} = 95°$, $m\overset{\frown}{BC} = 115°$. Find $m\overset{\frown}{AC}$ (the minor arc) and $m\overset{\frown}{ABC}$.
$m\overset{\frown}{ABC} = 95 + 115 = 210°$ (major arc from $A$ to $C$ through $B$)
Minor arc $\overset{\frown}{AC} = 360 - 210 = 150°$