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# Triangle Sum Theorem

## Interior angles add to 180 degrees

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Triangle Sum Theorem
License: CC BY-NC 3.0

How can the contractor know what the measurement of the third angle should be before he measures 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 \begin{align*}40^o\end{align*} and the second angle measures \begin{align*}70^o\end{align*}.

In this concept, you will learn about the sum of the angles of triangles.

### Triangle Sum Theorem

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.

Notice that triangles \begin{align*}a, b, c\end{align*} and \begin{align*}d\end{align*} 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 \begin{align*}180^\circ\end{align*}.

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 \begin{align*}180^\circ\end{align*}.

The adjacent angle to the interior angle is \begin{align*}120^\circ\end{align*}.  Use the relationship between interior and exterior angles to set up an equation and solve for the measure of \begin{align*}x\end{align*}.

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

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

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 \begin{align*}50^\circ\end{align*} and \begin{align*}70^\circ\end{align*}. Their sum is also \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.

### Examples

#### Example 1

Earlier, you were given a problem about the contractor and the triangle shaped trim.

If the first angle is \begin{align*}70^o\end{align*} and the second angle is \begin{align*}40^o\end{align*}, 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 + \begin{align*}70^o + 40^o = 180^o\end{align*}. \begin{align*}180^o - 110^o = 70^o\end{align*}.

Then determine the measure of the third angle.

\begin{align*}70^o\end{align*} = the third angle.

#### Example 2

Solve the following problem.

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

First, determine if angle\begin{align*}S\end{align*} is an interior or an exterior angle.

Angle \begin{align*}S\end{align*} 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 angle \begin{align*}S\end{align*}.

Angle \begin{align*}S\end{align*} = \begin{align*}30^o + 35^o\end{align*} = \begin{align*}65^o\end{align*}

The answer is that the measure of angle \begin{align*}S\end{align*} is \begin{align*}65^o\end{align*}.

#### Example 3

If the sum of two angles of a triangle is \begin{align*}150^\circ\end{align*}, 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 \begin{align*}180^o\end{align*}.

Then, calculate the sum of the third angle.

\begin{align*}150^o\end{align*} + the third angle = \begin{align*}180^o\end{align*}. \begin{align*}180^o - 150^o = 30^o\end{align*}.

The answer is that the third angle is \begin{align*}30^o\end{align*}.

#### Example 4

If the sum of two of the angles is \begin{align*}75^\circ\end{align*}, 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.

\begin{align*}75^o\end{align*} = the measure of the third angle's exterior angle.

The answer is that the measure of the third angle's exterior angle is \begin{align*}75^o\end{align*}.

#### Example 5

A triangle has three angles: angle \begin{align*}A\end{align*}, angle \begin{align*}B\end{align*} and angle \begin{align*}C\end{align*}. Angle \begin{align*}A = 33^\circ\end{align*}, Angle \begin{align*}B = 65^\circ\end{align*}, what is the measure of Angle \begin{align*}C\end{align*}?

First, determine if you are calculating an interior or exterior angle.

Angle \begin{align*}C\end{align*} is an interior angle.

Next, remember the relationship between the three interior angles.

Angle \begin{align*}C + 33^o + 65^o = 180^o\end{align*}. \begin{align*}180^o - 98^o = 82^o\end{align*}.

Then, determine the measure of the third angle.

\begin{align*}82^o =\end{align*} the measure of angle \begin{align*}C\end{align*}.

The answer is that the measure of angle \begin{align*}C\end{align*} is \begin{align*}82^o\end{align*}.

### Review

Find the measure of the missing angle in each triangle.

1. \begin{align*}20 + 70 + x = 180^\circ\end{align*}
2. \begin{align*}60 + 60 + x = 180^\circ\end{align*}
3. \begin{align*}90 + 15 + x = 180^\circ\end{align*}
4. \begin{align*}100 + 45 + x = 180^\circ\end{align*}
5. \begin{align*}10 + 105 + x = 180^\circ\end{align*}
6. \begin{align*}120 + 45 + x = 180^\circ\end{align*}
7. \begin{align*}145 + 5 + x = 180^\circ\end{align*}
8. \begin{align*}150 + 20 + x = 180^\circ\end{align*}
9. \begin{align*}110 + 45 + x = 180^\circ\end{align*}
10. \begin{align*}60 + 40 + x = 180^\circ\end{align*}
11. \begin{align*}70 + 50 + x = 180^\circ\end{align*}
12. \begin{align*}80 + 45 + x = 180^\circ\end{align*}
13. \begin{align*}50 + 45 + x = 180^\circ\end{align*}
14. \begin{align*}30 + 55 + x = 180^\circ\end{align*}
15. \begin{align*}75 + 55 + x = 180^\circ\end{align*}

To see the Review answers, open this PDF file and look for section 8.7.

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### Vocabulary Language: English

Exterior angles

An exterior angle is the angle formed by one side of a polygon and the extension of the adjacent side.

Interior angles

Interior angles are the angles inside a figure.

Triangle

A triangle is a polygon with three sides and three angles.

Triangle Sum Theorem

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