6.2: Exterior Angles in Convex Polygons
What if you were given a twelvesided regular polygon? How could you determine the measure of each of its exterior angles? After completing this Concept, you'll be able to use the Exterior Angle Sum Theorem to solve problems like this one.
Watch This
CK12 Foundation: Chapter6ExteriorAnglesinConvexPolygonsA
Watch the second half of this video.
James Sousa: Angles of Convex Polygons
Guidance
Recall that an exterior angle is an angle on the outside of a polygon and is formed by extending a side of the polygon.
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 colormatched angles are vertical angles and congruent. The Exterior Angle Sum Theorem stated that the exterior angles of a triangle add up to
Investigation: Exterior Angle TearUp
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 16.
 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
Exterior Angle Sum Theorem: The sum of the exterior angles of any polygon is
Proof of the Exterior Angle Sum Theorem:
Given: Any
Prove:
NOTE: The interior angles are
The exterior angles are
Statement  Reason 

1. Any 
Given 
2. 
Definition of a linear pair 
3. 
Linear Pair Postulate 
4. 
Definition of supplementary angles 
5. 
Sum of all interior and exterior angles in an 
6. 
Polygon Sum Formula 
7. 
Substitution PoE 
8. 
Distributive PoE 
9. 
Subtraction PoE 
Example A
What is
Example B
What is the measure of each exterior angle of a regular heptagon?
Because the polygon is regular, each interior angle is equal. This also means that all the exterior angles are equal. The exterior angles add up to
Example C
What is the sum of the exterior angles in a regular 15gon?
The sum of the exterior angles in any convex polygon, including a regular 15gon, is
Watch this video for help with the Examples above.
CK12 Foundation: Chapter6ExteriorAnglesinConvexPolygonsB
Concept Problem Revisited
The exterior angles of a regular polygon sum to
Vocabulary
An exterior angle is an angle that is formed by extending a side of the polygon. A regular polygon is a polygon in which all of its sides and all of its angles are congruent.
Guided Practice
Find the measure of each exterior angle for each regular polygon below:
1. 12gon
2. 100gon
3. 36gon
Answers:
For each, divide
1.
2.
3.
Practice
 What is the measure of each exterior angle of a regular decagon?
 What is the measure of each exterior angle of a regular 30gon?
 What is the sum of the exterior angles of a regular 27gon?
Find the measure of the missing variables:
 The exterior angles of a quadrilateral are
x∘,2x∘,3x∘, and4x∘. What isx ?
Find the measure of each exterior angle for each regular polygon below:
 octagon
 nonagon
 triangle
 pentagon
 50gon
 heptagon
 34gon

Challenge Each interior angle forms a linear pair with an exterior angle. In a regular polygon you can use two different formulas to find the measure of each exterior angle. One way is
360∘n and the other is180∘−(n−2)180∘n (180∘ minus Equiangular Polygon Formula). Use algebra to show these two expressions are equivalent. 
Angle Puzzle Find the measures of the lettered angles below given that
m  n .
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Image Attributions
Here you'll learn the Exterior Angle Sum Theorem that states that the exterior angles of a polygon always add up to \begin{align*}360^\circ\end{align*}.