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So this 2 times 1 half AB, that takes into consideration this bottom right triangle and this top one. And what's the area of this large one that I will color in in green? What's the area of this large one? Well, that's pretty straightforward. It's just 1 half C times C. So plus 1 half C times C, which is 1 half C squared. Now, let's simplify this thing and see what we come up with. And you might guess where all of this is going.
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Garfield's proof of the Pythagorean theorem Geometry Khan Academy.mp3
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Well, that's pretty straightforward. It's just 1 half C times C. So plus 1 half C times C, which is 1 half C squared. Now, let's simplify this thing and see what we come up with. And you might guess where all of this is going. So let's see what we get. So we can rearrange this. So this is 1 half times A plus B squared is going to be equal to 2 times 1 half.
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Garfield's proof of the Pythagorean theorem Geometry Khan Academy.mp3
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And you might guess where all of this is going. So let's see what we get. So we can rearrange this. So this is 1 half times A plus B squared is going to be equal to 2 times 1 half. Well, that's just going to be 1. So it's going to be equal to A times B plus 1 half C squared. Well, I don't like these 1 halves laying around, so let's multiply both sides of this equation by 2.
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Garfield's proof of the Pythagorean theorem Geometry Khan Academy.mp3
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So this is 1 half times A plus B squared is going to be equal to 2 times 1 half. Well, that's just going to be 1. So it's going to be equal to A times B plus 1 half C squared. Well, I don't like these 1 halves laying around, so let's multiply both sides of this equation by 2. I'm just going to multiply both sides of this equation by 2. On the left-hand side, I'm just left with A plus B squared. So let me write that.
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Garfield's proof of the Pythagorean theorem Geometry Khan Academy.mp3
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Well, I don't like these 1 halves laying around, so let's multiply both sides of this equation by 2. I'm just going to multiply both sides of this equation by 2. On the left-hand side, I'm just left with A plus B squared. So let me write that. A plus B squared. And on the right-hand side, I am left with 2AB. Trying to keep the color coding right.
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Garfield's proof of the Pythagorean theorem Geometry Khan Academy.mp3
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So let me write that. A plus B squared. And on the right-hand side, I am left with 2AB. Trying to keep the color coding right. And then 2 times 1 half C squared, that's just going to be C squared. Plus C squared. Well, what happens if you multiply out A plus B times A plus B?
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Garfield's proof of the Pythagorean theorem Geometry Khan Academy.mp3
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Trying to keep the color coding right. And then 2 times 1 half C squared, that's just going to be C squared. Plus C squared. Well, what happens if you multiply out A plus B times A plus B? What is A plus B squared? Well, it's going to be A squared plus 2AB plus B squared. And then our right-hand side is still going to be equal to all of this business.
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Garfield's proof of the Pythagorean theorem Geometry Khan Academy.mp3
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Well, what happens if you multiply out A plus B times A plus B? What is A plus B squared? Well, it's going to be A squared plus 2AB plus B squared. And then our right-hand side is still going to be equal to all of this business. And changing all of the colors is difficult for me. So let me copy and let me paste it. So it's still going to be equal to the right-hand side.
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Garfield's proof of the Pythagorean theorem Geometry Khan Academy.mp3
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And then our right-hand side is still going to be equal to all of this business. And changing all of the colors is difficult for me. So let me copy and let me paste it. So it's still going to be equal to the right-hand side. Well, this is interesting. How can we simplify this? Is there anything that we can subtract from both sides?
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Garfield's proof of the Pythagorean theorem Geometry Khan Academy.mp3
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So it's still going to be equal to the right-hand side. Well, this is interesting. How can we simplify this? Is there anything that we can subtract from both sides? Well, sure there is. You have a 2AB on the left-hand side. You have a 2AB on the right-hand side.
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Garfield's proof of the Pythagorean theorem Geometry Khan Academy.mp3
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Is there anything that we can subtract from both sides? Well, sure there is. You have a 2AB on the left-hand side. You have a 2AB on the right-hand side. Let's subtract 2AB from both sides. If you subtract 2AB from both sides, what are you left with? You are left with the Pythagorean theorem.
