Geometry Lesson 2.5 - Exterior Angles of a Triangle

At each vertex of a triangle, an exterior angle of the triangle may be formed by extending ONE SIDE of the triangle. See picture below.

Exterior Angles of a Triangle

We can use equations to represent what the picture is saying to us. For example,

x + y + z = 180

We know this is true, because the sum of the angles inside a triangle is 180 degrees. I should also point out that the measure of angle w + measure of angle z = 180 degrees, because they are a pair of supplementary angles. Notice how Z and W together make a straight line? That's 180 degrees.

If we combine the two equations above, we can say that the measure of angle w = x + y.

x + y + z = 180
w + z = 180

Now, rewrite the second equation as z = 180 - w and substitute that for z in the first equation:

x + y + 180 - w = 180
x + y - w = 0
x + y = w

There is a theorem called the TRIANGLE EXTERIOR ANGLE THEOREM and this is what is says:

The measure of an exterior angle (our w) of a triangle equals to the sum of the measures of the two remote interior angles (our x and y) of the triangle.

Let's try two samples.

Sample A:

If the measure of the exterior angle = (3x - 10) degrees and the measure of the two remote interiors angles are 25 degrees and (x + 15) degrees, find x.

First example of finding an exterior angle

To solve, we use the fact that W = X + Y. Note that here I'm referring to the angles W, X, and Y. Their names are not important. What is important is that the exterior angle equals the sum of the remote interior angles.

We equate and solve for x.

3x - 10 = 25 + x + 15
3x - 10 = x + 40
3x - x = 10 + 40
2x = 50
x = 50/2
x = 25

The angles, then, are 25, 40, and 65 degrees.

Sample B

The exterior angle given is 110 degrees. Two remote interior angles measure 50 and (2x + 30). Find x.

Remember: exterior = sum of remote interior angles

We equate 110 to (2x + 30) + 50 and solve for x.

110 = 2x + 30 + 50
110 = 2x + 80
110 - 80 = 2x
30 = 2x
30/2 = x
15 = x