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But also encapsulate things like patterns and three-dimensional shapes. So it's almost everything that we see, all of the visually mathematical things that we understand can in some way be categorized in geometry. Now with that out of the way, let's just start from the basics, a basic starting point from geometry. And then we can just grow from there. So if we just start at a dot, that dot right over there, it's just a point. It's just that little point on that screen right over there. We'd literally call that a point.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And then we can just grow from there. So if we just start at a dot, that dot right over there, it's just a point. It's just that little point on that screen right over there. We'd literally call that a point. And I'll call that a definition. And the fun thing about mathematics is that you can make definitions. We could have called this an armadillo.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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We'd literally call that a point. And I'll call that a definition. And the fun thing about mathematics is that you can make definitions. We could have called this an armadillo. But we decide to call this a point, which I think makes sense because it's what we would call it in just everyday language as well. That is a point. Now what's interesting about a point is that it is just a position, that you can't move on a point.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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We could have called this an armadillo. But we decide to call this a point, which I think makes sense because it's what we would call it in just everyday language as well. That is a point. Now what's interesting about a point is that it is just a position, that you can't move on a point. If you moved, if you were at this point, and if you moved in any direction at all, you would no longer be at that point. So you cannot move on a point. Now there are differences between points.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Now what's interesting about a point is that it is just a position, that you can't move on a point. If you moved, if you were at this point, and if you moved in any direction at all, you would no longer be at that point. So you cannot move on a point. Now there are differences between points. For example, that's one point there. Maybe I have another point over here. And then I have another point over here.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Now there are differences between points. For example, that's one point there. Maybe I have another point over here. And then I have another point over here. And then another point over there. And you want to be able to refer to the different points. And not everyone has the luxury of a nice colored pen like I do.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And then I have another point over here. And then another point over there. And you want to be able to refer to the different points. And not everyone has the luxury of a nice colored pen like I do. Otherwise, they could refer to the green point, or the blue point, or the pink point. And so in geometry, to refer to points, we tend to give them labels. And the labels tend to have letters.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And not everyone has the luxury of a nice colored pen like I do. Otherwise, they could refer to the green point, or the blue point, or the pink point. And so in geometry, to refer to points, we tend to give them labels. And the labels tend to have letters. So for example, this could be point A. This could be point B. This would be point C. And this right over here could be point D. So if someone says, hey, circle point C, you know which one to circle.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And the labels tend to have letters. So for example, this could be point A. This could be point B. This would be point C. And this right over here could be point D. So if someone says, hey, circle point C, you know which one to circle. You know that you would have to circle that point right over there. Well, that so far, it's kind of interesting. You have these things called points.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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This would be point C. And this right over here could be point D. So if someone says, hey, circle point C, you know which one to circle. You know that you would have to circle that point right over there. Well, that so far, it's kind of interesting. You have these things called points. You really can't move around on a point. All they do is specify a position. What if we want to move around a little bit more?
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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You have these things called points. You really can't move around on a point. All they do is specify a position. What if we want to move around a little bit more? What if we want to get from one point to another? So what if we started at one point, and we wanted all of the points, including that point, that connect that point and another point? So all of these points right over here.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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What if we want to move around a little bit more? What if we want to get from one point to another? So what if we started at one point, and we wanted all of the points, including that point, that connect that point and another point? So all of these points right over here. So what would we call this thing? All of the points that connect A and B along a straight, and I'll use everyday language here, along kind of a straight line like this. Well, we'll call this a line segment.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So all of these points right over here. So what would we call this thing? All of the points that connect A and B along a straight, and I'll use everyday language here, along kind of a straight line like this. Well, we'll call this a line segment. In everyday language, you might call it a line, but we'll call it a line segment. Because we'll see when we talk in mathematical terms, a line means something slightly different. So this is a line segment.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Well, we'll call this a line segment. In everyday language, you might call it a line, but we'll call it a line segment. Because we'll see when we talk in mathematical terms, a line means something slightly different. So this is a line segment. And if we were to connect D and C, this would also be another line segment. And once again, because we always don't have the luxury of colors, this one is clearly the orange line segment. This is clearly the yellow line segment.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So this is a line segment. And if we were to connect D and C, this would also be another line segment. And once again, because we always don't have the luxury of colors, this one is clearly the orange line segment. This is clearly the yellow line segment. We want to have labels for these line segments. And the best way to label the line segments are with its end points. And that's another word here.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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This is clearly the yellow line segment. We want to have labels for these line segments. And the best way to label the line segments are with its end points. And that's another word here. So a point is just literally A or B. But A and B are also the end points of these line segments, because it starts and ends at A and B. So let me write this.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And that's another word here. So a point is just literally A or B. But A and B are also the end points of these line segments, because it starts and ends at A and B. So let me write this. A and B are end points. Another definition right over here. Once again, we could have called them aardvarks or end armadillos.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So let me write this. A and B are end points. Another definition right over here. Once again, we could have called them aardvarks or end armadillos. But we as mathematicians decide to call them end points, because that seems to be a good name for it. And once again, we need a way to label these line segments that have the end points. And what's a better way to label a line segment than with its actual end points?
