[Music] all right here we go let's get it started we got our first uh unit of geometry tools for geometry so this is all about the foundational stuff we'll need for the rest of the year so it's a bit different than algebra we're going to do a lot of vocab and a lot of shapes and different things here so this is a hugely important uh start to the entire year we're going to start with points lines and planes oh my that's uh dorothy from wizard of oz if you've seen wizard classic movie don't freak out like dorothy did i think hers was lions tigers and bears uh don't freak out we can handle and do it but be ready take some notes uh so we have the vocab for the entire year so let's get it rolling this is how it's gonna look we're gonna do a lot of vocab this section so it'll tone down here in a while but for the first couple seconds we got some vocab to really pound out so i kind of filled in half of it i want you to fill it in with me feel free to pause anytime if you need to go back and recopy these notes so it really syncs and go back and just copy them again that's a great way to learn all right so a point is our first term that we need so we have a term we're going to name it and show a picture of it a point indicates what it actually indicates location so when we're talking about points we're talking about a location sometimes we plot these on a graph and whatnot we're talking about location what's interesting about a point is and jot this down please it has no size so even though we draw it we're just showing the location we're not saying that it's taking up any space we're just saying it's like this picture over here this dot that dot really has no size it's just showing that location on the screen or whatnot and we call it point a so to call it use a capital letter it's point a has no size it just indicates location awesome so that's the point only two more what about a line then what's a line well a line is represented by a straight path that extends in two opposite directions without end and has no thickness a line contains what so that is a lot that is a mouthful there but don't freak out you probably know what a line is i mean there's a picture of a line a line does it extends forever and ever in both directions no worries it goes forever and ever in both directions has no thickness so just like this has no size it has no thickness it's just kind of there a line contains infinite amount of points or infinitely many points is how we usually say that infinitely many chinese my best handwriting points so playing contains so on this i drew two points a and b but really there's an infinite amount of points on this i can draw billions and billions and billions because they have no size so between a and b there's infinite millions and millions of points in there it's just made up of tons and tons of points so this has infinitely infinite means forever so it's infinite many points awesome so how do i name this thing we have two options option ones is you take any two points on the line so if i look at my line here schedule a line i can say it's a b and what do we do to show that's the line we draw a little line on top ooh isn't that cool so you have to actually draw a line above it now i know you're talking about line a b is it cool if you did ba sure i don't care about order i don't i don't care which two points you pick as long as they're on that line it looks like that or we could have used a lower case letter so what about a lowercase letter well they usually do it out here it's this right here and they put at the end without points and it's usually cursive so we can say line in this case this is line l so and that's usually cursive so line l so you have options sometimes this isn't here but your points are usually always going to be there so pick one of those that works out well for you all right let's go do a plane then so a plane is represented by a flat surface that it's surface that extends without end and has no thickness plane contains infinitely many points so it also contains infinitely many points check this out over here i drew a picture for you that's how we normally draw it's just like a line but instead of going two directions it goes in all directions so that's why it's the shaded gray region here this is a parallelogram that i drew but you can draw a rectangle whatever you want showing that it's this plane it's like a table top or the floor or the ceiling or a wall it's that flat surface uh so that's what we're talking about a plane it goes on even though it's not drawn forever and ever and ever so if you look up at the ceiling just imagine that goes forever and ever and ever in every direction that's a plane so how do we name it it's any three points on the plane so if you look at this i've got four points r u s and t rust we've got some four points in there so if i want to name it i pick any three of those points i don't care i'm going to say it's plain r u t rut it doesn't matter pick s t u pick u t r i don't care as long as they're points or sometimes they'll do this for you they will draw a little capital letter down here in the corner for you just like the little cursive l there it's usually some kind of old script we could say this is plane b so that's another way to do it if they put so there's no point by it it's referring to the entire plane and everything on it fantastic so there's our three keys to this section the tools the foundations for really for the rest of the year all right just a quick picture it's not in your notes just check it out we've got points lines and planes look at this crazy so we're going to get wild three dimensional cool stuff going on where are some points here's some points here's a b and c so there are points on this we've got points a b c and d can you see them on there here's a point boom here's a point b here's point c here's point d so the points are definitely there are there lines on this thing sure i've got some lines look straight up and down here i've got line a b is a line anywhere else i've got cd so i've got lines on this crazy picture do i have planes sure i got three planes can you see them we've got the blue plane going this way we've got plane q going the the blue and it goes forever just because it is i drew a line here but it goes forever this direction it's being cut by what plane h and also by plane what plane m so we've got to get three dimensional we got to start doing visual it can be an adjustment it can be new don't freak out like dorothy over here uh you can handle it it just takes a little practice all right let's get back to the notes and we're gonna do more vocab awesome so let's check this out collinear points are what so here's a picture of it check out these points d e and g are co-linear are points that lie on the same line and my handwriting is not the best i apologize so points that lie on the same line so that's all they are so d e and g are co-linear points they lie on the same line fantastic so that is a great vocab that's going to keep coming up and coming