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

(nonadjacent 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.

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

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