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Garfield's proof of the Pythagorean theorem Geometry Khan Academy.mp3
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You have a 2AB on the right-hand side. Let's subtract 2AB from both sides. If you subtract 2AB from both sides, what are you left with? You are left with the Pythagorean theorem. So you're left with A squared plus B squared is equal to C squared. Very, very exciting. And for that, we have to thank the 20th president of the United States, James Garfield.
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Garfield's proof of the Pythagorean theorem Geometry Khan Academy.mp3
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Transformation C maps negative two, three, to four, negative one. So let me do negative two, comma, three. And it maps that to four, negative one. To four, negative one. And point negative five, comma, five. Negative five, comma, five. Negative five, comma, five.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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To four, negative one. And point negative five, comma, five. Negative five, comma, five. Negative five, comma, five. It maps that to seven, negative three. To seven, negative three. So seven, negative three.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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Negative five, comma, five. It maps that to seven, negative three. To seven, negative three. So seven, negative three. And so let's think about this a little bit. How could we get from this point to this point, and that point to that point? Now it's tempting to view this that maybe a translation is possible.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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So seven, negative three. And so let's think about this a little bit. How could we get from this point to this point, and that point to that point? Now it's tempting to view this that maybe a translation is possible. Because if you imagined a line like that, you could say, hey, let's just shift this whole thing down. And then to the right, these two things happen to have the same slope. They both have a slope of negative 2 3rds.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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Now it's tempting to view this that maybe a translation is possible. Because if you imagined a line like that, you could say, hey, let's just shift this whole thing down. And then to the right, these two things happen to have the same slope. They both have a slope of negative 2 3rds. And so this point would map to this point, and that point would map to that point. But that's not what we want. We don't want negative two, three to map to seven, negative three.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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They both have a slope of negative 2 3rds. And so this point would map to this point, and that point would map to that point. But that's not what we want. We don't want negative two, three to map to seven, negative three. We want negative two, three to map to four, negative one. So you could get this line over this line, but we won't map the points that we want to map. So this can't be, at least I can't think of a way, that this could actually be a translation.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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We don't want negative two, three to map to seven, negative three. We want negative two, three to map to four, negative one. So you could get this line over this line, but we won't map the points that we want to map. So this can't be, at least I can't think of a way, that this could actually be a translation. Now let's think about whether our transformation could be a reflection. Well, if we imagine a line that has, let's see, these both have a slope of negative 3. These both have a slope of negative 2 3rds.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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So this can't be, at least I can't think of a way, that this could actually be a translation. Now let's think about whether our transformation could be a reflection. Well, if we imagine a line that has, let's see, these both have a slope of negative 3. These both have a slope of negative 2 3rds. So if you imagine a line that had a slope of positive 3 halves, that was equidistant from both. And I don't know if this is, let's see, is this equidistant? Is this equidistant from both of them?
