In this lesson, we examine the Third Angle Theorem in order to prove the values of an unknown angle within a set of similar triangles. After the lesson and a couple of examples, you can test your understanding with a quiz.

## The Third Angle Theorem

By now you’re probably pretty familiar with triangles. After all, you’ve been playing with triangle blocks since preschool and have now learned a great deal about their length, area, and perimeter. But what about those pesky angles? Would it be so much for them to just all be like 45-45-90 or 30-60-90 triangles? At least that way you could remember everything you ever needed in order to deal with them.In this lesson, we’re going to make dealing with triangles and their difficult angles quite a bit easier. We’re going to prove the **Third Angle Theorem**, which states that if you are comparing two triangles and each has two angles that are the same as an angle in the different triangle, then the third angle of each will have the same value.

## All Triangles Have 180 Degrees

Let’s start by reviewing perhaps the most basic fact about triangles with respect to their angles. If you were to add up the values of all three angles in any triangle on a flat plane, you would always get 180 degrees.

I don’t care what the triangle looks like, if it’s acute, obtuse, right, or even scalene! If you add the values of the three angles up, you always get 180 degrees.In fact, the only time that changes is if you suddenly put that triangle on something that’s not a flat plane; however, the type of math that is used to solve those triangles is only really useful to engineers and math professors, so we don’t have to worry about it here!So, one last time, how many degrees is the sum of the three measures of a triangle? That’s right, 180.

## Similarity of Triangles

Now that’s not the only thing we need to review. Before we dig any deeper, let’s make sure that you understand what is meant by the term ‘**similar triangle**.’ By similar, I mean that the triangles have the same angle values. A 30-60-90 triangle is similar to every other 30-60-90 triangle. When I say that the angles are similar, I mean that they are the same.

Now, on to an example to prove this.

## Example

Let’s say that you were trying to solve two triangles named ABC and XYZ. The letters of each both refer to the three angles of the triangle. Angle A has a value of 45 degrees, while angle B has a value of 30 degrees. Meanwhile, angle X also has a value of 45 degrees and angle Y has a value of 30 degrees as well. So, are these two triangles similar?To be sure, we need to solve for the values of angle C and angle Z.

To find the value of C, subtract 45 and 30 from 180. Remember 180 is the number of degrees in any triangle. Do that, and you’ll get 105 degrees; that is the value of angle C.

Now on to angle Z. Again, the total number of degrees in a triangle is 180, so subtract 30 and 45 to represent angles X and Y. You end up with an answer of 105 degrees. Therefore, these triangles are similar because the angles of A, B, and C are the same as the angles of X, Y, and Z, respectively.

## A Known & Two Unknowns

That was a pretty easy example, so let’s try one that’s a little bit more difficult to make sure that you get the math in question. Let’s say now that you are looking at the two triangles DEF and NOP. Again, each letter represents an angle in each triangle.

The value of angle D is 20, while the value of angle E is 120. Meanwhile, the value of angle N is also 20, but the value of angle O is 40. In this one, you have to figure out the values of angles F and P in order to solve the triangle.Again, even if it doesn’t look like it, the same math applies. That means that for triangle DEF, we subtract the values of the lesser-known angles from 180.

That gives us an answer of 40. We do the same thing for triangle NOP, giving us a value of 120.With that, we can tell that the values of the angles in each triangle are 20, 40, and 120.

Don’t be put off that the letters are out of order. Since the values of the angles still match, the triangles are still similar.

## Lesson Summary

In this lesson, we proved the validity of the **Third Angle Theorem** to solve triangles.

The basis of this rule involves remembering that a triangle always has a sum of interior angles that equals 180 degrees. In our first example, we proved the math at play and in our second, we proved the theorem.