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Exterior Angles in Convex
Polygons
Bill Zahner
Lori Jordan
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Printed: January 6, 2016
AUTHORS
Bill Zahner
Lori Jordan
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C HAPTER
Chapter 1. Exterior Angles in Convex Polygons
1
Exterior Angles in Convex
Polygons
Here you’ll learn the Exterior Angle Sum Theorem that states that the exterior angles of a polygon always add up to
360◦ .
What if you were given a twelve-sided 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
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/136947
CK-12 Foundation: Chapter6ExteriorAnglesinConvexPolygonsA
Watch the second half of this video.
MEDIA
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URL: http://www.ck12.org/flx/render/embeddedobject/1267
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.
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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. The Exterior Angle Sum Theorem stated 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
1. Draw a hexagon like the hexagons above. Color in the exterior angles as well.
2. Cut out each exterior angle and label them 1-6.
3. 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: Any n−gon with n sides, n interior angles and n exterior angles.
Prove: n exterior angles add up to 360◦
NOTE: The interior angles are x1 , x2 , . . . xn .
The exterior angles are y1 , y2 , . . . yn .
TABLE 1.1:
Statement
1. Any n−gon with n sides, n interior angles and n
exterior angles.
2
Reason
Given
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Chapter 1. Exterior Angles in Convex Polygons
TABLE 1.1: (continued)
Statement
2. xn◦ and y◦n are a linear pair
3. xn◦ and y◦n are supplementary
4. xn◦ + y◦n = 180◦
5. (x1◦ + x2◦ + . . . + xn◦ ) + (y◦1 + y◦2 + . . . + y◦n ) = 180◦ n
6. (n − 2)180◦ = (x1◦ + x2◦ + . . . + xn◦ )
7. 180◦ n = (n − 2)180◦ + (y◦1 + y◦2 + . . . + y◦n )
8. 180◦ n = 180◦ n − 360◦ + (y◦1 + y◦2 + . . . + y◦n )
9. 360◦ = (y◦1 + y◦2 + . . . + y◦n )
Reason
Definition of a linear pair
Linear Pair Postulate
Definition of supplementary angles
Sum of all interior and exterior angles in an n−gon
Polygon Sum Formula
Substitution PoE
Distributive PoE
Subtraction PoE
Example A
What is y?
y is an exterior angle, as well as all the other given angle measures. Exterior angles add up to 360◦ , so set up an
equation.
70◦ + 60◦ + 65◦ + 40◦ + y = 360◦
y = 125◦
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 360◦ , so each angle is 360
7 ≈ 51.43 .
Example C
What is the sum of the exterior angles in a regular 15-gon?
The sum of the exterior angles in any convex polygon, including a regular 15-gon, is 360◦ .
Watch this video for help with the Examples above.
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CK-12 Foundation: Chapter6ExteriorAnglesinConvexPolygonsB
Concept Problem Revisited
The exterior angles of a regular polygon sum to 360◦ . The measure of each exterior angle in a dodecagon (twelve◦
◦
sided regular polygon) is 360
12 = 30 .
Guided Practice
Find the measure of each exterior angle for each regular polygon below:
1. 12-gon
2. 100-gon
3. 36-gon
Answers:
For each, divide 360◦ by the given number of sides.
1. 30◦
2. 3.6◦
3. 10◦
Explore More
1. What is the measure of each exterior angle of a regular decagon?
2. What is the measure of each exterior angle of a regular 30-gon?
3. What is the sum of the exterior angles of a regular 27-gon?
Find the measure of the missing variables:
4.
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Chapter 1. Exterior Angles in Convex Polygons
5.
6. The exterior angles of a quadrilateral are x◦ , 2x◦ , 3x◦ , and 4x◦ . What is x?
Find the measure of each exterior angle for each regular polygon below:
7.
8.
9.
10.
11.
12.
13.
14.
octagon
nonagon
triangle
pentagon
50-gon
heptagon
34-gon
Challenge Each interior angle forms a linear pair with an exterior angle. In a regular polygon you can use two
◦
(n−2)180◦
◦
different formulas to find the measure of each exterior angle. One way is 360
n and the other is 180 −
n
(180◦ minus Equiangular Polygon Formula). Use algebra to show these two expressions are equivalent.
15. Angle Puzzle Find the measures of the lettered angles below given that m || n.
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 6.2.
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