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Possible transformations example Transformations Geometry Khan Academy.mp3
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These both have a slope of negative 2 3rds. So if you imagine a line that had a slope of positive 3 halves, that was equidistant from both. And I don't know if this is, let's see, is this equidistant? Is this equidistant from both of them? It's either going to be that line or this line right over, or that line. Actually, that line looks better. So that one.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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Is this equidistant from both of them? It's either going to be that line or this line right over, or that line. Actually, that line looks better. So that one. And once again, I'm just eyeballing it. So a line that has slope of positive 3 halves. So this one looks right in between the two.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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So that one. And once again, I'm just eyeballing it. So a line that has slope of positive 3 halves. So this one looks right in between the two. Or actually, it could be someplace in between. But either way, we just have to think about it qualitatively. So imagine a line that looked something like that.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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So this one looks right in between the two. Or actually, it could be someplace in between. But either way, we just have to think about it qualitatively. So imagine a line that looked something like that. And if you were to reflect over this line, then this point would map to this point, which is what we want. And this purple point, negative 5 comma 5, would map to that point. It would be reflected over.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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So imagine a line that looked something like that. And if you were to reflect over this line, then this point would map to this point, which is what we want. And this purple point, negative 5 comma 5, would map to that point. It would be reflected over. So it's pretty clear that this could be a reflection. Now, rotation actually makes even more sense, or at least in my brain, makes a little more sense. If you were to rotate around this point right over here, this point would map to that point.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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It would be reflected over. So it's pretty clear that this could be a reflection. Now, rotation actually makes even more sense, or at least in my brain, makes a little more sense. If you were to rotate around this point right over here, this point would map to that point. And that point would map to that point. So a rotation also seems like a possibility for transformation C. Now let's think about transformation D. We are going from 4, negative 1 to 7, negative 3. Actually, maybe I'll put that in magenta as well, to 7, negative 3, just like that.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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If you were to rotate around this point right over here, this point would map to that point. And that point would map to that point. So a rotation also seems like a possibility for transformation C. Now let's think about transformation D. We are going from 4, negative 1 to 7, negative 3. Actually, maybe I'll put that in magenta as well, to 7, negative 3, just like that. And we want to go from negative 5, 5 to negative 2, 3. So I could definitely imagine a translation right over here. This point went 3 to the right and 2 down.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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Actually, maybe I'll put that in magenta as well, to 7, negative 3, just like that. And we want to go from negative 5, 5 to negative 2, 3. So I could definitely imagine a translation right over here. This point went 3 to the right and 2 down. This point went 3 to the right and 2 down. So a translation definitely makes sense. Now let's think about a reflection.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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This point went 3 to the right and 2 down. This point went 3 to the right and 2 down. So a translation definitely makes sense. Now let's think about a reflection. So it would be tempting to get from this point to this point. I could reflect around that. But that won't help this one over here.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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Now let's think about a reflection. So it would be tempting to get from this point to this point. I could reflect around that. But that won't help this one over here. And to get from that point to that point, I could reflect around that. But once again, that's not going to help that point over there. So a reflection really doesn't seem to do the trick.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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But that won't help this one over here. And to get from that point to that point, I could reflect around that. But once again, that's not going to help that point over there. So a reflection really doesn't seem to do the trick. And what about a rotation? Well, to go from this point to this point, we could rotate around this point. We could go there.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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So a reflection really doesn't seem to do the trick. And what about a rotation? Well, to go from this point to this point, we could rotate around this point. We could go there. But that won't help this point right over here. While this is rotating there, this point is going to rotate around like that. And it's going to end up someplace out here.
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Possible transformations example Transformations Geometry Khan Academy.mp3
