10.2 Arcs and Central Angles
Notes
A central angle has its vertex at the center. Its measure equals the measure of its intercepted arc.
- Minor arc: less than $180^\circ$, named with two letters, $\overset{\frown}{AB}$.
- Major arc: more than $180^\circ$, named with three letters, $\overset{\frown}{ACB}$. Its measure is $360^\circ$ minus the minor arc.
- A semicircle measures $180^\circ$.
- Arc Addition: adjacent arcs add.
- Example: $m\angle AOB = 70^\circ$, so $m\overset{\frown}{AB} = 70^\circ$ and the major arc is $290^\circ$.
Practice
Central angle $\angle POQ$ measures $85^\circ$. What is the measure of the major arc $\overset{\frown}{PRQ}$?
$360 - 85 = 275$.
In circle $O$, $\overline{AC}$ is a diameter and $m\angle AOB = 64^\circ$. Find $m\overset{\frown}{BC}$.
$\overset{\frown}{AB}$ and $\overset{\frown}{BC}$ make a semicircle: $180 - 64 = 116$.
A circle is divided into central angles of $x^\circ$, $2x^\circ$ and $3x^\circ$. What is the largest arc?
$6x = 360$, so $x = 60$ and $3x = 180$.
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