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CCSS.Math:

let's say we have two lines over here let's call this line right over here line a B so a and B boat sit on this line and let's say we have this other line over here and we'll call this line C D so it goes through Point C and it goes through point D and it just keeps on going forever and let's say that these lines both sit on the same plane and in this case the plane is our screen or this little piece of paper that we're looking at right over here and they never intersect they never intersect so they're on the same plane but they never intersect each other they never intersect if those two things are true then and where they're not the same line they never intersect and they can be on the same plane then we say that these lines are parallel we say that they are parallel they're moving in the same general direction it is in fact the exact same general direction if we were looking at it from an algebraic point of view we would say that they have the same slope but they have different y-intercepts they involve different points if we drew our coordinate axes here they would intersect that at a different point but they would have the same exact slope and what I want to do is think about how angles relate to parallel lines so right over here we have these two parallel lines we can say that a B line ay B is parallel is parallel to line C D is parallel to line C D sometimes you'll see it specified on G on geometric drawings like this they'll put a little arrow here to show that these two lines are parallel and if you've already used the single arrow they might put a double arrow to show that this line is parallel to that line right over there now with that out of the way what I want to do is draw a line that intersects both of these parallel lines so here's a line that intersects both of them let me draw it a little bit neater than that so let me draw that line right over there and I'll just call that I'll just call that line well actually I'll do some points over here well I'll just call that line L and this line that intersects both of these parallel lines we call that a transversal this is a transversal line it is transversing both of these parallel lines this is a transversal and what I want to think about out is the angles that are formed and how they relate to each other the angles that are formed at the intersection between this transversal line and the two parallel lines so we could first of all start off with this angle right over here that angle right over there and we could call that angle well if we made if we made some labels here that would be d this point and then something else but I'll just call it this angle right over here we know that that's going to be equal to its vertical angles so that this angle is vertical with that one so it's going to be equal to that angle right over there we also know that this angle right over here is going to be equal to the angle that is that it's vertical angle or the vet are the angle that is opposite the intersection so it's going to be equal that and sometimes you'll see it specified like this we will see a double angle mark like that or sometimes you'll see someone write this to show that these two are equal and these two are equal right over here now the other thing we know is we can do the exact same exercise up here that these two are going to be equal to each other and these two are going to be equal to each other they're all vertical angles what's interesting here is thinking about the relationship is thinking about the relationship between this angle right over here the relationship between that angle right over there and this angle right up over here and if you just look at it it is actually obvious what that relationship is that they are going to be the same exact angle that if you put a protractor here and measured it you would get the exact same measure up here and if I draw it drew parallel lines maybe I'll draw it straight left and right and it might be a little bit more obvious that if I draw it so if I assume that these two lines are parallel and I have a transversal here what I'm saying is that this angle is going to be the exact same measure as that angle there and to visualize that just imagine tilting this line and as you take different so it looks like it's the case over there if you take the line like this and you look at it over here it's clear that this is equal to this and there's actually no proof for this this is one of those things that are that a mathematician would say is intuitively obvious that if you look at it as you tilt this line you would say that they're going that these angles are the same or think about putting a protractor here to actually measure these angles if you put a protractor here this you would have one side of the angle at the zero degree and the other side would would specify that point and if you put the protractor over here you put the protractor over here the exact same thing would happen one side would be on this parallel line and the other side would point at the exact same point so given that we know that not only not only is this side equivalent to this side it is also equivalent to this side over here and that tells us that that's also equivalent to that side over there so all of these things in green are equivalent and by the same exact argument this side right over here is going or this angle is going to have the same measure as this angle and that's going to be the same as this angle because they are opposite now or their vertical angles now the important thing to realize is just what we've deduced here the vertical angles are equal and the corresponding angles at the same points of intersection are also equal and so that's a new word that I'm introducing right over here this angle and this angle are corresponding they represent kind of the top right corner in this example of where we intersected here they represent the eye still against the top or the top right corner of the intersection this would be the top left corner top left corner they're always going to be equal corresponding angles and once again it comes really just it's a I guess for lack of a better word it is a bit obvious now on top of that there are other words that people will see we've essentially just proven that not only is this angle equivalent to this angle but it's also equivalent to this angle right over here and these two angles maybe if I call this let me label them so that we can we can make some headway here so I'm going to use lowercase letters for the angles themselves so let's call this lowercase a lowercase B lowercase e so lowercase e for the angle lowercase D and then let me call this e F G H so we know from vertical angles that B is equal to C but we also know that B is equal to F because they are corresponding angles it's equal to F and that F is equal to G so vertical angles are equivalent corresponding angles are equivalent and so we also know obviously that B is equal to G and so we say that alternate interior angles are equivalent so you see that they're kind of on the interior of the interest there between the two lines but they're on all opposite sides of the transversal now you don't have to know that that fancy word alternate interior angles you really just have to deduce this what we just saw over here know that vertical angles are going to be equal and corresponding angles are going to be equal and you see it with the other ones too we know that a is going to be equal to D which is going to be equal to H which is going to be equal which is going to be equal to E