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# Exterior Angle Theorem

According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles of the triangle. That definition sounds more complicated than it is, so consider the following diagram:

Above, angle 4 is the exterior angle. Angle 4 is adjacent to angle 3, which means angles 1 and 2 are non-adjacent angles. Therefore, the sum of the measures of angles 1 and 2 is equivalent to the measure of angle 4. Expressed mathematically, we can say that $m\angle 4=m\angle 4+m\angle 4$ .

## Proving the Exterior Angle Theorem

We can prove why the Exterior Angle Theorem is true using concepts that you're familiar with. Let ΔPQR be a triangle with exterior angle ∠4 at vertex Q. Interior angles ∠1, ∠2, and ∠3 are inside the triangle.

Angles Table

Angle Location
∠1 Interior
∠2 Interior
∠3 Interior
∠4 Exterior

Apply the Triangle Sum Theorem to $∆\mathrm{PQR}$ :

$m\left(\angle 1\right)+m\left(\angle 2\right)+m\left(\angle 3\right)=180°$ .

Observe that ∠3 and ∠4 form a linear pair:

$m\left(\angle 3\right)+m\left(\angle 4\right)=180°$ (definition of a linear pair, and linear pairs are supplementary).

Use the Substitution Property to equate the two expressions for 180°:

$m\left(\angle 1\right)+m\left(\angle 2\right)+m\left(\angle 3\right)=m\left(\angle 3\right)+m\left(\angle 4\right)$

Apply the Subtraction Property to isolate m(∠4):

$m\left(\angle 4\right)=m\left(\angle 1\right)+m\left(\angle 2\right)$

This completes the proof of the Exterior Angle Theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

## Exterior Angle Theorem practice problems

a. What is the measure of the exterior angle of an equilateral triangle?

We know that equilateral triangles have 3 sides each measuring 60 degrees, meaning we don't need to know where the exterior angle is to answer this question. We know that the exterior angle will be equal to the combined measures of the two remote interior angles due to the Exterior Angles Theorem, which must be 120 degrees since all three angles measure 60 degrees. Therefore, the exterior angle will measure 120 degrees for all 3 exterior angles.

b. $∆\mathrm{ABC}$ is an isosceles triangle and the measure of $\angle C$ is 100 degrees. What is the measure of the exterior angle adjacent to $\angle A$ ?

An isosceles triangle has 2 congruent sides, so ΔABC has 2 equivalent interior angles. Two angles measuring 100 degrees would exceed the 180 degrees prescribed by the Triangle Sum Theorem, meaning the other 2 angles must be the equivalent ones. We can calculate that the congruent angles each measure 40 degrees so the sum of its interior angles is 180 as $40+40+100=180$ . Per the Exterior Angle Theorem, the exterior angle adjacent to $\angle A$ will have the same measure as $\angle B$ (40 degrees) $+\angle C$ (100 degrees) . Therefore, the exterior angle measures 140 degrees.

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