CK-12 Geometry - Second Edition

http://www.ck12.org Chapter 6. Polygons and Quadrilaterals

Exterior Angle Sum Theorem:The sum of the exterior angles of any polygon is 360◦.

Proof of the Exterior Angle Sum Theorem

Given: Anyn−gon withnsides,ninterior angles andnexterior angles.

Prove:nexterior angles add up to 360◦

NOTE: The interior angles arex 1 ,x 2 ,...xn.

The exterior angles arey 1 ,y 2 ,...yn.

TABLE6.2:

Statement Reason

1. Anyn−gon withnsides,ninterior angles andn
   exterior angles.

Given

2.x◦nand y◦nare a linear pair Definition of a linear pair

3.x◦nand y◦nare supplementary Linear Pair Postulate

4.x◦n+y◦n= 180 ◦ Definition of supplementary angles

5.(x◦ 1 +x 2 ◦+...+x◦n)+(y◦ 1 +y◦ 2 +...+y◦n) = 180 ◦n Sum of all interior and exterior angles in ann−gon

6.(n− 2 ) 180 ◦= (x◦ 1 +x◦ 2 +...+x◦n) Polygon Sum Formula

7. 180◦n= (n− 2 ) 180 ◦+(y◦ 1 +y◦ 2 +...+y◦n) Substitution PoE


8. 180◦n= 180 ◦n− 360 ◦+(y◦ 1 +y◦ 2 +...+y◦n) Distributive PoE


9. 360◦= (y◦ 1 +y◦ 2 +...+y◦n) Subtraction PoE

Example 5:What isy?

Solution:yis an exterior angle, as well as all the other given angle measures. Exterior angles add up to 360◦, so set
up an equation.