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# Polygon Interior Angles Theorem

The sum of the measures of the interior angles of a polygon with vertices is given by the following formula.

The formula can be applied to any polygon, not just regular polygons.

### Proof

In the proof, a pentagon will be considered. However, the proof is valid for any polygon.

For any pentagon, two non-intersecting diagonals can be drawn to divide the pentagon into three triangles. In the case of an arbitrary polygon with vertices, non-intersecting diagonals can be drawn to divide the polygon into triangles.

Let be the sum of the measures of the angles of and and be the sums of the measures of angles of and respectively. It can be noted that the sum of angle measures of the pentagon is equal to the total of the sums of angle measures of the three triangles.
By the Triangle Angle Sum Theorem, the sum of angle measures of each triangle is equal to Then, can be substituted for and
This means that is equal to or In the case of an arbitrary polygon with vertices, there are triangles, so the sum of the angle measures of the polygon is This proves the theorem.