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Exterior Angles Theorems

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What if you knew that two of the exterior angles of a triangle measured 130^\circ ? 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

CK-12 Exterior Angles Theorems

James Sousa: Introduction to the Exterior Angles of a Triangle

Then watch this video.

James Sousa: Proof that the Sum of the Exterior Angles of a Triangle is 360 Degrees

Finally, watch this video.

James Sousa: Proof of the Exterior Angles Theorem

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 counterclockwise.

Notice that the interior angle and its adjacent exterior angle form a linear pair and add up to 180^\circ .

m\angle1 + m\angle2 = 180^\circ

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^\circ .

m\angle{1} + m\angle{2}+m\angle{3} &= 360^\circ\\m\angle{4} + m\angle{5} + m\angle{6} & = 360^\circ .

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.)

m\angle{A} + m\angle{B}=m\angle{ACD} .

Example A

Find the measure of \angle{RQS} .

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

Set up an equation to solve for the missing angle.

112^\circ + m\angle{RQS} & = 180^\circ\\m\angle{RQS} &= 68^\circ

Example B

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

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

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

We also know that the three interior angles must add up to 180^\circ by the Triangle Sum Theorem.

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

\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

Example C

What is the value of p in the triangle below?

First, we need to find the missing exterior angle, which we will call x . Set up an equation using the Exterior Angle Sum Theorem.

130^\circ + 110^\circ + x &= 360^\circ\\x& = 360^\circ-130^\circ-110^\circ\\x& = 120^\circ

x and p add up to 180^\circ because they are a linear pair.

x + p & = 180^\circ\\120^\circ + p & = 180^\circ\\p & = 60^\circ

CK-12 Exterior Angles Theorems

Guided Practice

1. Find m\angle{C} .

2. Two interior angles of a triangle are 40^\circ and 73^\circ . What are the measures of the three exterior angles of the triangle?

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

Answers:

1. Using the Exterior Angle Theorem

m\angle{C} + 16^\circ & = 121^\circ\\m\angle{C} & = 105^\circ

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

So, the measures of the three exterior angles are 140 , 107 and 113 .

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

&(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

Substitute in 20 for x to find each angle.

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

Practice

Determine m\angle{1} .

Use the following picture for the next three problems:

  1. What is m\angle{1}+m\angle{2}+m\angle{3} ?
  2. What is m\angle{4}+m\angle{5}+m\angle{6} ?
  3. What is m\angle{7}+m\angle{8}+m\angle{9} ?

Solve for x .

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