Angles Within a Circle

Introduction: A circle is all points equidistant from one point called the center of the circle. Segments drawn within the circle create angles which we define and measure.

We show circle O below. A circle is named based on the name of the point which is the center. The segment OA is a radius of the circle.

Definition: A radius is the segment connecting (sometimes referred to as the “distance between”) the center and the circle itself.

Important facts: If points C, D, and E are also on this circle, then the following we know the following information:

• OC is a radius.


• .


• Arc .


• Arc .


• The measure of the central angle is the same as the arc of the circle intercepted by this angle.

Important fact: The measure of a central angle is the same as the measure of the intercepted arc.

Definition: The diagram below shows an additional angle within the circle O.The angle has a vertex F on the circle. This is called an interior angle.

Important fact: The measure of an interior angle is one half of the measure of the intercepted arc.

Therefore .

1. In the diagram below, circle O is given with angle . What are the measures of arc and angle ?

Since , we have =because the measure of a central angle is the same as the measure of the intercepted arc.

Since , we have intercepting an arc of 100º. This inscribed angle has a measure of half the intercepted arc which is 50º.

2. The diagram given below shows circle O with central angle . Find the measures of the following: , , ,

since it is intercepted by the central angle .

To find the measure of , notice that AE is a diameter and the arc from A to E must be 180º. This leaves of arc from C to E and therefore . We could also note that is supplementary to .

is an inscribed angle intercepting an arc of 65º. Therefore .

is also 32.5º since triangle ACO is isosceles because both OA and OC are radii of the same circle and must have the same lengths.

3. In circle O at right, arc and . Find the measures of all the numbered angles.

Angle 1 is because it is an inscribed angle intercepting an arc of 98º.

Similarly angle 3 is 34º.

Angle 4 is 98º because it is a central angle intercepting an arc of 98º.

This makes angle 5 82º because it is supplementary to angle 4.

Angle 6 is because it is an inscribed angle intercepting the arc from Q to A which is one half of the circle minus .

Angle 2 is 90º because it is an inscribed angle intercepting half the circle, which is 180º.