Exterior Angles of Triangles
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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 = 45º, and m<R = 72º.  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 45º and 72º.

x = 45 + 72
x = 117

So, an exterior angle at P measures 117º.

 

 

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 40º.

First.....the 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 110º.