To help you see what the sum of all exterior angles of a polygon is, we will use a square and then a regular pentagon. Since it is very easy to see what the sum is for a square, we will start with the square. You can also use rectangles!

Notice that an exterior angle is formed by a side of the square and an extension of an adjacent side. For example in the figure below, angle x, angle y, angle z, and angle w are all exterior angles.

Each interior angle in a square is equal to 90 degrees. Notice that the sum of an interior angle plus the adjacent exterior angle is equal to 180 degrees.

**Interior angle + adjacent exterior angle = 180 degrees.**

In fact, the sum of (one interior angle plus the adjacent exterior angle) of any polygon always add up to 180 degrees. This is so because when you extend any side of a polygon, what you are really doing is extending a straight line and a straight line is always equal to 180 degrees.

For example, 90 degrees + w = 180 degrees

90 degrees - 90 degrees + w = 180 degrees - 90 degrees

0 + w = 90 degrees

w = 90 degrees

Since there are 4 exterior angles, 4 x 90 degrees = 360 degrees.

In the figure or pentagon above, we use a to represent the interior angle of the pentagon and we use x,y,z,v, and w to represents the 5 exterior angles.

To find the measure of the interior angle of a pentagon, we just need to use this formula.

The interior angle of any polygon = [(n - 2 ) 180] / n

Since n is equal to 5, [(n - 2 ) 180] / n = [(5 - 2) 180] / 5 = [3 x 180] / 5 = 540 / 5 = 108

∠x and ∠a are adjacent (common side and common vertex) and supplementary.

Again, interior angle + adjacent exterior angle = 180 degrees.

108 degrees + adjacent exterior angle = 180 degrees

108 degrees - 108 degrees + adjacent exterior angle = 180 degrees - 108 degrees

0 + adjacent exterior angle = 180 degrees - 108 degrees

Adjacent exterior angle = 72 degrees

Since there are 5 exterior angles, 5 x 72 = 360 degrees.

It does not matter how many sides the polygon has!

**Exterior angle sum theorem**

The sum of all exterior angles of a polygon is always equal to 360 degrees.

As already shown, we know that interior angle + adjacent exterior angle = 180 degrees.

interior angle + adjacent exterior angle = 180 degrees

interior angle - interior angle + adjacent exterior angle = 180 degrees - interior angle

adjacent exterior angle = 180 degrees - interior angle

Since a square has 4 exterior angles, just multiply the equation above by 4 to get the sum of its exterior angles.

4(adjacent exterior angle) = 4(180 degrees - interior angle)

4(adjacent exterior angle) = 4(180 degrees) - 4(interior angle)

The sum of the exterior angles = 4(adjacent exterior angle)

The sum of the exterior angles = 4(180 degrees) - 4(90 degrees)

The sum of the exterior angles = 720 degrees - 360 degrees

The sum of the exterior angles = 360 degrees

Similarly, since a pentagon has 5 exterior angles, we get the following equation

The sum of the exterior angles = 5(180 degrees) - 5(interior angle)

The sum of the exterior angles = 5(180 degrees) - 5(108)

The sum of the exterior angles = 900 - 540

The sum of the exterior angles = 360

Suppose the number of sides of a polygon is n. Show that the sum of all exterior angles of the polygon is 360 degrees.

n(adjacent exterior angle) = n(180 degrees - interior angle)

n(adjacent exterior angle) = n(180 degrees) - n(interior angle)

n(adjacent exterior angle) = sum of exterior angles of any polygon

interior angle = [(n - 2 )180] / n

The sum of exterior angles of any polygon = n(180 degrees) - (n)[(n - 2)180] / n

The sum of exterior angles of any polygon = 180n - (n - 2)180

The sum of exterior angles of any polygon = 180n - (180n - 360)

The sum of exterior angles of any polygon = 180n - 180n + 360

The sum of exterior angles of any polygon = 360

Notice that the sum of the interior angles = (n)[(n - 2)180] / n = (n - 2)180