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A lot of what geometry is about is proving things about the world. In order to really prove things about the world, we have to be very careful, very precise, very exact with our language so that we know what we're proving and we know what we're assuming and what type of deductions we are making as we prove things. And so to get some practice being precise and exact with our language, I'm going to go through some exercises from the geometric definitions exercise on Khan Academy. So this first one says, three students attempt to define what it means for lines L and M to be perpendicular. Can you match the teacher's comments to the definitions? All right, so it looks like three different students attempt definitions of what it means to be perpendicular, and then there's these teacher's comments that we can move around. So we're going to, I guess, pretend that we're the teacher.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So this first one says, three students attempt to define what it means for lines L and M to be perpendicular. Can you match the teacher's comments to the definitions? All right, so it looks like three different students attempt definitions of what it means to be perpendicular, and then there's these teacher's comments that we can move around. So we're going to, I guess, pretend that we're the teacher. So Ruby's definition for being perpendicular, L and M, lines L and M, are perpendicular if they never meet. Well, that's not true. In fact, perpendicular lines, for sure, will intersect.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So we're going to, I guess, pretend that we're the teacher. So Ruby's definition for being perpendicular, L and M, lines L and M, are perpendicular if they never meet. Well, that's not true. In fact, perpendicular lines, for sure, will intersect. So in fact, they intersect at right angles. So that is not going to be correct. And so, actually, this looks right.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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In fact, perpendicular lines, for sure, will intersect. So in fact, they intersect at right angles. So that is not going to be correct. And so, actually, this looks right. Were you thinking of parallel lines? Because that looks like what she was trying to define. If things are on the same plane and they never intersect, then you are talking about parallel lines.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And so, actually, this looks right. Were you thinking of parallel lines? Because that looks like what she was trying to define. If things are on the same plane and they never intersect, then you are talking about parallel lines. I also get Shreya's definition. L and M are perpendicular if they meet at one point and one of the angles at their point of intersection is a right angle. Well, that seems spot on.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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If things are on the same plane and they never intersect, then you are talking about parallel lines. I also get Shreya's definition. L and M are perpendicular if they meet at one point and one of the angles at their point of intersection is a right angle. Well, that seems spot on. So let's see. I would say, woo-hoo, nice work. I couldn't have said it better myself.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Well, that seems spot on. So let's see. I would say, woo-hoo, nice work. I couldn't have said it better myself. Now let's just make sure this comment matches for this definition. Abhishek says, L and M are perpendicular if they meet at a single point such that the two lines make a T. Well, that's, in a hand-wavy way, kind of right. When you imagine perpendicular lines, you could imagine them kind of forming a cross or I guess a T would be part of it.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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I couldn't have said it better myself. Now let's just make sure this comment matches for this definition. Abhishek says, L and M are perpendicular if they meet at a single point such that the two lines make a T. Well, that's, in a hand-wavy way, kind of right. When you imagine perpendicular lines, you could imagine them kind of forming a cross or I guess a T would be part of it. But I think this comment is spot on, the teacher's comment. Your definition is kind of correct, but it lacks mathematical precision. What does he mean by a T?
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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When you imagine perpendicular lines, you could imagine them kind of forming a cross or I guess a T would be part of it. But I think this comment is spot on, the teacher's comment. Your definition is kind of correct, but it lacks mathematical precision. What does he mean by a T? What does it mean to make a T? Shreya's definition is much more precise. They're perpendicular if they meet at one point and one of their angles at their point of intersection is a right angle, is a 90-degree angle.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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What does he mean by a T? What does it mean to make a T? Shreya's definition is much more precise. They're perpendicular if they meet at one point and one of their angles at their point of intersection is a right angle, is a 90-degree angle. Let's check our answer. So let's do a few more of these. This is actually a lot of fun.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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They're perpendicular if they meet at one point and one of their angles at their point of intersection is a right angle, is a 90-degree angle. Let's check our answer. So let's do a few more of these. This is actually a lot of fun. So once again, we're gonna have three students attempting to define, but now they're going to define an object called an angle. Can you match the teacher's comments to the definitions? So Ruby, well, these same three students, Ruby says the amount of turn between two straight lines that have a common vertex.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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This is actually a lot of fun. So once again, we're gonna have three students attempting to define, but now they're going to define an object called an angle. Can you match the teacher's comments to the definitions? So Ruby, well, these same three students, Ruby says the amount of turn between two straight lines that have a common vertex. Well, this is kind of getting there. The definition of an angle, we typically talk about two rays with a common vertex. She's talking about two lines with a common vertex.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So Ruby, well, these same three students, Ruby says the amount of turn between two straight lines that have a common vertex. Well, this is kind of getting there. The definition of an angle, we typically talk about two rays with a common vertex. She's talking about two lines with a common vertex. And she's talking about the amount of turn. So she's really talking about more of kind of the measure of an angle. So let's see what comment here.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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She's talking about two lines with a common vertex. And she's talking about the amount of turn. So she's really talking about more of kind of the measure of an angle. So let's see what comment here. So you seem to be getting at the idea of a measure of an angle and not the definition of an angle itself. So this is actually right. I would put this one right here.