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Once again, we could have called them aardvarks or end armadillos. But we as mathematicians decide to call them end points, because that seems to be a good name for it. And once again, we need a way to label these line segments that have the end points. And what's a better way to label a line segment than with its actual end points? So we would refer to this line segment over here. We would put its end points there. And to show that it's a line segment, we would draw a line over it, just like that.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And what's a better way to label a line segment than with its actual end points? So we would refer to this line segment over here. We would put its end points there. And to show that it's a line segment, we would draw a line over it, just like that. This line segment down here, we would write it like this. And we could have just as easily written it like this. CD with a line over it would have referred to that same line segment.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And to show that it's a line segment, we would draw a line over it, just like that. This line segment down here, we would write it like this. And we could have just as easily written it like this. CD with a line over it would have referred to that same line segment. BA with a line over it would refer to that same line segment. Now you might be saying, well, I'm not satisfied just traveling in between A and B. And this is actually another interesting idea.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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CD with a line over it would have referred to that same line segment. BA with a line over it would refer to that same line segment. Now you might be saying, well, I'm not satisfied just traveling in between A and B. And this is actually another interesting idea. When you were just on A, when you were just on a point, and you couldn't travel at all, you couldn't travel in any direction while staying on that point, that means you have zero options to travel in. You can't go up or down, left or right, in or out of the page, and still be on that point. And so that's why we say a point has zero dimensions.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And this is actually another interesting idea. When you were just on A, when you were just on a point, and you couldn't travel at all, you couldn't travel in any direction while staying on that point, that means you have zero options to travel in. You can't go up or down, left or right, in or out of the page, and still be on that point. And so that's why we say a point has zero dimensions. Zero dimensions. Now all of a sudden, we have this thing, this line segment here. And this line segment, we can at least go to the left and the right along this line segment.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And so that's why we say a point has zero dimensions. Zero dimensions. Now all of a sudden, we have this thing, this line segment here. And this line segment, we can at least go to the left and the right along this line segment. We can go towards A or towards B. So we can go back or forward in one dimension. So the line segment is a one dimensional idea, almost, or one dimensional object, although these are more kind of abstract ideas.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And this line segment, we can at least go to the left and the right along this line segment. We can go towards A or towards B. So we can go back or forward in one dimension. So the line segment is a one dimensional idea, almost, or one dimensional object, although these are more kind of abstract ideas. There is no such thing as a perfect line segment. Because everything, a line segment, you can't move up or down on this line segment while being on it. While in reality, anything that we think is a line segment, even a stick of some type, a very straight stick, or a string that is taut, that still will have some width.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So the line segment is a one dimensional idea, almost, or one dimensional object, although these are more kind of abstract ideas. There is no such thing as a perfect line segment. Because everything, a line segment, you can't move up or down on this line segment while being on it. While in reality, anything that we think is a line segment, even a stick of some type, a very straight stick, or a string that is taut, that still will have some width. But the geometrical pure line segment has no width. It only has a length here. So you can only move along the line.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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While in reality, anything that we think is a line segment, even a stick of some type, a very straight stick, or a string that is taut, that still will have some width. But the geometrical pure line segment has no width. It only has a length here. So you can only move along the line. And that's why we say it's one dimensional. A point, you can't move at all. A line segment, you can only move in that back and forth along that same direction.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So you can only move along the line. And that's why we say it's one dimensional. A point, you can't move at all. A line segment, you can only move in that back and forth along that same direction. Now I just hinted that it can actually have a length. How do you refer to that? Well, you refer to that by not writing that line on it.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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A line segment, you can only move in that back and forth along that same direction. Now I just hinted that it can actually have a length. How do you refer to that? Well, you refer to that by not writing that line on it. So if I write AB with a line on top of it like that, that means I'm referring to the actual line segment. If I say that, let me do this in a new color, if I say that AB is equal to 5 units, it might be centimeters or meters or whatever, just the abstract units 5, that means that the distance between A and B is 5, that the length of line segment AB is actually 5. Now let's keep on extending it.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Well, you refer to that by not writing that line on it. So if I write AB with a line on top of it like that, that means I'm referring to the actual line segment. If I say that, let me do this in a new color, if I say that AB is equal to 5 units, it might be centimeters or meters or whatever, just the abstract units 5, that means that the distance between A and B is 5, that the length of line segment AB is actually 5. Now let's keep on extending it. Let's say we want to just keep going in one direction. So let's say that I started A, let me do this in a new color, let's say I started A and I want to go to D, but I want the option of keep on going. So I can't go further in A's direction than A, but I can go further in D's direction.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Now let's keep on extending it. Let's say we want to just keep going in one direction. So let's say that I started A, let me do this in a new color, let's say I started A and I want to go to D, but I want the option of keep on going. So I can't go further in A's direction than A, but I can go further in D's direction. So this idea that I just showed, essentially it's like a line segment, but I can keep on going past this end point, we call this a ray. And the starting point for a ray is called the vertex, not a term that you'll see too often. You'll see vertex later on in other contexts, but it's good to know.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So I can't go further in A's direction than A, but I can go further in D's direction. So this idea that I just showed, essentially it's like a line segment, but I can keep on going past this end point, we call this a ray. And the starting point for a ray is called the vertex, not a term that you'll see too often. You'll see vertex later on in other contexts, but it's good to know. This is the vertex of the ray. It's not the vertex of this line segment, so maybe I shouldn't label it just like that. And what's interesting about a ray, it's once again a one-dimensional figure, but you could keep on going in one of the direct, you can keep on going past one of the end points.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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You'll see vertex later on in other contexts, but it's good to know. This is the vertex of the ray. It's not the vertex of this line segment, so maybe I shouldn't label it just like that. And what's interesting about a ray, it's once again a one-dimensional figure, but you could keep on going in one of the direct, you can keep on going past one of the end points. And the way that we would specify a ray is we would say, we would call it AD, and we would put this little arrow over on top of it to show that it is a ray. And in this case, it matters the order that we put the letters in. If I put DA as a ray, this would mean a different ray.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And what's interesting about a ray, it's once again a one-dimensional figure, but you could keep on going in one of the direct, you can keep on going past one of the end points. And the way that we would specify a ray is we would say, we would call it AD, and we would put this little arrow over on top of it to show that it is a ray. And in this case, it matters the order that we put the letters in. If I put DA as a ray, this would mean a different ray. That would mean that we're starting at D and then we're going past A. So this is not ray DA, this is ray AD. Now the last idea that I'm sure you're thinking about is well, what if I could keep on going in both directions?
