Back To CourseGeometry: High School
15 chapters | 160 lessons
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Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.
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 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.
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.
Now we have two angles and a side! So we can say that triangle MNQ is congruent to triangle POQ because of the AAS theorem. And therefore, MN is congruent to OP because corresponding parts of congruent triangles are congruent, or CPCTC.
Okay, let's lose the bow tie and get back to our game. Let's try to find some matches and identify how they're congruent.
Here are two triangles: DEF and JKL. Are they matches? Let's see. EF is congruent to KL. That's a pair of sides. And we have E congruent to K and F congruent to L, so two angles and a side. Should we use angle-angle-side? Well, this time we have the included side, so it's most accurately angle-side-angle.
How about these two? Oh, we don't have any congruent angles, do we? Do we have any vertical angles? No. Hmm. Well, we have MN and LO and LM and NO. So that's two pairs of sides. Oh, and LN is congruent to LN because of the reflexive property, which is just fancy math-speak for 'it's the same line.' So we have three congruent sides. That's side-side-side!
Here's another pair. This time, we can see that DF and RS are congruent. So are DE and ST. And angles D and S? Those are the included angles between our congruent sides. So this is side-angle-side, or SAS.
How about one more? These upside-down pyramids have one pair of congruent sides: AB and DE. Then we have angles A and D as well as angles C and F. That's two angles and a non-included side. So this one is our theorem du jour, angle-angle-side, or AAS. We're awesome at this game!
In summary, we added to our matching skills when we're trying to find congruent triangles. We already knew side-side-side, side-angle-side, and angle-side-angle.
Here, we learned about the angle-angle-side theorem, or AAS. This theorem states 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.
This theorem is an adaptation of angle-side-angle, or ASA. When we have a non-included side, we know it could be an included side due to the sum of the interior angles of a triangle always being 180.
After watching this lesson, you should be able to remember what the the angle-angle-side (AAS) theorem states, understand when to use it, and explain how it helps determine whether triangles are congruent to each other.
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Back To CourseGeometry: High School
15 chapters | 160 lessons