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So let's see what comment here. So you seem to be getting at the idea of a measure of an angle and not the definition of an angle itself. So this is actually right. I would put this one right here. We just got lucky this was already aligned. So Shreya's definition, two lines that come together. So once again, this is kind of, the definition of an angle is two rays with a common vertex.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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I would put this one right here. We just got lucky this was already aligned. So Shreya's definition, two lines that come together. So once again, this is kind of, the definition of an angle is two rays with a common vertex. So two lines that come together, this is just intersecting lines. Now when that happens, you might be forming some angles, but I would just say, were you thinking of intersecting lines? And let's see what Abhishek says.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So once again, this is kind of, the definition of an angle is two rays with a common vertex. So two lines that come together, this is just intersecting lines. Now when that happens, you might be forming some angles, but I would just say, were you thinking of intersecting lines? And let's see what Abhishek says. A figure composed of two rays sharing a common endpoint. The common endpoint is known as the vertex. Yep, that's a good definition of an angle.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And let's see what Abhishek says. A figure composed of two rays sharing a common endpoint. The common endpoint is known as the vertex. Yep, that's a good definition of an angle. So Abhishek got it this time. Let's do another one. So three students are now attempting to define what it means for two lines to be parallel.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Yep, that's a good definition of an angle. So Abhishek got it this time. Let's do another one. So three students are now attempting to define what it means for two lines to be parallel. So now let's match the teacher's comments. So Daniela says two lines are parallel if they are distinct and one can be translated on top of the other. All right, so that actually seems pretty interesting.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So three students are now attempting to define what it means for two lines to be parallel. So now let's match the teacher's comments. So Daniela says two lines are parallel if they are distinct and one can be translated on top of the other. All right, so that actually seems pretty interesting. That's actually not the first way that I would have defined parallel lines. I would have said, hey, if they're on the same plane and they don't intersect, then they are parallel. But this seems pretty good because if you're translating something, you're not, you aren't going to rotate it.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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All right, so that actually seems pretty interesting. That's actually not the first way that I would have defined parallel lines. I would have said, hey, if they're on the same plane and they don't intersect, then they are parallel. But this seems pretty good because if you're translating something, you're not, you aren't going to rotate it. You're not going to change its direction, I guess one way to think about it. And so if you're translating one, if you can, if they're two different lines, but you can shift them without changing their direction, which is what translation is all about, on top of each other, that actually feels pretty good. So I'll put that right over there.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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But this seems pretty good because if you're translating something, you're not, you aren't going to rotate it. You're not going to change its direction, I guess one way to think about it. And so if you're translating one, if you can, if they're two different lines, but you can shift them without changing their direction, which is what translation is all about, on top of each other, that actually feels pretty good. So I'll put that right over there. So Ori says two lines are parallel if they are close together but don't intersect. So if you're trying to define parallel lines, parallel lines, it doesn't matter if they're close together or not. They just have to be in the same plane and not intersect.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So I'll put that right over there. So Ori says two lines are parallel if they are close together but don't intersect. So if you're trying to define parallel lines, parallel lines, it doesn't matter if they're close together or not. They just have to be in the same plane and not intersect. They could be very far apart and they could still be parallel. So this isn't an incorrect statement. You could have two lines that are close together and don't intersect on the same plane and they are going to be parallel.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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They just have to be in the same plane and not intersect. They could be very far apart and they could still be parallel. So this isn't an incorrect statement. You could have two lines that are close together and don't intersect on the same plane and they are going to be parallel. But this isn't a good definition because you can also have parallel lines that are far apart. And so, actually I'd go with this statement right here. Part of your definition is correct, but the other part is not.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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You could have two lines that are close together and don't intersect on the same plane and they are going to be parallel. But this isn't a good definition because you can also have parallel lines that are far apart. And so, actually I'd go with this statement right here. Part of your definition is correct, but the other part is not. Parallel lines don't have to be close together. So this isn't a good definition of parallel lines. And let's see, Kaori.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Part of your definition is correct, but the other part is not. Parallel lines don't have to be close together. So this isn't a good definition of parallel lines. And let's see, Kaori. Two lines are parallel as long as they aren't perpendicular. Well, that's just not true because you can intersect. You can have two lines that intersect at non-right angles and they're not parallel and they're also not perpendicular.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And let's see, Kaori. Two lines are parallel as long as they aren't perpendicular. Well, that's just not true because you can intersect. You can have two lines that intersect at non-right angles and they're not parallel and they're also not perpendicular. So this is, you know, sorry, your definition is incorrect. This is actually a lot of fun pretending to be the teacher. Let's do another one.