Triangle Sum Theorem
Jen Kershaw
Kimberly Hopkins
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Printed: April 12, 2016
AUTHORS
Jen Kershaw
Kimberly Hopkins
www.ck12.org
C HAPTER
Chapter 1. Triangle Sum Theorem
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Triangle Sum Theorem
In this concept, you will learn about the sum of the angles of triangles.
Let’s Think About It
A contractor works on a house. He buys small boards of wood to trim the outline of the triangular shaped small roof
over the door. After the pieces are in place, the contractor measures the angles to make sure that the trim is a perfect
triangle. The first angle measures 70o and the second angle measures 40o . How can the contractor know what the
measurement of the third angle should be before he measures it?
In this concept, you will learn about the sum of the angles of triangles.
Guidance
A triangle is a figure with three sides and three angles.
In a triangle there is a relationship between the interior angles of the triangle. Interior angles are the angles inside
the triangle.
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Notice that triangles a, b, c and d are all different, they have different angle measures and different side lengths. If
you add up the measures of the three angles, they always equal 180◦ .
Now let’s look at triangles a little differently. In geometry, a triangle can be formed by the intersection of three lines.
First, notice that the lines create the three interior angles of the triangle.
Next notice that if you extend any side of the triangle, then it stretches beyond the triangle. This creates a pair of
angles, an interior angle and an exterior angle. An exterior angle is the angle formed outside of the edge of the
triangle.
Here is a clearer example of an exterior angle.
As you can see, the interior angle and the exterior angle form a line. Therefore their sum must be 180◦ .
The adjacent angle to the interior angle is 120◦ . Use the relationship between interior and exterior angles to set up
an equation and solve for the measure of x.
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Chapter 1. Triangle Sum Theorem
120 + x = 180
x = 180 − 120
x = 60◦
The missing measure of the interior angle is 60◦ .
There is another way to calculate the value of the missing interior angle. Take a look at the other given interior
angles of the triangle. They are 50◦ and 70◦ . Their sum is also 120◦ . In fact, the sum of any two interior angles in a
triangle is always equal to the exterior angle of the third angle.
Guided Practice
Solve the following problem.
What is the measure of angle S in the figure below?
First, determine if angleS is an interior or an exterior angle.
AngleS is an exterior angle.
Next, remember the relationship between the exterior angle of a missing angle and the two provided interior angles.
The other two interior angles have a sum equal to the measure of the third angle’s exterior angle.
Then, calculate the value of angleS.
Angle S = 30o + 35o = 65o
The answer is that the measure of angleS is 65o .
Examples
Solve the following problem.
Example 1
If the sum of two angles of a triangle is 150◦ , then what is the measure of the third angle?
First, determine if you are calculating an interior or exterior angle.
Interior
Next, remember the relationship between interior angles.
The sum of three interior angles is 180o .
Then, calculate the sum of the third angle.
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150o + the third angle = 180o . 180o − 150o = 30o .
The answer is that the third angle is 30o .
Example 2
If the sum of two of the angles is 75◦ , then what is the measure of the third angle’s exterior angle?
First, determine if you are calculating an interior or exterior angle.
Exterior
Next, remember the relationship between two interior angles and the measure of the third angle’s exterior angle.
The sum of the two interior angles is equal to the measure of the third angle’s exterior angle.
Then, determine the measure of the third angle.
75o = the measure of the third angle’s exterior angle.
The answer is that the measure of the third angle’s exterior angle is 75o .
Example 3
A triangle has three angles: angleA, angleB and angleC. Angle A = 33◦ , Angle B = 65◦ , what is the measure of
Angle C?
First, determine if you are calculating an interior or exterior angle.
AngleC is an interior angle.
Next, remember the relationship between the three interior angles.
Angle C + 33o + 65o = 180o . 180o − 98o = 82o .
Then, determine the measure of the third angle.
82o = the measure of angleC.
The answer is that the measure of angleC is 82o .
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Chapter 1. Triangle Sum Theorem
Follow Up
Remember the contractor and the triangle shaped trim? If the first angle is 70o and the second angle is40o , what is
the measure of the third angle?
First, determine if you are calculating an interior or exterior angle.
The third angle is the interior angle.
Next, remember the relationship between the three interior angles.
The third angle + 70o + 40o = 180o . 180o − 110o = 70o .
Then determine the measure of the third angle.
70o = the third angle.
Video Review
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/166915
Explore More
Find the measure of the missing angle in each triangle.
1. 20 + 70 + x = 180◦
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2. 60 + 60 + x = 180◦
3. 90 + 15 + x = 180◦
4. 100 + 45 + x = 180◦
5. 10 + 105 + x = 180◦
6. 120 + 45 + x = 180◦
7. 145 + 5 + x = 180◦
8. 150 + 20 + x = 180◦
9. 110 + 45 + x = 180◦
10. 60 + 40 + x = 180◦
11. 70 + 50 + x = 180◦
12. 80 + 45 + x = 180◦
13. 50 + 45 + x = 180◦
14. 30 + 55 + x = 180◦
15. 75 + 55 + x = 180◦
References
1. By rufus (flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons. http
s://commons.wikimedia.org/wiki/File%3AChattel_house_with_red_roof_front.jpg .
2. By rufus (flickr) [CC BY 2.0 (http://creativecommons.org/licenses/by/2.0)], via Wikimedia Commons. http
s://commons.wikimedia.org/wiki/File%3AChattel_house_with_red_roof_front.jpg .
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