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8.7: Triangle Sum Theorem

Difficulty Level: Basic Created by: CK-12
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Corrina looked at the following sculpture and loved the bright blue triangles in the middle. The sculpture was called "180". Corrina figured out that this is because the three interior angles of a triangle add up to be 180\begin{align*}180^\circ\end{align*}.

There are two angles at the bottom of the triangle. One is 40\begin{align*}40^\circ\end{align*} and the other is 75\begin{align*}75^\circ\end{align*}.

If this is the case, what is the measure of the angle at the top of the triangle?

Can you figure this out?

Use this Concept to learn about triangles and their angles. Then you will know how to solve this dilemma at the end of the Concept.

Guidance

This Concept is all about triangles. You have been learning about triangles for a long time. It is one of the first shapes that small children learn to recognize. Mathematically speaking, we know that the prefix “tri” means three and the rest of the word is “angles.” Therefore, 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. What are interior angles?

Interior angles are the angles inside the triangle. There are three of them and we can learn about the relationship between the interior angles of a triangle by looking at a few examples.

Notice that triangles a,b,c\begin{align*}a, b, c\end{align*} and d\begin{align*}d\end{align*} are all different, they have different angle measures and different side lengths. Look closely, though. If you add up the measures of the three angles, they always equal 180\begin{align*}180^\circ\end{align*}!

Write this down in your notebooks.

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. And as we know, those three angles have a sum of 180\begin{align*}180^\circ\end{align*}.

Next notice that if we extend any side of the triangle, then it stretches beyond the triangle. Now we have 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\begin{align*}180^\circ\end{align*}.

The adjacent angle to the interior angle is 120\begin{align*}120^\circ\end{align*}. If the exterior and the interior angle form a straight line, then their sum is 180\begin{align*}180^\circ\end{align*}. We can set up an equation and solve for the measure of x\begin{align*}x\end{align*}.

120+xxx=180=180120=60\begin{align*}120 + x &= 180\\ x &= 180 - 120\\ x &= 60^\circ\end{align*}

The missing measure of the interior angle is 60\begin{align*}60^\circ\end{align*}.

We could also figure this out another way. Take a look at the other given interior angles of the triangle. They are 50\begin{align*}50^\circ\end{align*} and 70\begin{align*}70^\circ\end{align*}. Their sum is also 120\begin{align*}120^\circ\end{align*}!

In fact, the sum of any two interior angles in a triangle is always equal to the exterior angle of the third angle.

But, we can use this information to figure out the missing third interior angle. If the sum of the two interior angles is 120, we can use the same equation to solve for the third missing angle.

120+xxx=180=180120=60\begin{align*}120 + x &= 180\\ x &= 180 -120\\ x &= 60^\circ\end{align*}

Notice that both methods will help you to find the correct measure of a missing interior angle.

What is the measure of angle S\begin{align*}S\end{align*} in the figure below?

We can see that angle S\begin{align*}S\end{align*} is an exterior angle. However, we do not know what its adjacent interior angle is. Can we still find the measure of angle S\begin{align*}S\end{align*}? We can. As we have just learned, the other two interior angles have a sum equal to the measure of the third angle’s exterior angle. Therefore angle S\begin{align*}S\end{align*} must be equal to the sum of the two angles we have been given.

SS=30+35=65\begin{align*}S &= 30 + 35\\ S &= 65^\circ\end{align*}

Incidentally, we can also find the measure of the third angle in the triangle by using the exterior angle. We know that the sum of this angle and angle S\begin{align*}S\end{align*} must be 180\begin{align*}180^\circ\end{align*}. If S\begin{align*}S\end{align*} is 65\begin{align*}65^\circ\end{align*}, then the angle must be 18065=115\begin{align*}180 - 65 = 115^\circ\end{align*}.

We also know that the sum of the three interior angles of a triangle is 180\begin{align*}180^\circ\end{align*}, so we could also find the missing angle by adding the two known angles and then subtracting from 180\begin{align*}180^\circ\end{align*}.

1+2+330+35+365+333=180=180=180=18065=115\begin{align*}\angle 1 + \angle 2 + \angle 3 &= 180^\circ\\ 30 + 35 + \angle 3 &= 180^\circ\\ 65 + \angle 3 &= 180^\circ\\ \angle 3 &= 180 - 65\\ \angle 3 &= 115^\circ\end{align*}

Use what you have learned to answer each question.

