“Finding Angle and Arc Measures in a Circle”

by

Christopher R. Whitworth

I am going to prove that when you are given a circle with an inscribed triangle with vertexes that intersects the circle, you can determine the measure of the arc by the measure of the angle angles, and vice versa.  The theorem that an angle’s measure is half of what its corresponding arc angle is will be proven.  The first step will be to create a circle with an inscribed triangle as shown below and to place a point between each vertex of the triangle on the circle.

Then, you must label the points the triangle.

Finally, you will measure the angles of the triangle either using Geometers Sketchpad 4.0 or a protractor.

Notice that none of the angles are equal to each other.  Since this is true, none of the arc angle measurements should be the same.  The next step will be to hide the angle measurements and to measure the arcs of the figure by using Geometers Sketchpad 4.0.

Then, you will want to show the angle measurements again so that you can compare each angle measurement with its corresponding arc measurement.

Then, you should divide each arc measurement by its corresponding angle measurement.  If your final answer is approximately 2.00, then the theorem has been tested and proven.

The theorem has been proven.  An arc’s measurement is twice as much as its corresponding angle’s measurement.