Exterior Angle Theorem: Definition & Formula

While it may sound scary at first, the exterior angle theorem is a simple formula that helps us solve for the missing angle of a triangle. Rather than examining a formal proof of the theorem, we’ll focus on the formula itself and how it can be applied.
Exterior Angle Theorem Background
So we all know that a straight line measures 180 degrees, and when two angles form to create a straight line, they too measure 180 degrees. What if I told you, that there is a cool mathematical trick that you can use when working with a triangle that applies this simple fact?

Now, before you get all anxious over the idea that this lesson is about a theorem, just know that in this case, a theorem is just the fancy way of saying rule. You can go ahead and breathe a sigh of relief to know that we aren’t going to work though any two-column proofs or try to prove this theorem by contradiction. Instead, we will focus on the formula itself and how we can apply the theorem to solve for missing angles of a triangle.

Definition & Formula
The exterior angle theorem states that the exterior angle formed when you extend the side of a triangle is equal to the sum of its non-adjacent angles. Remember, our non-adjacent angles are those that don’t touch the angle we are working with. So, when we extend the side of an angle, creating a straight line that goes beyond the triangle, we have created an exterior, or outside, angle that equals the sum of the two angles inside of the triangle that it does not touch.

Let’s look at the following diagram to put this in simpler terms.

Diagram of a triangle with an exterior angle

The exterior angle theorem tells us that the measure of angle D is equal to the sum of angles A and B.

In formula form: m<D = m<A + m<B

Think back to our common fact that a straight line is equal to 180 degrees. We must also remember that the angles of a triangle equal 180 degrees as well.

m<A + m<B + m<C = 180

m<C + m<D = 180

You might notice that each equation equals 180. Because of this, we can set them equal to each other.

m<A + m<B + m<C = m<C + m<D

Now, we can cancel out the m?C because it appears on both sides of our equation to simplify it down to:

m<A + m<B = m<D

And just like that, we are back to our original formula for the exterior angle theorem. It is that easy. Now, it's not likely that you will ever need to really prove this theorem, but it is always good to understand why something works before you begin applying it.

Applying the Exterior Angle Theorem

Diagram with angles labeled

Looking at this diagram, we are given the task to solve for the missing angle x. Let's begin by first plugging in our known information to the exterior angle theorem formula.

x + 51 = m<D

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