Example A

If the sum of two angles of a triangle is 150\begin{align*}150^\circ\end{align*}, then what is the sum of the third angle?

Solution: 30\begin{align*}30^\circ\end{align*}

Example B

If the sum of two of the angles is 75\begin{align*}75^\circ\end{align*}, then what is the measure of the third angle’s exterior angle?

Solution: 105\begin{align*}105^\circ\end{align*}

Example C

Angle A=33\begin{align*}A = 33^\circ\end{align*}, Angle B=65\begin{align*}B = 65^\circ\end{align*}, what is the measure of Angle C\begin{align*}C\end{align*}?

Solution: 82\begin{align*}82^\circ\end{align*}

Here is the original problem once again.

Corrina looked at the following sculpture and loved the bright blue triangles in the middle. The sculpture was called "180". Corrina figured out that this is because the three interior angles of a triangle add up to be 180\begin{align*}180^\circ\end{align*}.

There are two angles at the bottom of the triangle. One is 40\begin{align*}40^\circ\end{align*} and the other is 75\begin{align*}75^\circ\end{align*}.

If this is the case, what is the measure of the angle at the top of the triangle?

Can you figure this out?

To figure this out, let's write an equation to show the three angles. We will use x\begin{align*}x\end{align*} to represent the unknown angle.

40+75+x=180\begin{align*}40 + 75 + x = 180\end{align*}

115+x=180\begin{align*}115 + x = 180\end{align*}

Next, we solve it using an inverse operation.

x=180115\begin{align*}x = 180 - 115\end{align*}

x=65\begin{align*}x = 65\end{align*}

The missing angle's measure is 65\begin{align*}65^\circ\end{align*}.

Vocabulary

Here are the vocabulary words in this Concept.

Triangle
a figure with three sides and three angles.
Interior angles
the three inside angles of a triangle.
Exterior angles
the angles outside of a triangle formed by intersecting lines

Guided Practice

Here is one for you to try on your own.

What is the missing angle measure?

Now we can tell that this is a right triangle and that one of the angles is equal to 90 degrees. To figure out the measure of the missing angle, we have used a variable to represent the unknown quantity. Here is our equation.

55+90+x145+x180145x=180=180=x=35.\begin{align*}55 + 90 + x & = 180 \\ 145 + x & = 180 \\ 180 - 145 & = x \\ x & = 35^\circ .\end{align*}

Our answer is 35\begin{align*}35^\circ\end{align*}.

Video Review

Here is a video for review.

Practice

Directions: Find the measure of the missing angle in each triangle.

1. 20+70+x=180\begin{align*}20 + 70 + x = 180^\circ\end{align*}

2. 60+60+x=180\begin{align*}60 + 60 + x = 180^\circ\end{align*}

3. 90+15+x=180\begin{align*}90 + 15 + x = 180^\circ\end{align*}

4. 100+45+x=180\begin{align*}100 + 45 + x = 180^\circ\end{align*}

5. 10+105+x=180\begin{align*}10 + 105 + x = 180^\circ\end{align*}

6. 120+45+x=180\begin{align*}120 + 45 + x = 180^\circ\end{align*}

7. 145+5+x=180\begin{align*}145 + 5 + x = 180^\circ\end{align*}

8. 160+20+x=180\begin{align*}160 + 20 + x = 180^\circ\end{align*}

9. 110+45+x=180\begin{align*}110 + 45 + x = 180^\circ\end{align*}

10. 60+60+x=180\begin{align*}60 + 60 + x = 180^\circ\end{align*}

11. 70+50+x=180\begin{align*}70 + 50 + x = 180^\circ\end{align*}

12. 80+45+x=180\begin{align*}80 + 45 + x = 180^\circ\end{align*}

13. 50+45+x=180\begin{align*}50 + 45 + x = 180^\circ\end{align*}

14. 30+55+x=180\begin{align*}30 + 55 + x = 180^\circ\end{align*}

15. 75+55+x=180\begin{align*}75 + 55 + x = 180^\circ\end{align*}

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Color Highlighted Text Notes

Vocabulary Language: English

Triangle Sum Theorem

The Triangle Sum Theorem states that the three interior angles of any triangle add up to 180 degrees.

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