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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If I put DA as a ray, this would mean a different ray. That would mean that we're starting at D and then we're going past A. So this is not ray DA, this is ray AD. Now the last idea that I'm sure you're thinking about is well, what if I could keep on going in both directions? So let's say I can keep going in, let me, my diagram is getting messy, so let me introduce some more points. So let's say I have point E, and then I have point F right over here. And let's say that I have this object that goes through both E and F, but just keeps on going in both directions.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Now the last idea that I'm sure you're thinking about is well, what if I could keep on going in both directions? So let's say I can keep going in, let me, my diagram is getting messy, so let me introduce some more points. So let's say I have point E, and then I have point F right over here. And let's say that I have this object that goes through both E and F, but just keeps on going in both directions. This is, when we talk in geometry terms, this is what we call a line. Now notice, a line never ends. You can keep going in either direction.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And let's say that I have this object that goes through both E and F, but just keeps on going in both directions. This is, when we talk in geometry terms, this is what we call a line. Now notice, a line never ends. You can keep going in either direction. A line segment does end, it has end points. A line does not. And actually a line segment can sometimes be called just a segment.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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You can keep going in either direction. A line segment does end, it has end points. A line does not. And actually a line segment can sometimes be called just a segment. And so you would specify line EF, you would specify line EF with these arrows just like that. Now the thing that you're gonna see most typically when we're studying geometry are these right over here. Because we're gonna be concerned with sides of shapes, distances between points, and when you're talking about any of those things, things that have finite length, things that have an actual length, things that don't go off forever in one or two directions, then you're talking about a segment or a line segment.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And actually a line segment can sometimes be called just a segment. And so you would specify line EF, you would specify line EF with these arrows just like that. Now the thing that you're gonna see most typically when we're studying geometry are these right over here. Because we're gonna be concerned with sides of shapes, distances between points, and when you're talking about any of those things, things that have finite length, things that have an actual length, things that don't go off forever in one or two directions, then you're talking about a segment or a line segment. Now, if we go back to a line segment, just to kind of keep talking about new words that you might confront in geometry, if we go back talking about a line, and I was drawing a ray. So let's say I have point X and point Y, and so this is line segment XY, so I could specify, denote it just like that. If I have another point, let's say I have another point right over here, let's call that point Z.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Because we're gonna be concerned with sides of shapes, distances between points, and when you're talking about any of those things, things that have finite length, things that have an actual length, things that don't go off forever in one or two directions, then you're talking about a segment or a line segment. Now, if we go back to a line segment, just to kind of keep talking about new words that you might confront in geometry, if we go back talking about a line, and I was drawing a ray. So let's say I have point X and point Y, and so this is line segment XY, so I could specify, denote it just like that. If I have another point, let's say I have another point right over here, let's call that point Z. And I'll introduce another word. X, Y, and Z are on the same, they all lie on the same line, if you would imagine that a line could keep going on and on forever and ever. So we can say that X, Y, and Z are collinear.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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If I have another point, let's say I have another point right over here, let's call that point Z. And I'll introduce another word. X, Y, and Z are on the same, they all lie on the same line, if you would imagine that a line could keep going on and on forever and ever. So we can say that X, Y, and Z are collinear. So those three points are collinear. They all sit on the same line, and they also all sit on line segment XY. Now let's say we know, we're told, that XZ is equal to ZY, and they are all collinear.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So we can say that X, Y, and Z are collinear. So those three points are collinear. They all sit on the same line, and they also all sit on line segment XY. Now let's say we know, we're told, that XZ is equal to ZY, and they are all collinear. So that means this is telling us that the distance between X and Z is the same as the distance between Z and Y. So sometimes we can mark it like that. This distance is the same as that distance over there.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Now let's say we know, we're told, that XZ is equal to ZY, and they are all collinear. So that means this is telling us that the distance between X and Z is the same as the distance between Z and Y. So sometimes we can mark it like that. This distance is the same as that distance over there. So that tells us that Z is exactly halfway between X and Y. So in this situation, we would call Z the midpoint, the midpoint of line segment XY, because it's exactly halfway between. Now to finish up, we've talked about things that have zero dimensions, points.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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This distance is the same as that distance over there. So that tells us that Z is exactly halfway between X and Y. So in this situation, we would call Z the midpoint, the midpoint of line segment XY, because it's exactly halfway between. Now to finish up, we've talked about things that have zero dimensions, points. We've talked about things that have one dimension, a line, a line segment, or a ray. You might say, well, what has two dimensions? Well, in order to have two dimensions, that means I can go backwards and forwards in two different directions.