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# Exterior Angles Theorems

## Exterior angles equal the sum of the remote interiors.

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Exterior Angles in Given Triangles

Learning Goal

By the end of the lesson I will be able to . . . describe the properties and relationships of the exterior angles of triangles

What if you knew that two of the exterior angles of a triangle measured 130\begin{align*}130^\circ\end{align*}? 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

Then watch this video.

Finally, watch this video.

### 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 counter-clockwise.

Notice that the interior angle and its adjacent exterior angle form a linear pair and add up to 180\begin{align*}180^\circ\end{align*}.

m1+m2=180\begin{align*}m\angle1 + m\angle2 = 180^\circ\end{align*}

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\begin{align*}360^\circ\end{align*}.

m1+m2+m3m4+m5+m6=360=360\begin{align*}m\angle{1} + m\angle{2}+m\angle{3} &= 360^\circ\\ m\angle{4} + m\angle{5} + m\angle{6} & = 360^\circ\end{align*}.

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.)

mA+mB=mACD\begin{align*}m\angle{A} + m\angle{B}=m\angle{ACD}\end{align*}.

#### Example A

Find the measure of RQS\begin{align*}\angle{RQS}\end{align*}.

Notice that 112\begin{align*}112^\circ\end{align*} is an exterior angle of RQS\begin{align*}\triangle{RQS}\end{align*} and is supplementary to RQS\begin{align*}\angle{RQS}\end{align*}.

Set up an equation to solve for the missing angle.

112+mRQSmRQS=180=68\begin{align*}112^\circ + m\angle{RQS} & = 180^\circ\\ m\angle{RQS} &= 68^\circ\end{align*}

#### Example B

Find the measures of the numbered interior and exterior angles in the triangle.

We know that m1+92=180\begin{align*}m\angle{1} + 92^\circ = 180^\circ\end{align*} because they form a linear pair. So, m1=88\begin{align*}m\angle{1} = 88^\circ\end{align*}.

Similarly, m2+123=180\begin{align*}m\angle{2} + 123^\circ = 180^\circ\end{align*} because they form a linear pair. So, m2=57\begin{align*}m\angle{2} = 57^\circ\end{align*}.

We also know that the three interior angles must add up to 180\begin{align*}180^\circ\end{align*} by the Triangle Sum Theorem.

m1+m2+m388+57+m3m3=180by the Triangle Sum Theorem.=180=35\begin{align*}m\angle{1} + m\angle{2} +m\angle{3} & = 180^\circ \qquad \text{by the Triangle Sum Theorem.}\\ 88^\circ + 57^\circ + m\angle{3} &= 180\\ m\angle{3} & = 35^\circ\end{align*}

Lastly, m3+m435+m4m4=180because they form a linear pair.=180=145\begin{align*}\text{Lastly}, \ m\angle{3} + m\angle{4} & = 180^\circ \qquad \text{because they form a linear pair}.\\ 35^\circ + m\angle{4} &= 180^\circ\\ m\angle{4} &= 145^\circ\end{align*}

#### Example C

What is the value of p\begin{align*}p\end{align*} in the triangle below?

First, we need to find the missing exterior angle, which we will call x\begin{align*}x\end{align*}. Set up an equation using the Exterior Angle Sum Theorem.

130+110+xxx=360=360130110=120\begin{align*}130^\circ + 110^\circ + x &= 360^\circ\\ x& = 360^\circ-130^\circ-110^\circ\\ x& = 120^\circ\end{align*}

x\begin{align*}x\end{align*} and p\begin{align*}p\end{align*} add up to 180\begin{align*}180^\circ\end{align*} because they are a linear pair.

x+p120+pp=180=180=60\begin{align*}x + p & = 180^\circ\\ 120^\circ + p & = 180^\circ\\ p & = 60^\circ\end{align*}

### Guided Practice

1. Find mC\begin{align*}m\angle{C}\end{align*}.

2. Two interior angles of a triangle are 40\begin{align*}40^\circ\end{align*} and 73\begin{align*}73^\circ\end{align*}. What are the measures of the three exterior angles of the triangle?

3. Find the value of x\begin{align*}x\end{align*} and the measure of each angle.

1. Using the Exterior Angle Theorem

mC+16mC=121=105\begin{align*}m\angle{C} + 16^\circ & = 121^\circ\\ m\angle{C} & = 105^\circ\end{align*}

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\begin{align*}180^\circ\end{align*}) with an exterior angle. So, since one of the interior angles is 40\begin{align*}40^\circ\end{align*} that means that one of the exterior angles is 140\begin{align*}140^\circ\end{align*} (because 40+140=180\begin{align*}40+140=180\end{align*}). Similarly, since another one of the interior angles is 73\begin{align*}73^\circ\end{align*}, one of the exterior angles must be 107\begin{align*}107^\circ\end{align*}. 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\begin{align*}140^\circ\end{align*} and 107\begin{align*}107^\circ\end{align*}, then the third Exterior Angle must be 113\begin{align*}113^\circ\end{align*} since 140+107+113=360\begin{align*}140+107+113=360\end{align*}.

So, the measures of the three exterior angles are 140\begin{align*}140\end{align*}, 107\begin{align*}107\end{align*} and 113\begin{align*}113\end{align*}.

3. Set up an equation using the Exterior Angle Theorem.

(4x+2)+(2x9)=(5x+13)remote interior anglesexterior angle (6x7)=(5x+13)  x=20\begin{align*}&(4x+2)^\circ + (2x-9)^\circ = (5x+13)^\circ\\ & \quad \uparrow \qquad \qquad \nearrow \qquad \qquad \qquad \uparrow\\ & \text{remote interior angles} \qquad \qquad \text{exterior angle}\\ & \qquad \qquad \quad \ (6x-7)^\circ = (5x+13)^\circ\\ & \qquad \qquad \qquad \qquad \ \ x = 20\end{align*}

Substitute in 20\begin{align*}20\end{align*} for x\begin{align*}x\end{align*} to find each angle.

[4(20)+2]=82[2(20)9]=31Exterior angle: [5(20)+13]=113\begin{align*}[4(20)+2]^\circ=82^\circ && [2(20)-9]^\circ=31^\circ && \text{Exterior angle:} \ [5(20)+13]^\circ=113^\circ\end{align*}

### Practice

Determine m1\begin{align*}m\angle{1}\end{align*}.

Use the following picture for the next three problems:

1. What is m1+m2+m3\begin{align*}m\angle{1}+m\angle{2}+m\angle{3}\end{align*}?
2. What is m4+m5+m6\begin{align*}m\angle{4}+m\angle{5}+m\angle{6}\end{align*}?
3. What is \begin{align*}m\angle{7}+m\angle{8}+m\angle{9}\end{align*}?

Solve for \begin{align*}x\end{align*}.

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