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CCSS.Math:

we're asked to order the side lengths of the triangle from shortest to longest and we have the three sides here and we can use this little tool to order them in some way and if we look at the triangle we've been given the interior angles of the triangle and but 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 the the order of the lengths of the sides of the 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 shorter 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 or nor the shortest so a and then 65 degrees that opens up on to side C or the opposite side of that angle is C and so C is going to be the longest side and 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 the angles of a triangle have to add up to 180 degrees if this one's getting bigger these are going to 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 and so the general principle I'm not giving you any formal proof here but the intuition is is that the order of the angles help 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 could check our answer make sure we got it right now let's let's do a 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 sum Ollis tangle 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 on to 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 has its it's angle B right over here so angle B and then angle C opens up onto the large side so it's going to be the long largest angle and so we're done this is we've ordered the angles of the triangle from smallest to largest