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Now to finish up, we've talked about things that have zero dimensions, points. We've talked about things that have one dimension, a line, a line segment, or a ray. You might say, well, what has two dimensions? Well, in order to have two dimensions, that means I can go backwards and forwards in two different directions. So this page right here, or this video, or the screen that you're looking at is a two-dimensional object. I can go right, left, that is one dimension, or I can go up, down. And so this surface of the monitor you're looking at is actually two dimensions.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Well, in order to have two dimensions, that means I can go backwards and forwards in two different directions. So this page right here, or this video, or the screen that you're looking at is a two-dimensional object. I can go right, left, that is one dimension, or I can go up, down. And so this surface of the monitor you're looking at is actually two dimensions. Two dimensions. You can go backwards or forwards in two directions. And things that are two dimensions, we call them planar, or we call them planes.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And so this surface of the monitor you're looking at is actually two dimensions. Two dimensions. You can go backwards or forwards in two directions. And things that are two dimensions, we call them planar, or we call them planes. So if you took a piece of paper that extended forever, it just extended in every direction forever, that in the geometrical sense was a plane. The piece of paper itself, the thing that's finite, and you'll never see this talked about in a typical geometry class, but I guess if we were to draw the analogy, you could call a piece of paper maybe a plane segment, because it's a segment of an entire plane. If you had a third dimension, then you're talking about kind of our three-dimensional space and three-dimensional space, not only could you move left or right along the screen or up and down, you could also move in and out of the screen.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And things that are two dimensions, we call them planar, or we call them planes. So if you took a piece of paper that extended forever, it just extended in every direction forever, that in the geometrical sense was a plane. The piece of paper itself, the thing that's finite, and you'll never see this talked about in a typical geometry class, but I guess if we were to draw the analogy, you could call a piece of paper maybe a plane segment, because it's a segment of an entire plane. If you had a third dimension, then you're talking about kind of our three-dimensional space and three-dimensional space, not only could you move left or right along the screen or up and down, you could also move in and out of the screen. You could also have this dimension that I'll try to draw. You could go into the screen or you could go out of the screen like that. And as we go into higher and higher mathematics, although it becomes very hard to visualize, you'll see that we can even start to study things that have more than three dimensions.
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Basic geometry language and labels Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So that's one line, and then let me draw another line that is parallel to that. I'm claiming that these are parallel lines. And now I'm gonna draw some transversals here. So first let me draw a horizontal transversal. So just like that. And then let me do a vertical transversal. So just like that.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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So first let me draw a horizontal transversal. So just like that. And then let me do a vertical transversal. So just like that. And I'm assuming that the green one is horizontal and the blue one is vertical. So we assume that they are perpendicular to each other, that these intersect at right angles. And from this, I'm going to figure out, I'm gonna use some parallel line angle properties to establish that this triangle and this triangle are similar, and then use that to establish that both of these lines, both of these yellow lines have the same slope.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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So just like that. And I'm assuming that the green one is horizontal and the blue one is vertical. So we assume that they are perpendicular to each other, that these intersect at right angles. And from this, I'm going to figure out, I'm gonna use some parallel line angle properties to establish that this triangle and this triangle are similar, and then use that to establish that both of these lines, both of these yellow lines have the same slope. So actually let me label some points here. So let's call that point A, point B, point C, point D, and point E. So let's see. First of all, we know that angle CED is going to be congruent to angle AEB because they're both right angles.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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And from this, I'm going to figure out, I'm gonna use some parallel line angle properties to establish that this triangle and this triangle are similar, and then use that to establish that both of these lines, both of these yellow lines have the same slope. So actually let me label some points here. So let's call that point A, point B, point C, point D, and point E. So let's see. First of all, we know that angle CED is going to be congruent to angle AEB because they're both right angles. So that's a right angle, and then that is a right angle right over there. We also know some things about corresponding angles where a transversal intersects parallel lines. This angle corresponds to this angle if we look at the blue transversal as it intersects those two lines.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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First of all, we know that angle CED is going to be congruent to angle AEB because they're both right angles. So that's a right angle, and then that is a right angle right over there. We also know some things about corresponding angles where a transversal intersects parallel lines. This angle corresponds to this angle if we look at the blue transversal as it intersects those two lines. And so they're going to have the same measure. They're going to be congruent. Now this angle on one side of this point B is going to also be congruent to that because they are vertical angles, and we've seen that multiple times before.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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This angle corresponds to this angle if we look at the blue transversal as it intersects those two lines. And so they're going to have the same measure. They're going to be congruent. Now this angle on one side of this point B is going to also be congruent to that because they are vertical angles, and we've seen that multiple times before. And so we know that this angle, angle ABE, is congruent to angle ECD. Sometimes this is called alternate interior angles of a transversal and parallel lines. Well, if you look at triangle CED and triangle ABE, we see they already have two angles in common.