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# Exterior Angle Inequality

The exterior angle inequality theorem states that a triangle's exterior angle is greater than either of the non-adjacent interior angles. Before delving deeper into this topic, let's take a closer look at the exterior angle theorem.

## The exterior angle theorem

The exterior angle theorem states that the measure of an exterior angle of a triangle equals the sum of the measures of the two non-adjacent interior angles of the triangle. Remember, a triangle has 3 internal angles that always add up to 180 degrees. The triangle also has six exterior angles. The exterior angle theorem applies to each of the 6 exterior angles.

Let's take a look at the following example:

In this figure, the exterior angle is $\angle 4$ , the adjacent angle is $\angle 3$ , and the non-adjacent interior angles are $\angle 1$ and $\angle 2$ . According to the exterior angle theorem, $m\angle 4=m\angle 1+m\angle 2$ .

Let's look at how to prove this:

Proof:

Given: $∆PQR$

To Prove: $m\angle 4=m\angle 1+m\angle 2$

 Statement Reason 1 $∆PQR$ is a triangle Given 2 $m\angle 1+m\angle 2+m\angle 3=180°$ Triangle Sum Theorem 3 $\angle 3$ and $\angle 4$ forms a linear pair. Definition of a linear pair. 4 $\angle 3$ and $\angle 4$ are supplementary. If two angles form a linear pair, they are supplementary. 5 $m\angle 3+m\angle 4=180°$ Definition of supplementary angles. 6 $m\angle 3+m\angle 4=m\angle 1+m\angle 2+m\angle 3$ Statements 2, 5, and Substitution Property. 7 $m\angle 4=m\angle 1+m\angle 2$ Subtraction Property

## The exterior angle inequality theorem

Remember, the exterior angle inequality theorem states that any exterior angle of a triangle is greater than either of the opposite (non-adjacent) interior angles. Let's take a look at the following example:

In the above figure, the exterior angle is $\angle QRS$ . Its adjacent angle is $\angle R$ and the non-adjacent angles are $\angle Q$ and $\angle P$ . You can see that $m\angle QRS$ is larger than $m\angle Q$ . Since the exterior angle theorem states that $m\angle QRS=m\angle Q+m\angle P$ , $m\angle QRS$ must also be greater than $m\angle P$ .

Another way to state this is $m\angle QRS>m\angle Q$ and $m\angle QRS>m\angle P$ .

## Practice questions on the exterior angle theorem and exterior angle inequality

a. In the above figure, express $m\angle QRS$ in terms of the interior angles.

$m\angle Q+m\angle P$

b. In the above figure, which angle is adjacent to $\angle QRS$ ?

$\angle QRP$

c. In the above figure, what two angles are not adjacent to $\angle QRS$ ?

$\angle Q$ and $\angle P$

d. True or false: In the above figure, $m\angle Q>m\angle QRS$ ?

False

e. In the above figure, what is the exterior angle?

$\angle 4$

f. Which angle in the above figure is adjacent to the exterior angle?

$\angle 3$

g. In the above figure, the measures of which two angles (combined) equal m∠4?

$m\angle 4=m\angle 1+m\angle 2$

h. True or false: In the above figure, $m\angle 1 ?

True

## Get help learning about exterior angle inequality

Getting familiar with the exterior angle inequality theorem can be exciting, but also challenging. Whether your student wants to more easily identify non-adjacent angles or would like to brush up on their knowledge of the exterior angle theorem, it helps to have the assistance of a professional educator. Working alongside a tutor can make a positive difference in your student's academic progress. Find out how tutoring sessions can get your student on the right track in their geometry studies by contacting the Educational Directors at Varsity Tutors today.

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