Circles: Special Angles and Arcs in Circles

Central Angles

A central angle is an angle with its vertex at the center of the circle.  The measure of a central angle equals the measure of its intercepted arc.


Take a look at the circle below.

Circle C with points A and B on the circle, angle ACB is drawn
Circle C

If angle ACB equals 101 degrees, then arc AB equals 101 degrees.

Example:
Circle P has the following properties:

the measure of angle QPR equals the quantity 10 x minus 2, degrees; the measure of arc QR equals the quantity 8 x plus 14, degrees

Circle P with points Q and R on the circle, angle QPR is drawn
Circle P

Find the value of x.

Solution:
The measure of the central angle is equal to the measure of its intercepted arc.

the measure of angle QPR equals the measure of arc QR
10x – 2 = 8x + 14
10x = 8x + 16
2x = 16
x = 8