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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You can have two lines that intersect at non-right angles and they're not parallel and they're also not perpendicular. So this is, you know, sorry, your definition is incorrect. This is actually a lot of fun pretending to be the teacher. Let's do another one. All right. So three students attempt to define what a line segment is. And we have a depiction of a line segment right over here.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Let's do another one. All right. So three students attempt to define what a line segment is. And we have a depiction of a line segment right over here. We have point P, point Q, and the line segment is all the points in between P and Q. So let's match the teacher's comments to the definitions. Ivy's definition.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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And we have a depiction of a line segment right over here. We have point P, point Q, and the line segment is all the points in between P and Q. So let's match the teacher's comments to the definitions. Ivy's definition. All of the points in line with P and Q extending infinitely in both directions. Well, that would be the definition of a line. That would be the line PQ.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Ivy's definition. All of the points in line with P and Q extending infinitely in both directions. Well, that would be the definition of a line. That would be the line PQ. That would be if you're extending infinitely in both directions. So I would say are you thinking of a line instead of a line segment? Ethan's definition.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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That would be the line PQ. That would be if you're extending infinitely in both directions. So I would say are you thinking of a line instead of a line segment? Ethan's definition. The exact distance from P to Q. Well, that's just the length of a line segment. That's not exactly what a line segment is.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Ethan's definition. The exact distance from P to Q. Well, that's just the length of a line segment. That's not exactly what a line segment is. Let's see, Ibuka's definition. The points P and Q, which are called endpoints, and all of the points in a straight line between points P and Q. Yep, that looks like a good definition for a line segment. So we can just check our answer.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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That's not exactly what a line segment is. Let's see, Ibuka's definition. The points P and Q, which are called endpoints, and all of the points in a straight line between points P and Q. Yep, that looks like a good definition for a line segment. So we can just check our answer. So looking good. Let's do one more of this. I'm just really enjoying pretending to be a teacher.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So we can just check our answer. So looking good. Let's do one more of this. I'm just really enjoying pretending to be a teacher. All right, three students attempted to find what a circle is. Define what a circle is. Can you match the teacher's comments to the definitions?
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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I'm just really enjoying pretending to be a teacher. All right, three students attempted to find what a circle is. Define what a circle is. Can you match the teacher's comments to the definitions? Duru, the set of all points in a plane that are the same distance away from some given point, which we call the center. That actually seems like a pretty good definition of a circle. So, stupendous, well done.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Can you match the teacher's comments to the definitions? Duru, the set of all points in a plane that are the same distance away from some given point, which we call the center. That actually seems like a pretty good definition of a circle. So, stupendous, well done. Oliver's definition. The set of all points in 3D space that are the same distance from a center point. Well, if we're talking about 3D space and the set of all points that are equidistant from that point in 3D space, now we're talking about a sphere, not a circle.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So, stupendous, well done. Oliver's definition. The set of all points in 3D space that are the same distance from a center point. Well, if we're talking about 3D space and the set of all points that are equidistant from that point in 3D space, now we're talking about a sphere, not a circle. And so, you seem to be confusing a circle with a sphere. And then finally, a perfectly round shape. Well, that's kinda true, but if you're talking about three dimensions, you could be talking about a sphere.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Well, if we're talking about 3D space and the set of all points that are equidistant from that point in 3D space, now we're talking about a sphere, not a circle. And so, you seem to be confusing a circle with a sphere. And then finally, a perfectly round shape. Well, that's kinda true, but if you're talking about three dimensions, you could be talking about a sphere. If you go beyond three dimensions, hypersphere, whatever else. In two dimensions, yeah, a perfectly round shape. Most people would call it a circle, but that doesn't have a lot of precision to it.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Well, that's kinda true, but if you're talking about three dimensions, you could be talking about a sphere. If you go beyond three dimensions, hypersphere, whatever else. In two dimensions, yeah, a perfectly round shape. Most people would call it a circle, but that doesn't have a lot of precision to it. It doesn't give us a lot that we can work with from a mathematical point of view. So, I would say, actually, what the teacher's saying. Your definition needs to be much more precise.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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Most people would call it a circle, but that doesn't have a lot of precision to it. It doesn't give us a lot that we can work with from a mathematical point of view. So, I would say, actually, what the teacher's saying. Your definition needs to be much more precise. Duru's definition is much, much more precise. The set of all points that are equidistant in a plane that are equidistant away from a given point, which we call the center. So, yep, Carlos could use a little bit more precision.