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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Now this angle on one side of this point B is going to also be congruent to that because they are vertical angles, and we've seen that multiple times before. And so we know that this angle, angle ABE, is congruent to angle ECD. Sometimes this is called alternate interior angles of a transversal and parallel lines. Well, if you look at triangle CED and triangle ABE, we see they already have two angles in common. So if they have two angles in common, well, their third angle has to be in common because this third angle's just going to be 180 minus these other two. And so this third angle's just going to be 180 minus the other two. And so just like that, we notice we have all three angles are the same in both of these triangles, or they're not all the same, but all of the corresponding angles, I should say, are the same.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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Well, if you look at triangle CED and triangle ABE, we see they already have two angles in common. So if they have two angles in common, well, their third angle has to be in common because this third angle's just going to be 180 minus these other two. And so this third angle's just going to be 180 minus the other two. And so just like that, we notice we have all three angles are the same in both of these triangles, or they're not all the same, but all of the corresponding angles, I should say, are the same. Angle, this blue angle has the same measure as this blue angle. This magenta angle has the same measure as this magenta angle. And then the other angles are right angles.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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And so just like that, we notice we have all three angles are the same in both of these triangles, or they're not all the same, but all of the corresponding angles, I should say, are the same. Angle, this blue angle has the same measure as this blue angle. This magenta angle has the same measure as this magenta angle. And then the other angles are right angles. These are right triangles here. So we could say triangle AEB, triangle AEB, is similar, similar, is similar to triangle DEC, triangle DEC, by, and we could say by angle, angle, angle. All the corresponding angles are congruent.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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And then the other angles are right angles. These are right triangles here. So we could say triangle AEB, triangle AEB, is similar, similar, is similar to triangle DEC, triangle DEC, by, and we could say by angle, angle, angle. All the corresponding angles are congruent. So we are dealing with similar triangles. And so we know similar triangles, the ratio of corresponding sides are going to be the same. So we could say that the ratio of, let's say the ratio of BE, the ratio of BE, let me write this down.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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All the corresponding angles are congruent. So we are dealing with similar triangles. And so we know similar triangles, the ratio of corresponding sides are going to be the same. So we could say that the ratio of, let's say the ratio of BE, the ratio of BE, let me write this down. This is this side right over here. The ratio of BE to AE, to AE, to AE, is going to be equal to, so that side over that side, well what is the corresponding side? The corresponding side to BE is side CE.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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So we could say that the ratio of, let's say the ratio of BE, the ratio of BE, let me write this down. This is this side right over here. The ratio of BE to AE, to AE, to AE, is going to be equal to, so that side over that side, well what is the corresponding side? The corresponding side to BE is side CE. So that's going to be the same as the ratio between CE and DE, and DE. And this just comes out of similar, the similarity of the triangles, CE to DE. So once again, once we establish these triangles are similar we can say the ratio of corresponding sides are going to be the same.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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The corresponding side to BE is side CE. So that's going to be the same as the ratio between CE and DE, and DE. And this just comes out of similar, the similarity of the triangles, CE to DE. So once again, once we establish these triangles are similar we can say the ratio of corresponding sides are going to be the same. Now what is the ratio between BE and AE? The ratio between BE and AE. Well that is the slope of this top line right over here.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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So once again, once we establish these triangles are similar we can say the ratio of corresponding sides are going to be the same. Now what is the ratio between BE and AE? The ratio between BE and AE. Well that is the slope of this top line right over here. We could say that's the slope of line AB. Slope, slope of line connecting, connecting A to B. Or actually let me just use, I could write it like this.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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Well that is the slope of this top line right over here. We could say that's the slope of line AB. Slope, slope of line connecting, connecting A to B. Or actually let me just use, I could write it like this. That is slope of, slope of A, slope of line AB. Remember slope is, when you're going from A to B, it's change in Y over change in X. So when you're going from A to B, your change in X is AE, and your change in Y is BE, or EB, however you want to refer to it.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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Or actually let me just use, I could write it like this. That is slope of, slope of A, slope of line AB. Remember slope is, when you're going from A to B, it's change in Y over change in X. So when you're going from A to B, your change in X is AE, and your change in Y is BE, or EB, however you want to refer to it. So this right over here is change in Y, and this over here is change in X. Well now let's look at this second expression right over here, CE over DE. CE over DE.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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So when you're going from A to B, your change in X is AE, and your change in Y is BE, or EB, however you want to refer to it. So this right over here is change in Y, and this over here is change in X. Well now let's look at this second expression right over here, CE over DE. CE over DE. Well now this is going to be change in Y over change in X between point C and D. So this is, this right over here, this is the slope of line, of line CD. And so just like that, by establishing similarity, we were able to see the ratio of corresponding sides are congruent, which shows us that the slopes of these two lines are going to be the same. And we are done.
