The AAS (Angle-Angle-Side) Theorem: Proof and Examples

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  • 0:05 Finding Matches
  • 1:28 The AAS Theorem
  • 3:12 Practice Proof
  • 4:20 Finding Congruence
  • 5:52 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

When trying to find out if triangles are congruent, it's helpful to have as many tools as possible. In this lesson, we'll add to our congruence toolbox by learning about the AAS theorem, or angle-angle-side.

Finding Matches

Why is the AAS theorem important? Well, did you ever play the game where you turn over two cards trying to find matching pairs? Maybe you're looking for two pictures of bananas, or lions, or lions eating bananas. I bet that pair is elusive.

That game is like what we're trying to do here with triangles. We want to find triangles that match, or are congruent. That means they have three congruent sides and three congruent angles.

There are several ways to determine if two triangles are congruent. If all three sides match, we call that side-side-side, or SSS. If we have a side, an included angle, and another side, then it's side-angle-side, or SAS. The included angle is the angle between the sides. You could also have an angle, included side, and another angle. We call that, yep, angle-side-angle, or ASA.


But what about this pair? Can we use side-side-side? No, we only know that AB is congruent to XY. What about side-angle-side? Again, we only have one side. There's still angle-side-angle. What about that? Still no. With ASA, we need the included side, or the side between the congruent angles.

This is where angle-angle-side comes in.

The AAS Theorem

The angle-angle-side Theorem, or AAS, tells us that if two angles and any side of one triangle are congruent to two angles and any side of another triangle, then the triangles are congruent.

So, SSS, SAS, ASA and now AAS? It seems like we're getting pretty liberal with what makes triangles congruent, doesn't it? It's like playing the matching card game and saying this cow and this cheeseburger match because cheeseburgers are made from cows. What would the cow think about that?

But AAS makes more sense than it might seem. In our two triangles here, angles B and Y are congruent. And angles C and Z are congruent. Let's say angle B is 30 degrees. And let's say angle C is 80 degrees. What is angle A? The sum of the interior angles of a triangle is 180, so 180 minus 30 minus 80 is 70. A is 70.

Well, if B is 30, then so is Y. And if C is 80, then so is Z. That means that both A and X are 70. A and X must also be congruent. That means that we know a pair of angles, B and Y, an included side, AB and XY, and then the angles on the other side, A and X. That's angle-side-angle.

Since triangles have three angles and their angles always add up to 180, if we know angle-angle-side, then we also know angle-side-angle. That's why we only need to know two angles and any side to establish congruence.

So AAS isn't really like saying a cow and a cheeseburger are a match. It's really just another way of saying two identical cows are a match.

Practice Proof

Let's take a break from the matching game and see AAS in action in a proof.

Here's a bow tie. It's also two triangles. We're given that NQ is congruent to OQ. We're also given that angle M is congruent to angle P. Can we prove that MN is congruent to OP?

In order to prove that, we want to prove that the triangles are congruent. Let's start our proof by stating that NQ is congruent to OQ. That's given. And angle M is congruent to angle P. Again, that's given. Do we have enough to establish that the triangles are congruent yet? We have one side and one angle. Alas, we're not there yet.

But we can say that angle MQN is congruent to angle OQP. Why? They're vertical angles. Vertical angles are always congruent.

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