Angles in a Circle

By Noelle Francell

The vertex of an angle can appear on, inside, or outside a circle. How does the location and measure of the vertex angle compare with the measure(s) of the arc(s) it intercepts?

In the case of an angle that is formed on the outside of a circle, the measure of the difference between the two arcs that are formed by the intersection of the lines is twice as much as the measure of the vertex angle formed by this intersection.  This can be shown by drawing a circle on GSP and then drawing two lines which intersect outside the circle (see above).

Next, measure the vertex angle by clicking on the points and then measure angle.

Once this is done, construct a point on the arc between the two intersections. Construct and arc between three points, and then measure them.

Now, calculate the difference between the larger and the smaller arcs that are created by the intersections of the secant lines.

Now compare this value to the measure of the angle that is formed by the intersection of these two lines outside the circle (<ABC)

This shows the relationship between the difference in arc measures that the vertex angle measure. By changing the position of the vertex angle, it is clear that the ratio remains the same.

In the case of an angle formed on the interior of the circle, the sum of the arcs that are formed from the intersection of the two lines is twice the measure of  the vertex of the interior angle.

Finally, when the vertex of the angle lies on the circle, the measure of the arc that is formed by the two intersecting lines is exactly have the measure of the angle formed.

Again, moving the vertex along the circle does not change the ratio.

In closing, when an angle is formed by the intersection of two lines outside the circle, the measure of the sum of the two arcs formed by the intersecting lines is exactly twice as much as the measure of the vertex angle. When an angle is formed by the intersection of two lines inside the circle, the measure of the difference of the two arcs formed by the intersecting lines is exactly twice as much as the measure of the vertex angle. When the vertex of the angle lies on the circle, the measure of the arc formed is exactly twice as much as the angle formed by the intersecting lines.