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Exterior Angles Theorems
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
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Printed: November 7, 2015
AUTHORS
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
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C HAPTER
Chapter 1. Exterior Angles Theorems
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Exterior Angles Theorems
Here you’ll learn what an exterior angle is as well as two theorems involving exterior angles: that the sum of the
exterior angles is always 360◦ and that in a triangle, an exterior angle is equal to the sum of its remote interior angles.
What if you knew that two of the exterior angles of a triangle measured 130◦ ? How could you find the measure of
the third exterior angle? After completing this Concept, you’ll be able to apply the Exterior Angle Sum Theorem to
solve problems like this one.
Watch This
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CK-12 Exterior Angles Theorems
MEDIA
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James Sousa: Introduction to the Exterior Angles of a Triangle
Then watch this video.
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James Sousa: Proof that the Sum of the Exterior Angles of a Triangle is 360 Degrees
Finally, watch this video.
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James Sousa: Proof of the Exterior Angles Theorem
Guidance
An Exterior Angle is the angle formed by one side of a polygon and the extension of the adjacent side.
In all polygons, there are two sets of exterior angles, one that goes around clockwise and the other goes around
counterclockwise.
Notice that the interior angle and its adjacent exterior angle form a linear pair and add up to 180◦ .
m6 1 + m6 2 = 180◦
There are two important theorems to know involving exterior angles: the Exterior Angle Sum Theorem and the
Exterior Angle Theorem.
The Exterior Angle Sum Theorem states that the exterior angles of any polygon will always add up to 360◦ .
m6 1 + m6 2 + m6 3 = 360◦
m6 4 + m6 5 + m6 6 = 360◦
.
The Exterior Angle Theorem states that an exterior angle of a triangle is equal to the sum of its remote interior
angles. ( Remote Interior Angles are the two interior angles in a triangle that are not adjacent to the indicated
exterior angle.)
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Chapter 1. Exterior Angles Theorems
m6 A + m6 B = m6 ACD
.
Example A
Find the measure of 6 RQS.
Notice that 112◦ is an exterior angle of 4RQS and is supplementary to 6 RQS.
Set up an equation to solve for the missing angle.
112◦ + m6 RQS = 180◦
m6 RQS = 68◦
Example B
Find the measures of the numbered interior and exterior angles in the triangle.
We know that m6 1 + 92◦ = 180◦ because they form a linear pair. So, m6 1 = 88◦ .
Similarly, m6 2 + 123◦ = 180◦ because they form a linear pair. So, m6 2 = 57◦ .
We also know that the three interior angles must add up to 180◦ by the Triangle Sum Theorem.
m6 1 + m6 2 + m6 3 = 180◦
◦
by the Triangle Sum Theorem.
◦
88 + 57 + m6 3 = 180
m6 3 = 35◦
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Lastly, m6 3 + m6 4 = 180◦
◦
35 + m6 4 = 180
because they form a linear pair.
◦
m6 4 = 145◦
Example C
What is the value of p in the triangle below?
First, we need to find the missing exterior angle, which we will call x. Set up an equation using the Exterior Angle
Sum Theorem.
130◦ + 110◦ + x = 360◦
x = 360◦ − 130◦ − 110◦
x = 120◦
x and p add up to 180◦ because they are a linear pair.
x + p = 180◦
120◦ + p = 180◦
p = 60◦
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CK-12 Exterior Angles Theorems
–>
Guided Practice
1. Find m6 C.
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Chapter 1. Exterior Angles Theorems
2. Two interior angles of a triangle are 40◦ and 73◦ . What are the measures of the three exterior angles of the
triangle?
3. Find the value of x and the measure of each angle.
Answers:
1. Using the Exterior Angle Theorem
m6 C + 16◦ = 121◦
m6 C = 105◦
If you forget the Exterior Angle Theorem, you can do this problem just like Example C.
2. Remember that every interior angle forms a linear pair (adds up to 180◦ ) with an exterior angle. So, since one
of the interior angles is 40◦ that means that one of the exterior angles is 140◦ (because 40 + 140 = 180). Similarly,
since another one of the interior angles is 73◦ , one of the exterior angles must be 107◦ . The third interior angle is
not given to us, but we could figure it out using the Triangle Sum Theorem. We can also use the Exterior Angle
Sum Theorem. If two of the exterior angles are 140◦ and 107◦ , then the third Exterior Angle must be 113◦ since
140 + 107 + 113 = 360.
So, the measures of the three exterior angles are 140, 107 and 113.
3. Set up an equation using the Exterior Angle Theorem.
(4x + 2)◦ + (2x − 9)◦ = (5x + 13)◦
↑
%
↑
remote interior angles
exterior angle
◦
(6x − 7) = (5x + 13)◦
x = 20
Substitute in 20 for x to find each angle.
[4(20) + 2]◦ = 82◦
[2(20) − 9]◦ = 31◦
Exterior angle: [5(20) + 13]◦ = 113◦
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Explore More
Determine m6 1.
1.
2.
3.
4.
5.
6.
Use the following picture for the next three problems:
7. What is m6 1 + m6 2 + m6 3?
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Chapter 1. Exterior Angles Theorems
8. What is m6 4 + m6 5 + m6 6?
9. What is m6 7 + m6 8 + m6 9?
Solve for x.
10.
11.
12.
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 4.2.
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