http://www.ck12.org Chapter 6. Polygons and Quadrilaterals
As you can see, there are two sets of exterior angles for any vertex on a polygon. It does not matter which set you
use because one set is just the vertical angles of the other, making the measurement equal. In the picture above, the
color-matched angles are vertical angles and congruent. TheExterior Angle Sum Theoremstated that the exterior
angles of a triangle add up to 360◦. Let’s extend this theorem to all polygons.
Investigation: Exterior Angle Tear-Up
Tools Needed: pencil, paper, colored pencils, scissors
- Draw a hexagon like the hexagons above. Color in the exterior angles as well.
- Cut out each exterior angle and label them 1-6.
- Fit the six angles together by putting their vertices together. What happens?
The angles all fit around a point, meaning that the exterior angles of a hexagon add up to 360◦, just like a triangle.
We can say this is true for all polygons.
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.
TABLE6.3:
Statement Reason- Anyn−gon withnsides,ninterior angles andn
exterior angles.
Given