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Proof parallel lines have the same slope High School Math Khan Academy.mp3
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So side, side, side works. What about angle, angle, angle? So let me do that over here. What about angle, angle, angle? So what I'm saying is, is if, let's say I have a triangle like this, like I have a triangle like that, and I have a triangle like this, and if we know, if we know that this angle is congruent to that angle, if this angle is congruent to that angle, which means that their measures are equal, or and, I should say and, and that angle is congruent to that angle, can we say that this is, that these are two congruent triangles, and in the first case, it looks like maybe it is, at least the way I drew it here. But when you think about it, you can have the exact same corresponding angles being, having the same measure or being congruent, but you can actually scale one of these triangles up and down and still have that property. For example, if I had this triangle right over here, this triangle right over here, it looks similar, and I'm using that in just the everyday language size, it has the same shape as these triangles right over here, and it has the same, it has the same angles, that angle is congruent to that angle, this angle down here is congruent to this angle over here, and this angle over here is congruent to this angle over here.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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What about angle, angle, angle? So what I'm saying is, is if, let's say I have a triangle like this, like I have a triangle like that, and I have a triangle like this, and if we know, if we know that this angle is congruent to that angle, if this angle is congruent to that angle, which means that their measures are equal, or and, I should say and, and that angle is congruent to that angle, can we say that this is, that these are two congruent triangles, and in the first case, it looks like maybe it is, at least the way I drew it here. But when you think about it, you can have the exact same corresponding angles being, having the same measure or being congruent, but you can actually scale one of these triangles up and down and still have that property. For example, if I had this triangle right over here, this triangle right over here, it looks similar, and I'm using that in just the everyday language size, it has the same shape as these triangles right over here, and it has the same, it has the same angles, that angle is congruent to that angle, this angle down here is congruent to this angle over here, and this angle over here is congruent to this angle over here. So all of the angles in all three of these triangles are the same, the corresponding angles have the same measure but clearly, clearly this triangle right over here is not the same, it is not congruent to the other two, the sides have a very different length. This side is much shorter than this side right over here, this side is much shorter than that side over there, and this side is much shorter over here. So with just angle, angle, angle, you cannot say that a triangle has the same size and shape.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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For example, if I had this triangle right over here, this triangle right over here, it looks similar, and I'm using that in just the everyday language size, it has the same shape as these triangles right over here, and it has the same, it has the same angles, that angle is congruent to that angle, this angle down here is congruent to this angle over here, and this angle over here is congruent to this angle over here. So all of the angles in all three of these triangles are the same, the corresponding angles have the same measure but clearly, clearly this triangle right over here is not the same, it is not congruent to the other two, the sides have a very different length. This side is much shorter than this side right over here, this side is much shorter than that side over there, and this side is much shorter over here. So with just angle, angle, angle, you cannot say that a triangle has the same size and shape. It does have the same shape but not the same size. So this does not imply, this does not imply congruency. So angle, angle, angle does not imply congruency.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So with just angle, angle, angle, you cannot say that a triangle has the same size and shape. It does have the same shape but not the same size. So this does not imply, this does not imply congruency. So angle, angle, angle does not imply congruency. What it does imply, and we haven't talked about this yet, is that these are similar triangles. So angle, angle, angle implies similar. So let me write it over here, it implies similar triangles.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So angle, angle, angle does not imply congruency. What it does imply, and we haven't talked about this yet, is that these are similar triangles. So angle, angle, angle implies similar. So let me write it over here, it implies similar triangles. And similar, you probably are used to the word in just everyday language, but similar has a very specific meaning in geometry, and similar things have the same shape but not necessarily the same size. So anything that is congruent, because it has the same size and shape, is also similar, but not everything that is similar is also congruent. So for example, this triangle is similar, all of these triangles are similar to each other, but they aren't all congruent.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So let me write it over here, it implies similar triangles. And similar, you probably are used to the word in just everyday language, but similar has a very specific meaning in geometry, and similar things have the same shape but not necessarily the same size. So anything that is congruent, because it has the same size and shape, is also similar, but not everything that is similar is also congruent. So for example, this triangle is similar, all of these triangles are similar to each other, but they aren't all congruent. These two are congruent if their sides are the same, I didn't make that assumption, but if we know that their sides are the same, then we can say that they're congruent, but neither of these are congruent to this one right over here because this is clearly much larger, has the same shape but a different size. So we can't have an AAA postulate or an AAA axiom to get to congruency. What about side-angle side?
