 This video is called Triangle Similarity Theorms. We have three similarity theorems to prove that triangles are similar. In the past, we had triangle congruence theorems, like SSS, SAS, ASA, and AAS. Now we have a statement called AA with that little symbol after it called a tilde, which means AA similarity. Another statement is SSS similarity, and the third statement we have is SAS similarity. So let's go piece by piece and explain what these mean. The first theorem we'll talk about is AA similarity. And that states if two angles of one triangle are congruent to two angles of another triangle, then we can prove that the two triangles are similar. Remember, the word similar means they have the same shape, the same angles, but the sides are proportional. So let's talk about an example of AA similarity. In these two triangles, let's pretend that we knew that these two angles are congruent and also that these two angles are congruent. That's enough to prove that these two triangles are congruent. Sorry, not congruent, similar. The angles are congruent, the triangles are similar. So the statement that we would use to show that these two triangles are similar, well let's give these triangles names. So if the two triangles that we have are called ABC and LMN, the statement we would use to show that they're similar is that triangle ABC is similar to triangle LM. So that demonstrates AA similarity. The next similarity statement we want to examine is SSS similarity. And that states if the measures of corresponding sides of two triangles are proportional, then the sides are similar. Now that word proportional is extremely important. Proportions are essentially ratios that are set equal to each other. In other words, fractions of sides. Let's examine what it means for sides to be proportional. So if we knew something about the side lengths of this triangle, then we could maybe prove that they are similar using the SSS similarity statement. So let's put in a couple of examples. Let's pretend like these shortest side lengths are let's say AC is 6 units long and LM is 8 units long. Let's also pretend that the medium side lengths are 9 and 12 and that the longest side lengths, let's say they are 12 and 15. Now in order to determine whether the sides are proportional, we need to divide small triangle by big triangle and see if those fractions are all the same. So we'll set up our ratios by putting the small triangle ABC divided by the bigger triangle LMN. It doesn't matter exactly how you choose to set up your ratios, just make sure you keep them organized. So the shortest side length of triangle ABC is 6. The shortest side length of triangle LMN is 8. And we want to see whether that fraction is equal to the medium sides. So the medium side of triangle ABC is 9 units. The medium side of triangle LMN is 12. And we want to see again if the longest sides of the two triangles follow that same proportion. So the longest side of triangle ABC is 12. The longest side of triangle LMN is 15. So are these indeed equal fractions? If you take the red fraction 6 and 8 and you plug that into your calculator, 6 divided by 8 is 0.75. 9 divided by 12 is also 0.75. And 12 divided by 15, you guessed it, 0.75. And so that means all three pairs of sides are proportional because the fractions 6 divided by 8 is equal to 9 divided by 12 is equal to 12 divided by 15. So that's SSS similarity. By the way, I just realized I made a mistake. This side length shouldn't be 15. It should be 16. And that means that this fraction should be 12 divided by 16. Because then that does indeed equal 0.75. Oops, sorry about that. The last similarity theorem we have to deal with is SAS similarity. And that states if two of the sides are proportional to two corresponding sides and the included angles are congruent, then the triangles are similar. Let's investigate that with a sample. So here we have two triangles and let's again pretend that these angles are congruent, corresponding angles. And then let's also pretend that the side lengths, let's say that this short leg here is 4, this short leg here is 6, this short leg is let's say it's also 6 and this is equal to 9. So we know right off the bat that we have a pair of angles congruent. That refers to these angles. And so we want to check to see whether the sides are proportional or not. Because we want to potentially use SAS similarity. So to verify that the sides are proportional, we'll need to check proportions. In other words, the shortest side of the small triangle is 4, the shortest side of the large triangle is 6, and does that equal the short side, pardon me, the longer side of the small triangle divided by the longer side of the big triangle. Well, I know that 4 divided by 6, that simplifies to 2 thirds. And likewise 6 ninths, you divide top and bottom by 3, that simplifies to 2 thirds as well. So that means that the sides are indeed proportional, and so therefore these two triangles are indeed similar.