6.1. Angles in Polygons http://www.ck12.org
Solution:From our investigation, we found that a quadrilateral has 360◦. We can write an equation to solve forx.
89 ◦+( 5 x− 8 )◦+( 3 x+ 4 )◦+ 51 ◦= 360 ◦
8 x= 224 ◦
x= 28 ◦Exterior Angles in Convex Polygons
Recall that an exterior angle is an angle on the outside of a polygon and is formed by extending a side of the polygon
(Chapter 4).
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 to the left,
the color-matched angles are vertical angles and congruent.
In Chapter 4, we introduced the Exterior Angle Sum Theorem, which stated that the exterior angles of a triangle add
up to 360◦. Let’s extend this theorem to all polygons.
Investigation 6-2: 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.