The Arc Angle Theorem (Inscribed Angle Theorem)

A triangle divides its circumcircle into three arcs. The arc angle of each these arcs is twice the opposite angle of the triangle.

One says that an angle inscribed in a circle subtends the opposite arc. So one can rephrase the theorem:

If an angle is incribed in a circle, then the subtended arc angle is twice the angle.

This theorem is true whether the angle is acute or obtuse.

Important special case:

The hypotenuse of a right triangle is a diameter of the circumcircle.

Conversely, if one side of a triangle is a diameter of a circle and the opposite vertex of the triangle is also on the circle, then that vertex is a right angle.

Two points on a circle divide the circle into two arcs. The longer arc is called the major arc, and the shorter arc is called the minor arc.

Geometer’s Sketchpad can measure arc angles in at least three different ways:

(1) If you select a circle and two points on the circle, and choose Arc Angle in the Measure menu, GSP will measure the minor arc (between 0 and 180 degrees).

(2) If you select a circle and two points on the circle, and choose Arc On Circle in the Construct menu, GSP will construct the arc that goes counterclockwise from the first point selected to the second point selected. (This could be the major arc or the minor arc.) If you select this constructed arc and choose Arc Angle in the Measure menu, GSP will measure the arc angle of this arc, which could be bigger than 180 degrees or less than 180 degrees.

(3) If you select a circle and three points on the circle, and choose Arc Through 3 Points in the Construct menu, GSP will construct the arc between the first and last points that goes through the second point. If you select this constructed arc and choose Arc Angle in the Measure menu, GSP will measure the arc angle of this arc, which could be bigger than 180 degrees or less than 180 degrees.

Illustration of methods (1) and (2):

In class we used method (1) to measure arc angles, so we came up with two different cases of the arc angle theorem, depending on whether the angle being measured was acute or obtuse. But if we use method (2) or (3) to measure arc angles, and we make sure to measure the correct arc -- opposite the angle being measured -- then there’s no need for two cases. For example, you could use method (3) and construct a point on the arc using the bisector of the inscribed angle.