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So for example, this triangle is similar, all of these triangles are similar to each other, but they aren't all congruent. These two are congruent if their sides are the same, I didn't make that assumption, but if we know that their sides are the same, then we can say that they're congruent, but neither of these are congruent to this one right over here because this is clearly much larger, has the same shape but a different size. So we can't have an AAA postulate or an AAA axiom to get to congruency. What about side-angle side? So let's try this out, side-angle side. So let's start off with one triangle right over here. So let's start off with a triangle that looks like this, I have my blue side, I have my pink side, and I have my magenta side.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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What about side-angle side? So let's try this out, side-angle side. So let's start off with one triangle right over here. So let's start off with a triangle that looks like this, I have my blue side, I have my pink side, and I have my magenta side. And let's say that I have another triangle that has this blue side, it has the same length as that blue side, so let me draw it like that, it has the same length as that blue side, so that length and that length are going to be the same, it has a congruent angle right after that, so this angle and the next angle for this triangle are going to have the same measure, they're going to be congruent, and then the next side is going to have the same length of this one over here, so that's going to be the same length as this over here. So it's going to be the same length. And we don't know, and because we only know that two of the corresponding sides have the same length, and the angle between them, and this is important, the angle between the two corresponding sides also have the same measure, we can do anything we want with this last side on this one.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So let's start off with a triangle that looks like this, I have my blue side, I have my pink side, and I have my magenta side. And let's say that I have another triangle that has this blue side, it has the same length as that blue side, so let me draw it like that, it has the same length as that blue side, so that length and that length are going to be the same, it has a congruent angle right after that, so this angle and the next angle for this triangle are going to have the same measure, they're going to be congruent, and then the next side is going to have the same length of this one over here, so that's going to be the same length as this over here. So it's going to be the same length. And we don't know, and because we only know that two of the corresponding sides have the same length, and the angle between them, and this is important, the angle between the two corresponding sides also have the same measure, we can do anything we want with this last side on this one. We can essentially, it's going to have to start right over here. You could start from this point. And we can pivot it to form any triangle we want.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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And we don't know, and because we only know that two of the corresponding sides have the same length, and the angle between them, and this is important, the angle between the two corresponding sides also have the same measure, we can do anything we want with this last side on this one. We can essentially, it's going to have to start right over here. You could start from this point. And we can pivot it to form any triangle we want. But we can see the only way we can form a triangle is if we bring this side all the way over here and close this right over there. And so we can see just logically that if we have, if for two triangles, they have one side that has the length the same, the next side has the length the same, and the angle in between them. So this angle, let me do that in the same color.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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And we can pivot it to form any triangle we want. But we can see the only way we can form a triangle is if we bring this side all the way over here and close this right over there. And so we can see just logically that if we have, if for two triangles, they have one side that has the length the same, the next side has the length the same, and the angle in between them. So this angle, let me do that in the same color. This angle in between them, this is the angle. This a is this angle and that angle. It's the angle in between them.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So this angle, let me do that in the same color. This angle in between them, this is the angle. This a is this angle and that angle. It's the angle in between them. This first side is in blue, and the second side right over here is in pink. And well, it's already written in pink. So we can see that if two sides have the same length, two corresponding sides have the same length, and the corresponding angle between them, they have to be congruent.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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It's the angle in between them. This first side is in blue, and the second side right over here is in pink. And well, it's already written in pink. So we can see that if two sides have the same length, two corresponding sides have the same length, and the corresponding angle between them, they have to be congruent. There's no other place to put this third side. So SAS, and sometimes it's once again called a postulate, an axiom, or if it's kind of proven, sometimes it's called a theorem. This does imply that the two triangles are congruent.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So we can see that if two sides have the same length, two corresponding sides have the same length, and the corresponding angle between them, they have to be congruent. There's no other place to put this third side. So SAS, and sometimes it's once again called a postulate, an axiom, or if it's kind of proven, sometimes it's called a theorem. This does imply that the two triangles are congruent. So we will give ourselves this tool in our toolkit. We had the SSS postulate. Now we have the SAS postulate.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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This does imply that the two triangles are congruent. So we will give ourselves this tool in our toolkit. We had the SSS postulate. Now we have the SAS postulate. Two sides are equal and the angle in between them for two triangles, corresponding sides and angles, then we can say that these are congruent triangles. Now what about, and I'm just going to try to go through all of the different combinations here. What if I have something, what if I have angle, side, angle?
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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Now we have the SAS postulate. Two sides are equal and the angle in between them for two triangles, corresponding sides and angles, then we can say that these are congruent triangles. Now what about, and I'm just going to try to go through all of the different combinations here. What if I have something, what if I have angle, side, angle? So let me try that. So what happens if I have angle, side, angle? So let's go back to this one right over here.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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What if I have something, what if I have angle, side, angle? So let me try that. So what happens if I have angle, side, angle? So let's go back to this one right over here. So actually, let me just redraw a new one for each of these cases. So angle, side, angle. So I'll draw a triangle here.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So let's go back to this one right over here. So actually, let me just redraw a new one for each of these cases. So angle, side, angle. So I'll draw a triangle here. So I have this triangle. So this would be maybe the side. That would be the side.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So I'll draw a triangle here. So I have this triangle. So this would be maybe the side. That would be the side. Let me draw the whole triangle actually first. So I have this triangle. Let me draw one side over here.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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That would be the side. Let me draw the whole triangle actually first. So I have this triangle. Let me draw one side over here. And then let me draw one side over there. And then this angle right over here, I'll do it in orange. And this angle over here, I will do it in yellow.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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Let me draw one side over here. And then let me draw one side over there. And then this angle right over here, I'll do it in orange. And this angle over here, I will do it in yellow. So if I have another triangle that has one side having equal measure, so I'll use it as this blue side right over here. So it has one side that has equal measure. And the two angles on either side of that side or either end of that side are the same, will this triangle necessarily be congruent?
