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.