up and coming up how about coplanar their points and lines here's a picture of it here's a crazy picture i've got my blue plane here and i've got this lines crossing and points these are points that lie on what on the same plane so coplanar means uh they're all in the same plane so c a d b are all coplanar points i've got these two lines a b and a d or coplanar lines they're all on the same plane fantastic so that's collinear coplanar and then our last vocab that kind of deals with this is space what is space space is the final frontier i'm just kidding don't write that down space is not the final frontier well maybe if you're tricky it is uh but in geometry let's get rid of that it's really the set of all possible points so if you think of space is actually great think about it think of the stars floating around everything out there it's three dimensional it's a set of all possible points in the world in the universe all floating around out there so space is important we're always talking about things in space all right fantastic moving on here's a picture this one is in your notes we're going to answer a few questions here and see make sure we're good to go so i've got this nice picture drawn for you let's see if we can do it what are two other ways to name the line qt so here's qt remember all i need is two points on that so n is actually on that line see it down here so i could call it nt or what's another way i could call it tq or i could call it really i should probably look at qn because tq is what it's actually the same q t or q t q t or d q are the same so really that's the same as this they're the same thing this would be the two other ways n t or q n so i picked two different points on there named it that way good to go what are two other ways to name plane p so here's plane p it's labeled down here pick any points you got r q s and v to choose from two other ways i could say well i'm gonna call it plane rvq i could call it what pick some other points on the plane i'm safe as long as i stay away from n and t can you see how t sticking out the paper and ends like below it they're not on that plane so we're really looking at i don't know qvs is that a pharmacy or something i don't know and how about this can we name three co-linear points sure looking for three points online i have two options here i can say rqs those are collinea they're on this line here or check out this line t q n so you have options either one is right we'll count them good to go t q n and the last question here name a point not coplanar with r s and v so r s v what's not gonna play or cusco planer t is not coplanar or i could say n is not coplanar they're not on there uh not on the same plane fantastic moving on more vocab are you serious yes more vocab now at this point you may be getting like man i'm tired of vocab we had options you know i thought about you know setting this to a tune of a christmas carol maybe 12 days of christmas and singing it check this out this is one option on the third day of math class my teacher talked to me rey goes one way segment is a part and a line goes forever two ways on the fourth day of math class my teacher taught to me point is a dot ray goes one way segment is a part and a line goes forever two ways okay i'm not really down with doing it like that mr sullivan was really for that and maybe he'll sing it a little bit for you if you want uh he'll sing it for you i'm going to start typing at this point because my handwriting's so bad uh here we go let's do it last i think we got three here and then we're good on vocab a segment is a part of a line that consists of what it consists of two endpoints and all of the points in between them i'm really fast now there it is so segment is a part of the line so it's part of a line that consists of two endpoints and all of the points in between them so what does that look like well let's draw a picture over here i drew one there's a picture for us this is part of a line i just cut it off boom boom i cut off part of the line and it's got two end points what are the end points well the end points are a and b they end the little part so this is a segment so make sure you know boom that's a segment how do i name it i name it by its endpoint so i'm going to name it a b but remember how we put that little line above it i'm not going to put a line above i'm actually going to draw the line segment above it so no arrows it looks like i almost did an arrow there i want to really stress no arrow and i want so it's a b line on top so that is a line segment you could also if you want do ba order is not important here i'm just referring to the two endpoints so i'm good to go fantastic how about this what about a ray what in the world if that's a segment array is going to be maybe you want to see the picture first i like to look at the pictures that's array so that's what we're talking about right there it looks a like a line it looks like a line segment but it's a different it's a ray so it consists of what it has one endpoint and all of the points all of the points in i'm sorry all the points on the line in one direction so it's an end point it consists of an endpoint and all the points on the line in one direction so when i look at this what is my endpoint here my endpoint is a and what's going it's going in one direction through point b it's all of those points in that direction so how do we name it now order is important you name array by its endpoint and another point in the ray but the end point must come first so there's only one way to name this right here you must start with the a because the endpoint so when i look at this i know a is the end point it goes through b and i draw a little ray on top boom it's just half it's just an arrow pointing so now order is important that is the only way to name that ba is not correct in that case fantastic so that's a ray so what are opposite rays then well opposite rays are two rays that do what they actually share an endpoint and why do they share an endpoint because they form a line so they share an end point and form a line well that's cool so what does that look like what does two opposite rays look like it looks like this boom there's a picture so you when you look at it you can call it a line sure it is a line they form a line but we can also say oh yeah they're actually opposite rays they're pointing in opposite directions so it's all the points in both directions on the line in the opposite direction so how do i name them well you gotta name both ways so where is the endpoint they're sharing they're actually sharing c so when i name this i can say well it's ca there's the ray and what is its partner its partner would be this way cb so they're sharing c that's why i put it first that's the end point one where a is going this way through ca boom the other ray is going cb this way and what do they make they make a line so that's how you name them you gotta list both of them so really if you see a line you can break it into two opposite rays fantastic not too