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- [Voiceover] We're asked to order the side lengths of the triangle from shortest to longest. And we have the three sides here, and we could use this little tool to order them in some way. If we look at the triangle, we've been given the interior angles of the triangle, and they haven't told us the actual side lengths. So, how are we supposed to actually order them from shortest to longest? Well, the realization that you need to make here is that the order of the lengths of the sides of a triangle are related to the order of the measures of angles that open up onto those sides. What do I mean by that? Well, let's think about these three angles right over here. 57 degrees, that is the smallest of these three, and so the side that this angle opens up to, or you can think of it as the opposite side, is going to be the shortest side of the triangle. So, b is going to be the shortest side. So, the next largest angle is 58 degrees, and so a is going to be the middle side. It's not going to be the longest nor the shortest. So, a. Then 65 degrees, that opens up onto side c, or the opposite side of that angle is c. So, c is going to be the longest side. To get an intuition for why that is, imagine a world where the 65 degree angle, if we were to make it bigger. If we were to make the 65 degree angle bigger, maybe by moving this point out and that point out, what would happen? Well, side c would get bigger, and because the angles of a triangle have to add up to 180 degrees, if this one's getting bigger, these will have to get smaller. Likewise, if I were to take angle... let's say, if I were to take this 58 degree angle, and if I were to make it smaller, what's going to happen? Well, side a is going to get smaller. So, the general principle, I'm not giving you any formal proof here, but the intuition is, is that the order of the angles will tell you what the order of the sides are going to be. So, the smallest side is going to be opposite the smallest angle. The largest side is going to be opposite the largest angle. We can check our answer, make sure we got it right. Now, let's do one that goes the other way around. Here, we want to order the angles of the triangle from smallest to largest, and we're given the sides. Well, same, exact idea. The smallest angle is going to be opposite the smallest side or the shortest side. Well, the shortest side is this side of length 7.2. The angle that opens up onto it is angle a. So, that's going to be the smallest angle. Then, the next smallest side is the side of length 7.3, and the angle that opens up to it is angle b right over here. So, angle b. Then angle c opens up onto the largest side. So, it's going to be the largest angle. So, we're done. We've ordered the angles of the triangle from smallest to largest.