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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And this angle over here, I will do it in yellow. So if I have another triangle that has one side having equal measure, so I'll use it as this blue side right over here. So it has one side that has equal measure. And the two angles on either side of that side or either end of that side are the same, will this triangle necessarily be congruent? And we're just going to try to reason it out. These aren't formal proofs. We're really just trying to set up what are reasonable postulates or what are reasonable assumptions we can have in our toolkit as we try to prove other things.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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And the two angles on either side of that side or either end of that side are the same, will this triangle necessarily be congruent? And we're just going to try to reason it out. These aren't formal proofs. We're really just trying to set up what are reasonable postulates or what are reasonable assumptions we can have in our toolkit as we try to prove other things. So that angle, let's call it that angle right over there, is going to be the same measure in this triangle. And this angle right over here in yellow is going to have the same measure on this triangle right over here. So regardless, I'm not in any way constraining the sides over here.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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We're really just trying to set up what are reasonable postulates or what are reasonable assumptions we can have in our toolkit as we try to prove other things. So that angle, let's call it that angle right over there, is going to be the same measure in this triangle. And this angle right over here in yellow is going to have the same measure on this triangle right over here. So regardless, I'm not in any way constraining the sides over here. So this side right over here could have any length. But it has to form this angle with it. So it could have any length.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So regardless, I'm not in any way constraining the sides over here. So this side right over here could have any length. But it has to form this angle with it. So it could have any length. And it can just go as far as it wants to go. No way have we constrained what the length of that is. Actually, let me mark this off too.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So it could have any length. And it can just go as far as it wants to go. No way have we constrained what the length of that is. Actually, let me mark this off too. So this is the same as this. So that side can be anything. We haven't constrained it at all.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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Actually, let me mark this off too. So this is the same as this. So that side can be anything. We haven't constrained it at all. And once again, this side could be anything. We haven't constrained it at all, but we know it has to go at this angle. So it has to go at that angle.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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We haven't constrained it at all. And once again, this side could be anything. We haven't constrained it at all, but we know it has to go at this angle. So it has to go at that angle. Well, once again, there's only one triangle that can be formed this way. We can say all day that this length could be as long as we want or as short as we want. And this one could be as long as we want or as short as we want.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So it has to go at that angle. Well, once again, there's only one triangle that can be formed this way. We can say all day that this length could be as long as we want or as short as we want. And this one could be as long as we want or as short as we want. But the only way that they can actually touch each other and form a triangle and have these two angles is if they are the exact same length as these two sides right over here. So this side will actually have to be the same as that side. And this would have to be the same as that side.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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And this one could be as long as we want or as short as we want. But the only way that they can actually touch each other and form a triangle and have these two angles is if they are the exact same length as these two sides right over here. So this side will actually have to be the same as that side. And this would have to be the same as that side. Once again, this isn't a proof. I'd call it more of a reasoning through it or an investigation, really just to establish what are reasonable baselines or axioms or assumptions or postulates that we can have. So for my purposes, I think ASA does show us that two triangles are congruent.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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And this would have to be the same as that side. Once again, this isn't a proof. I'd call it more of a reasoning through it or an investigation, really just to establish what are reasonable baselines or axioms or assumptions or postulates that we can have. So for my purposes, I think ASA does show us that two triangles are congruent. Now let's try another one. Let's try angle-angle-side. Let's try angle-angle-side.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So for my purposes, I think ASA does show us that two triangles are congruent. Now let's try another one. Let's try angle-angle-side. Let's try angle-angle-side. And in some geometry classes, and maybe if you have to go through an exam quickly, you might memorize, OK, side-side-side implies congruency. And that's kind of logical. Side-angle-side implies congruency, and so on and so forth.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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Let's try angle-angle-side. And in some geometry classes, and maybe if you have to go through an exam quickly, you might memorize, OK, side-side-side implies congruency. And that's kind of logical. Side-angle-side implies congruency, and so on and so forth. I'm not a fan of memorizing it. It might be good for time pressure. It is good to sometimes even just go through this logic.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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Side-angle-side implies congruency, and so on and so forth. I'm not a fan of memorizing it. It might be good for time pressure. It is good to sometimes even just go through this logic. If you're like, wait, does angle-angle-angle work? Well, no. I can find this case that breaks down angle-angle-angle.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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It is good to sometimes even just go through this logic. If you're like, wait, does angle-angle-angle work? Well, no. I can find this case that breaks down angle-angle-angle. If these work, just try to verify for yourself that they make logical sense why they would imply congruency. Now let's try angle-angle-side. Let's try angle-angle-side.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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