Polygons and Angles
Rule

Polygon Exterior Angles Theorem

The sum of the measures of the exterior angles of a polygon, one angle at each vertex, is
Hexagon with all the exterior angles marked

Based on the diagram, the relation below holds true.

Proof

Convex Polygon

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, the last equation can be solved for
Solve 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.

Proof

Concave Polygon

Consider a concave polygon and one exterior angle at each vertex. Let be the sum of the measures of the exterior angles.

Concave polygon and its exterior angles.
Notice that lies inside the polygon. In this case, its measure is considered to be negative. Because of this, and are supplementary.
The rest of exterior angles form a linear pair with their corresponding interior angles. This means all those pair of angles are supplementary.
Next, add these four equations to Equation (I). Remember that the Polygon Interior Angles Theorem states that the sum of the measures of the interior angles of a polygon with sides is
Finally, solve the resulting equation for
Solve for
As can be seen, the sum of the measures of the exterior angles of the concave polygon is which completes the proof. Keep in mind that the same reasoning can be applied to any other concave polygon.
Exercises