Geometry Tool Belt: Inscribed Angles

Before beginning this lesson, you should have learned about the following:

• The interior angles of a triangle add up to 180 degrees
• The definition of a circle, an arc, a central angle, an arc length, and an inscribed angle.

• The measure of an arc length is the measure of its central angle
• The definition of supplementary angles: angles A and B are supplementary when A+B = 180 degrees. 

Given a circle with an inscribed angle like  with inscribed angle a and arc b, what is the relation ship between a and b?

The formula relating these two values is

That is, the measure of an inscribed angle is half the measure of the arc its endpoints define.  

So or example, if we have an inscribed angle with of 40 degrees, then the arc length is 80 degrees.

Likewise if we're given an arc length of 50 degrees, then any inscribed angle with line segments intersecting the end points of the arc, where ever its vertex happens to be on the circle, will have an angle measure of 25 degrees.

Now that we know the formula, we should know how it came about.  Working through the proof of this relationship will complete your understanding of the concept, and exercise your knowlege in the geometry you've learned so far.