bad all right let's do some problems here so we've got this going on i've got this nice picture can you name the three line segments here so there are three line segments sure here's one right here boom d e is a line segment so i know it's a line or it's two opposite rays one segment's d e another one is e f is there any more sure the whole thing from here to here is also another line segment so we can say df so those are three line segments on this picture how about four rays can i do it well here's e i can say there's a ray going this way so i'm looking at e d is array i can also do what i could say f goes through what f e is that crazy this is also array now i'm getting confused i got to erase some of that any more rays on here sure what about this way i could say df and did i miss any i'm missing one do you see it what about if i started here at e i could say ef so there are four different rays in there be careful it is very important how you name them with the end point and what not all right which rays are opposite rays so if i look at this do we see opposite rays sure just like last time e is the middle one i've got a ray going this way if you do array going this way it forms the line so i'm looking at ed and who's this partner ef these guys are opposite rays fantastic and the final question oh that is the final question awesome very good moving on all right so here we go we're going to postulates and then we're done so no more vocab but we're doing postulates postulates are the key to geometry we can't prove these things are true they just are true you have to believe me just like when your kids told you or your parents told you about santa claus trust me on this they're true uh that's you know the 12 days of christmas we got sam got all going on here postulates we're going to say they are true i can't prove it to you but that's why geometry works we have to assume that they are true and then it makes everything good to go so how about this it says through any two points is exactly what so matt let's draw two points boom if i draw any two points anywhere in space a and b what is there between them aha exactly one line so any two points you draw there is exactly one line fantastic so that's a postulate i can't prove it it's just true it's just how it is so we may need that postulate coming up later on so jot that down and these are just kind of common sense things like it just makes sense that's how it is how about this if two lines intersect they intersect in exactly what well let's draw two lines here is a line and it intersects this line where do they intersect they intersect in exactly one point fantastic so again no way to prove that's true that's just how we say it they're going to cross right here at one point so these two lines cross at that point all right how about the next one if two distinct planes intersect and they intersect at exactly what now i drew this picture because it's kind of intense see the yellow plane and the blue plane they're crossing where are they crossing at a point s no it's more than that uh because you can see the cross at t they actually cross at this entire line here so this is huge when two planes intersect it's this entire line so they intersect in exactly one line this is the tricky one this one kind of gets people a little bit they intersect in one line so i kind of drew in the picture i want you to be able to see that what is the line in this case in this case they intersect an st so we've got this line where they cross uh huge two planes always intersect in a line two lines intersect in a point fantastic one more postulate for this section we're good to go through three any i'm sorry through any three non-collinear points there is exactly what so if you draw three non-collinear so there's make sure they're not aligned here's a b and c so if i have this so no matter where they are what is it they always have a plane around them so through it three i'm sorry any three non-cleaner points there is exactly one plane so any three points are coplanar non-collinear points arc through one plane all right in case you didn't know mr kelly is a huge justin beer fan this is one of the rooms in his house wow pretty intense good work mr kelly i kind of want to look at these different shapes from geometry we've been talking about so if we imagine this room this wall over here the wall of the bieber room is actually a plane so or look around your room you're in right now there is a plane so here is one plane then look at the ceiling the ceiling is actually another plane let me slide in a ceiling here and now we can kind of see this look at this where did the where does the ceiling intersect the wall well it's the it's the crack there's the corner there well what is that shape now you can really see it it is that line there so boom two planes do intersect the line that's a very visual way to see it uh you could also see it down here where it intersects the other wall we've got other things going on so we've got these planes uh these two lines are going through here where do these two lines intersect they intersect at this point so in the very corner of the room where i look up in the corner of the room there's a point where two lines are intersecting two cracks actually it's three cracks isn't it you've got this wall over here coming in so we've got three lines intersecting we've got three planes intersecting but planes intersect at these lines so that's the big difference i want to make sure you're good on fantastic all right two more questions on this we've got this thing this math cube here so i drew a cube i want to know what is the intersection of plane cue well if you have a hard time visualizing this shade it in here is cue e right here yeah there is plane c u e now find plane ebt where's ebt it's actually the right side of this cube so i'm looking at these two things where in the world do these two planes cross aha they're going to intersect right here on eb but is it just the segment eb no even though my picture when i'm talking planes it looks like a segment but it's actually a line this actually goes on and on forever and ever and ever even though it's not drawn that's actually a line there oh it's the line e b fantastic then i had this one r point c e a and t coplanar so check them out we've got c e whoa eight here up here are they co-planar in this case they are man it's crazy but you could even though it's not drawn i could draw a plane that's like the diagonal of the cube it is there well i'm having a hard time controlling it but it is there check that out maybe that helps visualize it so from c to a boom to t back down here maybe i'll erase some of these the other ones to kind of help out there it is it's going through it so they actually are coplanar so we've got these weird diagonals going on but yes they are coplanar pretty cool fantastic and we are done with the first section good work uh i know the video is kind of long took a while but the practice goes much quicker here's mr kelly back from his justin