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Exterior Angles Theorems
Bill Zahner
Lori Jordan
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Printed: October 7, 2012
AUTHORS
Bill Zahner
Lori Jordan
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C ONCEPT
Concept 1. Exterior Angles Theorems
1
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
MEDIA
Click image to the left for more content.
CK-12 Foundation: Chapter4ExteriorAnglesTheoremsA
MEDIA
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James Sousa:Introduction totheExterior Anglesof a Triangle
MEDIA
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James Sousa:Proof that the Sum of the Exterior Angles of a Triangleis 360 Degrees
MEDIA
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James Sousa:Proof of the Exterior Angles Theorem
1
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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 going around the polygon clockwise and the other goes around
the polygon counterclockwise. By the definition, the interior angle and its adjacent exterior angle form a linear pair.
The Exterior Angle Sum Theorem states that each set of exterior angles of a polygon add up to 360◦ .
m6 1 + m6 2 + m6 3 = 360◦
m6 4 + m6 5 + m6 6 = 360◦
Remote interior angles are the two angles in a triangle that are not adjacent to the indicated exterior angle. 6 A and
6 B are the remote interior angles for exterior angle 6 ACD.
The Exterior Angle Theorem states that the sum of the remote interior angles is equal to the non-adjacent exterior
angle. From the picture above, this means that m6 A + m6 B = m6 ACD. Here is the proof of the Exterior Angle
Theorem. From the proof, you can see that this theorem is a combination of the Triangle Sum Theorem and the
Linear Pair Postulate.
Given: 4ABC with exterior angle 6 ACD
Prove: m6 A + m6 B = m6 ACD
TABLE 1.1:
Statement
1. 4ABC with exterior angle 6 ACD
2. m6 A + m6 B + m6 ACB = 180◦
3. m6 ACB + m6 ACD = 180◦
4. m6 A + m6 B + m6 ACB = m6 ACB + m6 ACD
5. m6 A + m6 B = m6 ACD
Reason
Given
Triangle Sum Theorem
Linear Pair Postulate
Transitive PoE
Subtraction PoE
Example A
Find the measure of 6 RQS.
112◦ is an exterior angle of 4RQS. Therefore, it is supplementary to 6 RQS because they are a linear pair.
112◦ + m6 RQS = 180◦
m6 RQS = 68◦
If we draw both sets of exterior angles on the same triangle, we have the following figure:
Notice, at each vertex, the exterior angles are also vertical angles, therefore they are congruent.
6
6
6
2
4∼
=6 7
5∼
=6 8
6∼
=6 9
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Concept 1. Exterior Angles Theorems
Example B
Find the measure of the numbered interior and exterior angles in the triangle.
m6 1 + 92◦ = 180◦ by the Linear Pair Postulate, so m6 1 = 88◦ .
m6 2 + 123◦ = 180◦ by the Linear Pair Postulate, so m6 2 = 57◦ .
m6 1 + m6 2 + m6 3 = 180◦ by the Triangle Sum Theorem, so 88◦ + 57◦ + m6 3 = 180◦ and m6 3 = 35◦ .
m6 3 + m6 4 = 180◦ by the Linear Pair Postulate, so m6 4 = 145◦ .
Example C
What is the value of p in the triangle below?
First, we need to find the missing exterior angle, we will call it x. Set up an equation using the Exterior Angle Sum
Theorem.
130◦ + 110◦ + x = 360◦
x = 360◦ − 130◦ − 110◦
x = 120◦
x and p are supplementary and add up to 180◦ .
x + p = 180◦
120◦ + p = 180◦
p = 60◦
Watch this video for help with the Examples above.
MEDIA
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CK-12 Foundation: Chapter4TriangleSumTheoremA
Concept Problem Revisited
The third exterior angle of the triangle below is 6 1.
By the Exterior Angle Sum Theorem:
m6 1 + 130◦ + 130◦ = 360◦
m6 1 = 100◦
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Vocabulary
Interior angles are the angles on the inside of a polygon while exterior angles are the angles on the outside of a
polygon. Remote interior angles are the two angles in a triangle that are not adjacent to the indicated exterior angle.
Two angles that make a straight line form a linear pair and thus add up to 180◦ . The Triangle Sum Theorem states
that the three interior angles of any triangle will always add up to 180◦ . The Exterior Angle Sum Theorem states
that each set of exterior angles of a polygon add up to 360◦ .
Guided Practice
1. Find m6 A.
2. Find m6 C.
3. Find the value of x and the measure of each angle.
Answers:
1. Set up an equation using the Exterior Angle Theorem. m6 A + 79◦ = 115◦ . Therefore, m6 A = 36◦ .
2. Using the Exterior Angle Theorem, m6 C + 16◦ = 121◦ . Subtracting 16◦ from both sides, m6 C = 105◦ .
3. Set up an equation using the Exterior Angle Theorem.
(4x + 2)◦ + (2x − 9)◦ = (5x + 13)◦
↑
%
↑
interior angles
exterior angle
◦
(6x − 7) = (5x + 13)◦
x = 20◦
Substituting 20◦ back in for x, the two interior angles are (4(20) + 2)◦ = 82◦ and (2(20) − 9)◦ = 31◦ . The exterior
angle is (5(20) + 13)◦ = 113◦ . Double-checking our work, notice that 82◦ + 31◦ = 113◦ . If we had done the problem
incorrectly, this check would not have worked.
Practice
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?
8. What is m6 4 + m6 5 + m6 6?
9. What is m6 7 + m6 8 + m6 9?
Solve for x.
4
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Concept 1. Exterior Angles Theorems
10.
11.
12.
13. Suppose the measures of the three angles of a triangle are x, y, and z. Explain why x + y + z = 180.
14. Suppose the measures of the three angles of a triangle are x, y, and z. Explain why the expression (180 − x) +
(180 − y) + (180 − z) represents the sum of the exterior angles of the triangle.
15. Use your answers to the previous two problems to help justify why the sum of the exterior angles of a triangle is
360 degrees. Hint: Use algebra to show that (180−x)+(180−y)+(180−z) must equal 360 if x+y+z = 180.
5