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 = 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.....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 110°.