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Exterior Angles Theorems

Exterior angles equal the sum of the remote interiors.

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Exterior Angles Theorems

Exterior Angle Theorems 

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.


Remote interior angles are the two angles in a triangle that are not adjacent to the indicated exterior angle. A and B are the remote interior angles for exterior angle 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 mA+mB=mACD. 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: ABC with exterior angle ACD

Prove: mA+mB=mACD

Statement Reason
1. ABC with exterior angle ACD Given
2. mA+mB+mACB=180 Triangle Sum Theorem
3. mACB+mACD=180 Linear Pair Postulate
4. mA+mB+mACB=mACB+mACD Transitive PoE
5. mA+mB=mACD Subtraction PoE

Measuring Angles in a Triangle 

1. Find the measure of RQS.

112 is an exterior angle of RQS. Therefore, it is supplementary to RQS because they are a linear pair.


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.


2. Find the measure of the numbered interior and exterior angles in the triangle.

m1+92=180 by the Linear Pair Postulate, so m1=88.

m2+123=180 by the Linear Pair Postulate, so m2=57.

m1+m2+m3=180 by the Triangle Sum Theorem, so 88+57+m3=180 and m3=35.

m3+m4=180 by the Linear Pair Postulate, so m4=145.

3. 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.


x and p are supplementary and add up to 180.


Using the Exterior Angle Sum Theorem 

The third exterior angle of the triangle below is 1.

By the Exterior Angle Sum Theorem:



Example 1

Find mA.

Example 2

Find mC.

Using the Exterior Angle Theorem, mC+16=121. Subtracting 16 from both sides, mC=105.

Example 3

Find the value of x and the measure of each angle.

Set up an equation using the Exterior Angle Theorem.

(4x+2)+(2x9)=(5x+13) interior anglesexterior angle(6x7)=(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 \begin{align*}(5(20)+13)^\circ=113^\circ\end{align*}. Double-checking our work, notice that \begin{align*}82^\circ + 31^\circ = 113^\circ\end{align*}. If we had done the problem incorrectly, this check would not have worked.

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

Use the following picture for the next three problems:

  1. What is \begin{align*}m\angle{1}+m\angle{2}+m\angle{3}\end{align*}?
  2. What is \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*}.

  1. Suppose the measures of the three angles of a triangle are x, y, and z. Explain why \begin{align*}x+y+z=180\end{align*}.
  2. Suppose the measures of the three angles of a triangle are x, y, and z. Explain why the expression \begin{align*}(180-x)+(180-y)+(180-z)\end{align*} represents the sum of the exterior angles of the triangle.
  3. 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 \begin{align*}(180-x)+(180-y)+(180-z)\end{align*} must equal 360 if \begin{align*}x+y+z=180\end{align*}.

Review (Answers)

To view the Review answers, open this PDF file and look for section 4.2. 

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Exterior Angle Sum Theorem Exterior Angle Sum Theorem states that the exterior angles of any polygon will always add up to 360 degrees.

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