Sum of the Exterior Angles of a Polygon

The sum of the exterior angles of a polygon is 360º regardless of how many sides the polygon has. The pentagon below shows the relationship between exterior and interior angles. The exterior angle is formed by extending one of the sides of the polygon past where the two sides indicate. It is also important to note that the interior and exterior angles always form a linear pair.Polygon with angles

To prove that the sum of the interior angles of a polygon equals 360º take a generic polygon with  number of sides, where is any whole numbers.

Since interior and exterior angles always form linear pairs then we can add each of interior and exterior angles together, and set each of them equal to 180º.

Sum Exterior Angles 1

The three dots in the middle of the equations indicate that there are equations there that I did not write. Since we are looking at a polygon with an indefinite number of sides we place those dots there as a place holder.

We can add all of the above equations together gets us the following equation:

Sum Exterior Angles 2

We can rewrite the above equation as the sum of the interior angles and the sum of the exterior angles. ∑ is a Greek letter which indicates to add a series of numbers up, in this case the interior and exterior angles of a polygon.

Sum Exterior Angles 3

Using the Sum of the Interior Angles of a Polygon Theorem we can rewrite the sum of the interior angles of a polygon as 180º(n-2):

Sum Exterior Angles 4

Distributing the 180º:

Sum Exterior Angles 5

Adding 360º and subtracting 180ºn from both sides proves that the sum of the exterior angles of a polygon is 360º, regardless of how many sides there are.

Sum Exterior Angles 6

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