Proof: Base angles of isosceles trapezoids

0:00- [Instructor] You might already know from your previous geometry knowledge that if we had an isosceles triangle, so this is a triangle where two of the sides have the same length, that these two base angles are going to have the same measure. Now, what I have drawn or this picture right over here, this is of an isosceles trapezoid where these two legs have the same length.

0:25And the question that we're going to try to address is: Can we make the same conclusion that these base angles, are they going to have the same measure? And if you get inspired either at the beginning or at any time this video to continue the proof yourself, feel free to do so.

0:43Well, to prove that what looks to be the case actually is the case, I'm actually going to construct a triangle here. And the general idea is if I can prove that this is, if this is an isosceles triangle, or if I can maybe use some similarity with an isosceles triangle, then maybe I can prove that these two angles are going to be the same for an isosceles trapezoid.

1:09So let me label this top point. Let's see, I have A, B, C, D, so let's call this one E right over here. Now the first thing I wanna show is that triangle ECD is similar to triangle EBA. E, B, A. Now, how can I do that? Well, one way is to show that at least two of the angles, if two of the angles, two corresponding angles are congruent, then the third one is going to be, and we could show that.

1:39For example, we know that CD and BA are parallel, and so CB can be viewed as a transversal. And so you could say that we have that angle B here or angle CBA and angle ECD are corresponding angles, so they're going to have the same angle measure. Well, we could also do that over here on this side.

2:03Angle A or angle EDB... Or, sorry, angle EAB and angle EDC would also have the same measure by that same argument. And so we already have two angles common between these two triangles, the smaller triangle and this larger one. And of course the third one is shared by both of them; that's this angle right over here.

2:26So because they have all the angles in common, we could say by angle-angle similarity, this is going to be true. Now that is helpful because if two triangles are similar, the ratio between corresponding sides is going to be the same. Now, why is that useful? Well, let's think about it a little bit.

2:50That means that EC, the ratio of EC to EB, to EB, and actually let me rewrite EB as EC plus CB, EC plus CB. That ratio is gonna be the same as the ratio of ED, ED, to this entire side, which we could say is EA, or we could say it is ED ED plus DA. So let's do a little bit of algebraic manipulation here.

3:28If we essentially cross multiply or we multiply both sides by ED plus DA and EC plus CB, we would get ED, so this times this, ED times EC plus CB is equal to this times that is equal to EC times ED; I don't wanna write over that; plus DA. Now we can distribute things.

3:57So we would get; I'm gonna write a little bit small so I have space; ED times EC plus ED times CB; I just distributed this; is going to be equal to EC times E times ED plus EC times DA. Let's see if we can simplify this. We have ED times EC twice, so we can subtract that from both sides of this equation.

4:26And then we're left with ED times CB and we have EC times DA. Now we know that CB and DA are congruent or they have the same length. So we could replace, let's say DA with CB, and then we have ED times CB. Let me rewrite this. ED times CB is equal to EC times CB.

4:54Well, if we divide both sides by the length of CB, we see that ED is equal to EC. ED is equal to EC, so this side has the same length as that side. Well, why is that useful? Well, we immediately know that then triangle ECD is isosceles, and if it's isosceles, then we know its base angles are going to be congruent.

5:21So one way to think about it is this is going to have the same measure or the base angles have the same measure, and by corresponding angles, this is going to have the same measure. You could also make an argument that the big triangle now is definitely an isosceles triangle and therefore the two base angles are going to be the same.

5:39So we've just proven that if you start, or we've just proven that an isosceles trapezoid, just like an isosceles triangle, the base angles have the same measure. And actually we could use that information to come up with other interesting things. For example, if we know that the measure here is x degrees, then we know that the measure here is x degrees.

6:02And then because these are parallel lines over here, we know that the measure here is x degrees, and then that is going to be supplementary to this angle right over here. I'll just do a bunch of lines. And so this is going to be 180 minus x degrees. So another way to think about it is if you have an isosceles trapezoid, your diagonal angles here or opposite angles across a diagonal like this, they are going to be supplementary.

6:35So other interesting things we can figure out based on that.