Exterior Angles Conjectures
An exterior angle for a polygon is formed by extending one side of the polygon from one if its endpoints. From this, we see that an exterior angle and interior angle form a linear pair of angles. If you extend each side of a polygon to form one exterior angle at each vertex, you get a set of exterior angles. This conjecture tells us that the sum of a set of exterior angles is 360 degrees.
This result, all by itself is not so exciting. The resulting corollaries about regular polygons are much more interesting. For example, the first corollary gives a formula for finding the measure of an exterior angle to a regular n-gon. Putting this together with the Linear Pairs Conjecture would give us a way to calculate the measure of each interior angle of a regular n-gon.
(See: Corollary to Polygon Sum Conjecture which states that the measure of each interior angle of a regular n-gon is 180*(n-2)/n.)
The precise statement of the conjecture is:
Conjecture (Exterior Angle Conjecture ):
The sum of the n exterior angles for any convex polygon with n sides is 360 degrees.
Corollary (Exterior Angle Measures for Regular n-gons ): Each exterior angle for a regular n-gon has measure equal to 360/n degrees.
Explanation of the Corollary: There are n exterior angles, each with the same measure, and which add up to 360 degrees. Therefore, each of them must have measure equal to 360 divided by n.
Interactive Sketch Pad Demonstrations:
Please feel free to try the activity sheet associated with this conjecture.
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