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. The Lesson:
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:
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: 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
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 . Let's Practice:
- 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.
- 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.
- 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.