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Proving that vertical angles are equal. Created by Sal Khan.
Video transcript
What I want to do in this video is prove to ourselves that vertical angles really are equal to each other, their measures are really equal to each other. So let's have a line here and let's say that I have another line over there, and let's call this point A, let's call this point B, point C, let's call this D, and let's call this right over there E. And so I'm just going to pick an arbitrary angle over here, let's say angle CB --what is this, this looks like an F-- angle CBE. What I want to do is if I can prove that angle CBE is always going to be equal to its vertical angle --so, angle DBA-- then I'd prove that vertical angles are always going to be equal, because this is just a generalilzable case right over here. So what I want to prove here is angle CBE is equal to, I could say the measure of angle CBE --you will see it in different ways-- actually this time let me write it without measure so that you get used to the different notations. I will just say prove angle CBE is equal to angle DBA. Is equal to angle DBA. So the first thing we know...the first thing we know so what do we know? We know that angle CBE, and we know that angle DBC are supplementary they are adjacent angles and their outer sides, both angles, form a straight angle over here. So we know that angle CBE and angle --so this is CBE-- and angle DBC are supplementary. I will just write "sup" for that. They are supplementary. Which means that angle CBE plus angle DBC is equal to 180 degrees. Fair enough. We also know --so let me see this is CBE, this is what we care about and we want to prove that this is equal to that-- we also know that angle DBA --we know that this is DBA right over here-- we also know that angle DBA and angle DBC are supplementary this angle and this angle are supplementary, their outer sides form a straight angle, they are adjacent so they are supplementary which tells us that angle DBA, this angle right over here, plus angle DBC, this angle over here, is going to be equal to 180 degrees. Now, from this top one, this top statement over here, we can subtract angle DBC from both sides and we get angle CBE is equal to 180 degrees minus angle DBC that's this information right over here, I just put the angle DBC on the right side or subtracted it from both sides of the equation and this right over here, if I do the exact same thing, subtract angle DBC from both sides of the equation, I get angle DBA is equal to 180 degrees --let me scroll over to the right a little bit-- is equal to 180 degrees minus angle DBC. So clearly, angle CBE is equal to 180 degrees minus angle DBC angle DBA is equal to 180 degrees minus angle DBC so they are equal to each other! They are both equal to the same thing so we get, which is what we wanted to get, angle CBE is equal to angle DBA. Angle CBE, which is this angle right over here, is equal to angle DBA and sometimes you might see that shown like this; so angle CBE, that's its measure, and you would say that this measure right over here is the exact same amount. And we have other vertical angles whatever this measure is, and sometimes you will see it with a double line like that, that you can say that THAT is going to be the same as whatever this angle right over here is. You will see it written like that sometimes, I like to use colors but not all books have the luxury of colors, or sometimes you will even see it written like this to show that they are the same angle; this angle and this angle --to show that these are different-- sometimes they will say that they are the same in this way. This angle is equal to this vertical angle, is equal to its vertical angle right over here and that this angle is equal to this angle that is opposite the intersection right over here. What we have proved is the general case because all I did here is I just did two general intersecting lines I picked a random angle, and then I proved that it is equal to the angle that is vertical to it.