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Geometric precision practice Introduction to Euclidean geometry Geometry Khan Academy.mp3
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So I just have an arbitrary triangle right over here, triangle ABC. What I'm going to do is I'm going to draw an angle bisector for this angle up here. We could have done it with any of the three angles, but I'll just do this one. It'll make our proof a little bit easier. So I'm just going to bisect this angle, angle ABC. So let's just say that that's the angle bisector angle ABC, and so this angle right over here is equal to this angle right over here. And let me call this point down here, let me call it point D. The angle bisector theorem tells us that the ratio between the sides that aren't this bisector, so when I put this angle bisector here, it created two smaller triangles out of that larger one.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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It'll make our proof a little bit easier. So I'm just going to bisect this angle, angle ABC. So let's just say that that's the angle bisector angle ABC, and so this angle right over here is equal to this angle right over here. And let me call this point down here, let me call it point D. The angle bisector theorem tells us that the ratio between the sides that aren't this bisector, so when I put this angle bisector here, it created two smaller triangles out of that larger one. The angle bisector theorem tells us the ratios between the other sides of these two triangles that we've now created are going to be the same. So it tells us that the ratio of AB to AD is going to be equal to the ratio of BC to, you could say, CD. So the ratio of, I'll color code it, the ratio of that, which is this, to this, is going to be equal to the ratio of this, which is that, to this right over here, to CD, which is that over here.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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And let me call this point down here, let me call it point D. The angle bisector theorem tells us that the ratio between the sides that aren't this bisector, so when I put this angle bisector here, it created two smaller triangles out of that larger one. The angle bisector theorem tells us the ratios between the other sides of these two triangles that we've now created are going to be the same. So it tells us that the ratio of AB to AD is going to be equal to the ratio of BC to, you could say, CD. So the ratio of, I'll color code it, the ratio of that, which is this, to this, is going to be equal to the ratio of this, which is that, to this right over here, to CD, which is that over here. So once you see the ratio of that to that, it's going to be the same as the ratio of that to that. So that's kind of a cool result, but you can't just left it on faith because it's a cool result. You want to prove it to ourselves.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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So the ratio of, I'll color code it, the ratio of that, which is this, to this, is going to be equal to the ratio of this, which is that, to this right over here, to CD, which is that over here. So once you see the ratio of that to that, it's going to be the same as the ratio of that to that. So that's kind of a cool result, but you can't just left it on faith because it's a cool result. You want to prove it to ourselves. And so you could imagine right over here we have some ratios set up. So we're going to prove it using similar triangles. And unfortunate for us, these two triangles right here aren't necessarily similar.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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You want to prove it to ourselves. And so you could imagine right over here we have some ratios set up. So we're going to prove it using similar triangles. And unfortunate for us, these two triangles right here aren't necessarily similar. We know that these two angles are congruent to each other, but we don't know whether this angle is equal to that angle or that angle. We don't know. We can't make any statements like that.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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And unfortunate for us, these two triangles right here aren't necessarily similar. We know that these two angles are congruent to each other, but we don't know whether this angle is equal to that angle or that angle. We don't know. We can't make any statements like that. So in order to actually set up this type of a statement, we'll have to construct maybe another triangle that will be similar to one of these right over here. And one way to do it would be to draw another line. And this is a bit of this proof wasn't obvious to me the first time that I thought about it, so don't worry if it's not obvious to you.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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We can't make any statements like that. So in order to actually set up this type of a statement, we'll have to construct maybe another triangle that will be similar to one of these right over here. And one way to do it would be to draw another line. And this is a bit of this proof wasn't obvious to me the first time that I thought about it, so don't worry if it's not obvious to you. Is that what happens is if we can continue this bisector, this angle bisector right over here. So let's just continue it. It just keeps going on and on and on.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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And this is a bit of this proof wasn't obvious to me the first time that I thought about it, so don't worry if it's not obvious to you. Is that what happens is if we can continue this bisector, this angle bisector right over here. So let's just continue it. It just keeps going on and on and on. And let's also, maybe we can construct a similar triangle to this triangle over here if we draw a line that's parallel to AB down here. So let's try to do that. So I'm just going to say, if C is not on AB, you can always find a point that goes through or a line that goes through C that is parallel to AB.