 In this video, we'll look at some pairs of triangles and see whether we can prove they're similar or not. In this first example, we know all three sides of both triangles. Because we know all three sides and nothing about angles, we want to check to see whether the triangles are similar using the SSS similarity statement. So we need to check to see whether all three sides are proportional. I'm going to set up my ratios by taking the triangle on the left and dividing them by triangle on the right. And so in the triangle on the left, the shortest side is 12. In the triangle on the right, the shortest side is 16. So I'll use those side lengths as one ratio. 12 on the left triangle, 16 on the right triangle. And I want to see if those sides, those ratios are equal to the medium side lengths. The medium side length in the left triangle is 15. The medium side in the right triangle is 20. So the ratio will be 15 divided by 20. And then lastly I want to see if that ratio is equal to the long side of the triangle on the left divided by the long side of the triangle on the right. So 18 divided by 24. So are these three fractions equal to each other? That's the question. If you take 12 divided by 16, that fraction simplifies to 3 fourths. 15 divided by 20, that fraction also simplifies to 3 fourths. And 18 divided by 24, well both top and bottom are divisible by 6. And so that also simplifies to 3 fourths. And that means that these ratios are indeed equal. And so all the sides are proportional. So we've just proven that these two triangles are indeed similar by SSS similarity. Here's another example. We want to see if these two triangles are similar. And we know that these two angles are congruent. So 110 is congruent to 110. And then we also have two pairs of sides. So we're going to check using the SAS similarity theorem. We already know we have the angles and in particular the included angles are similar. So now our job is to see whether the sides are proportional. I'll set up my ratios by dividing the big triangle part divided by the little triangle part. And again, if you chose to have little divided by big, that's fine as well. So in the big triangle, 35 is the bigger of the two sides. And so the long side of the big triangle, sorry, long side, I meant the short side. The short side of the big triangle is 35. The short side of the little triangle is 14. So we want to see whether that equals the longer side of the big triangle divided by the long of the little. So the fraction 35 14th is divisible by 7, and that simplifies to 5 over 2. 40 divided by 16, both 40 and 16 are divisible by 8, and that also simplifies to 5 over 2. And so that means that the two sides that we know about are proportional and the included angle, the included angles are congruent. And so therefore these triangles are indeed similar. In this pair of triangles, we know something about one pair of angles, 23 and 23, and we know two pair of sides. So it seems like we might want to check SAS similarity to see whether they're similar. The problem with this pair is that the angle that we're dealing with isn't the included angle. And by that I mean it only touches one of the pairs of sides that we know about. It only touches this 16 and 24 side pair. If instead we knew that let's say these angles were congruent, then we could maybe check using SAS. But since we don't know that those angles are congruent, we can't prove that these two triangles are similar. How about these two triangles? Are they similar? At first glance it seems like no, of course not. They only have one pair of angles congruent. 110 is congruent to 110. However, the triangle sum theorem says that in a triangle all the angles have to add up to 180. And so if this, let's say in this smaller triangle we know 110 and 41, that means this angle must be 29 degrees because 29 plus 41 plus 110 is equal to 180. And so now we've established that we have two pairs of angles that are congruent. 110 is congruent to 110 and 29 is congruent to 29. And so these two triangles are similar because of AA. In a similar way to the previous problem, these two triangles have right angles and they also have, well, one has 50 degrees, one has 40 degrees. However, the triangle sum theorem says that 90 plus 50 plus some other angle must add up to 180. And so that angle we'll see is indeed 40 degrees. And so we have two triangles with at least two angles congruent. Both have 90 degrees and both have 40 degrees. And so these triangles are indeed similar by AA similarity. How about these two triangles? Can we prove that they're similar? Since we don't know anything about angles, we want to check using SSS similarity. So in the triangle on the left, the shortest side is 18. In the triangle on the right, the shortest side is 15. So that's one ratio we'll use, 18 to 15. Again, in the triangle on the left, the medium side is 24. The medium side in the triangle on the right is 20. And then finally, the long side in the triangle on the left is 36. The long side in the triangle on the right is 28. And so we want to check whether these three fractions are equal to each other. So 18 divided by 15. 18, 15, both top and bottom are divisible by 3. And that simplifies to 6 fifths. 24 and 20, well, both at top and bottom are divisible by 4. And that simplifies to 6 fifths as well. So that means that these two sides are proportional. But 36, 28, hmm. Both of those are divisible by 4. And so that reduces to 9 sevenths. And so that means that that long side is not proportional. And so even though these first two sides, first two ratios are equal, since the last one isn't equal, we've just shown that these two triangles are not similar.