Question

Write a proof of the Polygon Exterior Angle-Sum Theorem. The sum of the measures of the exterior angles of a convex polygon, one at each vertex, is 360. The sum of the measures of n pairs of interior and exterior angles is

Answer 1

Weare asked to write a proof of the Exterior angle Theorem that states that the sum of exterior angles in any polygon adds to 360 degrees.

Consider first the theorem that states that the sum of the interior angles of a polygon of "n" sides is given by:

180 * (n-2)

We also know that since the exterior angles of the polygon are all "supplementary" angles to the interior angles, then the addition of ALL interior angles PLUS the addition of ALL exterior angles of a polygon of "n" sides must give:

Sum ALL interior Angles + Sum ALL exterior Angles = n * 180

We can now use the theorem mentioned above about the sum of the interior angles of a polygon of "n" sides as: 180 * (n - 2)

Then we can find the Sum of ALL experior angles by subtracting the formula as shown:

Sum ALL interior Angles + Sum ALL exterior Angles = n * 180

180 * (n - 2) + Sum ALL exterior Angles = n * 180

using distributive property

180 n - 360 + Sum ALL exterior Angles = n * 180

subtract 180 n from both sides

- 360 + Sum ALL exterior Angles = n * 180 - 180 n

- 360 + Sum ALL exterior Angles = 0

add 360 to both sides

Sum ALL exterior Angles = 0 + 360

Sum ALL exterior Angles = 360

which is what we wanted to prove.