## Exterior Angles of Triangle

**Theorem: **An measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.

(non-adjacent interior angles may also be referred to as remote interior angles)

**An exterior angle** is formed by one side of a triangle and the extension of an adjacent side of the triangle.

In the triangle at the right, ∠4 is an exterior angle.

The theorem above states that if ∠4 is an exterior angle, its measure is equal to the sum of the measures of the 2 interior angles to which it is not adjacent, namely, ∠2 and ∠3.

**m∠4 = m∠2 + m∠3**

Since the measure of an exterior angle equals the sum of its two non-adjacent interior angles, the exterior angle is also greater than either of the individual non-adjacent interior angles.

**m∠4 > m∠2 **and also** m∠4 > m∠3**

**Theorem:** The measure of an exterior angle of a triangle is greater than either of its two non-adjacent interior angles.

**Examples**

1. In ∆PQR, m∠Q = 45°, and m∠R = 72°. Find the measure of an exterior angle at P.

It is always helpful to draw a diagram and label it with the given information.

Then, using the first theorem above, set the exterior angle ( x ) equal to the sum of the two non-adjacent interior angles which are 45° and 72°.

x = 45 + 72

x = 117

So, an exterior angle at P measures **117°**.

2. In ∆DEF, an exterior angle at F is represented by 8x + 15. If the two non-adjacent interior angles are represented by 4x + 5, and 3x + 20, find the value of x.

First, draw and label a diagram.

Next, use the first theorem to set up an equation.

Then solve the equation for x.

8x + 15=(4x + 5)+(3x + 20)

8x + 15 = 7x + 25

8x = 7x + 10

x = 10

So, **x = 10**

3. Find the measure of an exterior angle at the base of an isosceles triangle whose vertex angle measures 40°.

First, draw and label a diagram.

You may choose to place the exterior angle at either vertex B or C. They will have the same measure.

Next, we have to find the measure of a base angle–

— let’s say ∠B.

Remember that the 2 base angles of an isosceles triangle are equal, so we’ll represent each as y.

Then, write an equation, using the fact that there are 180 degrees in a triangle.

Now we can solve for x using the exterior angle theorem. Set the measure of the exterior angle equal to the sum of the measures of the two non-adjacent interior angles.

y + y + 40 = 180

2y + 40 = 180

2y = 140

y = 70

x = 70 + 40

x = 110

So, an exterior angle at the base measures **110°**.