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Polygon Exterior Angles Theorem

Rule

Polygon Exterior Angles Theorem

The sum of the measures of the exterior angles of a convex polygon, on angle at each vertex, is

Hexagon with all the exterior angles marked

Based on the diagram above, the relation below holds true.

Proof

Consider a hexagon and all its exterior angles. Let be the sum of the measures of the exterior angles.

Hexagon with all the exterior angles marked

Notice that an interior angle and its exterior angle form a linear pair. Therefore, the sum of their measures is equal to By the Polygon Interior Angles Theorem, the sum of the measures of the interior angles of a polygon with sides is With this information, add all the equations in the system above. Finally, by solving the last equation for the desired result is obtained.

Although the proof above used a hexagon, the same reasoning can be applied no matter the number of sides of the polygon.