So by now, we've seen how to use the law of sines to solve for missing sides in a non-right triangle. One of the most common types of problems you'll run into is where a problem will give you some angles and sides, and they'll ask you to classify and then solve a triangle by finding all of the other missing information, the other missing sides and angles. Now if that seems kind of intimidating, don't worry, because what I'm going to show you in this video is that, really, to solve these problems, all we need is the law of sines that we just learned, plus another familiar equation we've seen a bunch of times before. So I'm going to walk you through a step-by-step process of how to solve these two specific types of triangles. But first, I want to introduce you to the three types or categories of triangles for which you'll use the law of sines. Let's just jump right in. Alright? So you're going to see a bunch of acronyms in your books and classes like ASA, SAA, and SSA. It's kind of just a soup of letters of s's and a's. And, really, all these types or acronyms just have to deal with is the information that you're given at the start of a problem. The first one is called an angle-side-angle. This is where you were given two angles and the side between them. For example, you were given big A, big C, and little b. The next one is called the side-angle-angle. This is where you're given two angles and a side that's not between them but adjacent to them. So an example of this would be big A, big C, and little c. And then finally, the last one is called side-side-angle. This is where you're given, you guessed it, two sides and an angle. An example of this would be little b, little c, and big C. We're actually not really going to talk about this one at all in this video. We're going to talk about this one in a later video. All we're going to cover, really, are these two in this specific video. Alright? So without further ado, let's just jump right into our example here because the first thing it asks us to do is classify this triangle. So let's just jump right in. Alright? So I've got this information here. A = 30, C = 70 degrees, and then little c = 6. Alright? Regardless of if you can't sort of tell off the bat what type of triangle it is, the first thing you're always going to do is just try to draw or attempt to sketch the triangle because then you'll be able to see what it is from there. Alright? So I'm just going to go ahead and sketch this. Your sketch doesn't have to look exactly like mine, but what I always like to do is look at the angles first because it's easiest to visualize what's going on there. I know that one of the angles is going to be 30 degrees. So what I'm going to do is I'm going to draw a little sort of leg like this, and the 30-degree angle is going to look something like that. So this is going to be 30 degrees. Alright? Now the next one, the next angle that I know is going to be 70 degrees, which is going to look something like this but then a little bit sort of tilted in, not exactly a 90-degree angle. So it's going to look something like this. Alright? So this is going to be my 70 degrees. And I'm basically just going to keep on going with those lines until they sort of intersect like this. And that's going to be my other angle. So this is sort of a sketch of my triangle. This is going to be angle A, and that's angle C based on what I've given, which means that by default, this has to be B. Right? And remember, the rule for the sides, they always have to go opposite of their corresponding angles. So this is little c, which we know is 6. This is going to be little a, and this is going to be little b. So our job is to now solve for this triangle by solving for these missing variables, which we don't know over here, big B, little a, and little b. However, what we can do right now is we can actually classify this triangle because we've already drawn the sketch. If you notice here, what we've got is we've got an angle and angle and one of the sides that's not between the angles but adjacent. So that means that this is actually exactly what this type of triangle set. Right? So the example for this is ACC, and that's exactly what we were given inside this problem. But we just drew it out, and we kind of confirmed this. Right? So this is going to be an SAA triangle. Right? Side-angle-angle. Alright? So if we are given an angle-side-angle or SAA triangle, we're going to go ahead and stick to these steps. And I'm just going to jump right into the second one. So the easiest thing to solve for in these types of problems is actually this third angle over here. How do we solve for the third angle? Well, we're actually just going to go ahead and use a familiar equation that we've seen before, which is the fact that all of the angles in a triangle have to add up to 180 degrees. This is sometimes referred to as the angle sum formula. Basically, since we know these two, we can always solve for the third one, which is by subtracting this from 180. So here's what I'm going to do. I'm going to set that A + B + C is equal to 180 degrees. And so, therefore, I can find B just by subtracting the other two angles that I know from 180. So this is going to be 180 - 70 - 30, which is really just -100. So therefore, B is equal to this is going to be 80 degrees. A lot of times, you'll just be able to use really quick mental math to figure this out, but you could sort of solve this out if the numbers aren't so nice. We just figured out one of our missing variables is 80 degrees. Perfect. Great. So now that we have all of the angles solved, how do we solve for the two missing sides, a and b? Well, we're just going to go ahead and jump to our third step over here, which is we're going to use the law of sines. That's how we solve for missing sides in a non-right triangle. Alright? So we're just going to have to do this twice. And, really, in this third step, it actually doesn't matter which one you go ahead and solve for first. It doesn't matter. You could do a or b. Alright? So remember how the law of sines works? We're going to have to go ahead and set up our sine A over a = sine B over b = sine C over c. Remember, I'm looking for a in this specific example here. So I'm going to have to sort of pick two out of these three ratios in which I can figure out three out of four variables. Alright? If I try to look at B, what happens is I know the angle, but I don't know the side. I know the angle A, but I don't know the side. So I can't pick these two because that's only two out of four. So instead, I'm going to ignore this one, and I'm going to pick these two over here because I know everything about c. I know the angle and the side. Okay? So we've actually seen this exact sort of setup of variables before, but, really, what happens is