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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 say I have a line here and let's say that I have another line over there and let's call this this point a let's call this point B Point C let's call this D and let's call this right over II and so I'm just going to pick an arbitrary angle over here let's say angle C B what is this CBE look like an F this is angle CBE what I 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 prove that vertical angles are always going to be equal because this is just a generalizable case right over here so what I want to prove what I want to prove here is angle C V E is equal to I could say the measure of angle CBE you'll see it different ways actually this time let me write it without measure so that you get used to the different notations I'll just say prove angle CBE is equal to angle DBA is equal to angle 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 are supplementary they are adjacent angles and their outer sides of both angles form a straight angle over here so we know that angle C be e and angle so this is CBE and angle DBC and angle DBC are supplementary I'll just write sup for that they're supplementary which means that angle CBE plus angle DBC is equal to 180 degrees fair enough we also know so let me say this is CB 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 D ba and angle DBC 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 R is going to be equal to 180 degrees now from this from this top one this top statement over here we can subtract angle DBC from both sides and we get angle C be e 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 I 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 D BA 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 C B is equal to angle D ba 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'll see it with a double line like that that you could say that that is going to be the same as whatever this angle right over here is you'll see it written like that sometimes I like to use colors but not all books have the luxury of colors or sometimes you'll even see it written like this to show that they're the same angle this angle and this angle and show that these are different sometimes they'll say that they're the same in this way this verdict this angle is equal to this vertical angle is equal to it's it's vertical angle right over here and this angle is equal to this angle that's opposite the intersection right over here what we've proved it for 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's equal to the angle that is vertical to it