Classifying triangles by angles

Voiceover: We've already seen that we can categorize triangles as being equilateral, isoceles, or scalene based on the lengths of the sides of the triangle. So, if none of the lengths are congruent, if you have something like this, we would consider this scalene. I am assuming that this side is not equal to that side and neither of these are equal to that side. So that would be scalene, and this is all review. If I have at least two of the sides being equal, so let's say that side is the same length as that side right over there, so I'll mark it off as these are the same length, this would be an isosceles triangle. Then if all three of them, all three sides, are congruent, if all three sides are the same length, we would call this equilateral. And in most circles, you could also say this is isosceles because isosceles would be at least two sides being equal. So this one definitely has at least two sides, it has all three. So this is, you could say, equilateral and isosceles. While this one, if we assume this third side is a different length, this would just be isosceles, not equilateral. Now all of that is review. Now, what I want to think about in this video is well what if we're not given the lengths of the sides and what if we're just given a few of the angles. So, for example, let's say we were given a triangle. Let's say that we were given a triangle where we're given a few of the angle measures. So let's say we are told that this angle right over here is 40 degrees and this angle right over here is 50 degrees. Now, based on that, could you somehow classify this as scalene, isosceles, or equilateral? Well, the key here to realize is if you know two interior angles of a triangle you can always figure out the third because the three need to add up to 180 degrees. So if this is 40 and that is 50, these two add up to 90. So to add up to 180 degrees, this one must be a 90-degree angle. We can even mark it as a right angle. So if you have a triangle where all of the interior angles are different, that means that all of the side lengths are going to be different. One way to think about it is if this angle became wider, then this length would have to become wider. If this angle became larger or smaller, then this side is going to have to become larger or smaller. If this angle became larger or smaller, then this side is going to become larger or smaller. So hopefully you appreciate that if you have three different angles you are going to have three different side lengths. So just based on the angles here, that we have three different angles, we can say that this is going to be a scalene triangle. Now we can look at a couple other examples. Let's do an interesting one. Let's say this angle right over here is 70 degrees and let's say this angle over here is 40 degrees. Now, based on the information I have given you, what kind of a triangle is this going to be? Can you even figure it out? Well, we use the same idea. The interior angles need to add up to 180. 70 plus 40 is 110, so you can say 110 plus what is equal to 180 degrees? Well, this is going to have to be 70 degrees. So this angle right over here is 70 degrees. So now we have a scenario where two of the angles have the same measure. One way to think about it, based on whether this angle is large or small, Is going to define the length of that side and this angle right over here, depending on how large or small it is, is going to define the length on this side. So because these two angles are congruent, because they have the same measure, their opposite sides are going to be congruent. So this is going to be the same as that. So just based on the information I had started giving you, because you can show that two angles are going to be the same, you can say that this is going to be an isosceles triangle. Now let's do a third example, and you could probably guess what I am going to do in this third example. Let's say this angle is 60 degrees. What type of a triangle is this going to be? Well, if this is 60 and this is 60, to make them add up to 180, that would have to be 60 degrees as well. If you have all of the angles congruent, that means that all of the sides are congruent and so now you are dealing with an equilateral. Now, as we said before, also you can view this as a subset as isosceles because you have at least two angles and you have two sides being congruent, but here it is all three, so this is an equilateral triangle.