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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It just keeps going on and on and on. And let's also, maybe we can construct a similar triangle to this triangle over here if we draw a line that's parallel to AB down here. So let's try to do that. So I'm just going to say, if C is not on AB, you can always find a point that goes through or a line that goes through C that is parallel to AB. So let's just by definition, let's just create another line right over here. And let's say, and let's call this point right over here F. And let's just pick this line in such a way that FC is parallel to AB. So this is parallel to that right over there.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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So I'm just going to say, if C is not on AB, you can always find a point that goes through or a line that goes through C that is parallel to AB. So let's just by definition, let's just create another line right over here. And let's say, and let's call this point right over here F. And let's just pick this line in such a way that FC is parallel to AB. So this is parallel to that right over there. So FC is parallel to AB. And we can just construct it that way. And now we have some interesting things.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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So this is parallel to that right over there. So FC is parallel to AB. And we can just construct it that way. And now we have some interesting things. And we did it that way so that we can make these two triangles be similar to each other. So let's see that. Let's see what happens.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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And now we have some interesting things. And we did it that way so that we can make these two triangles be similar to each other. So let's see that. Let's see what happens. So before we even think about similarity, let's think about what some of the angles or what we know about some of the angles here. We know that we have alternate interior angles. So just think about these two parallel lines.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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Let's see what happens. So before we even think about similarity, let's think about what some of the angles or what we know about some of the angles here. We know that we have alternate interior angles. So just think about these two parallel lines. So I could imagine AB keeps going like that. FC keeps going like that. And line BD right here is a transversal.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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So just think about these two parallel lines. So I could imagine AB keeps going like that. FC keeps going like that. And line BD right here is a transversal. Then whatever this angle is, this angle is going to be as well from alternate interior angles, which we've talked a lot about when we first talked about angles with transversals and all of that. So these two angles are going to be the same. But this angle and this angle are also going to be the same.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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And line BD right here is a transversal. Then whatever this angle is, this angle is going to be as well from alternate interior angles, which we've talked a lot about when we first talked about angles with transversals and all of that. So these two angles are going to be the same. But this angle and this angle are also going to be the same. Because this angle and that angle are the same. This is a bisector. So because this is a bisector, we know that angle ABD is the same as angle DBC.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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But this angle and this angle are also going to be the same. Because this angle and that angle are the same. This is a bisector. So because this is a bisector, we know that angle ABD is the same as angle DBC. So whatever this angle is, that angle is, and so is this angle. And that gives us some kind of an interesting result. Because here we have a situation where if you look at this larger triangle, BFC, we have two base angles that are the same, which means this must be an isosceles triangle.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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So because this is a bisector, we know that angle ABD is the same as angle DBC. So whatever this angle is, that angle is, and so is this angle. And that gives us some kind of an interesting result. Because here we have a situation where if you look at this larger triangle, BFC, we have two base angles that are the same, which means this must be an isosceles triangle. So BC must be the same as FC. So that was kind of cool. We just used the transversal and the alternate interior angles to show that these are isosceles and that BC and FC are the same thing.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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Because here we have a situation where if you look at this larger triangle, BFC, we have two base angles that are the same, which means this must be an isosceles triangle. So BC must be the same as FC. So that was kind of cool. We just used the transversal and the alternate interior angles to show that these are isosceles and that BC and FC are the same thing. And that could be useful. Because we know that we have a feeling that this triangle and this triangle are going to be similar. We haven't proven it yet.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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We just used the transversal and the alternate interior angles to show that these are isosceles and that BC and FC are the same thing. And that could be useful. Because we know that we have a feeling that this triangle and this triangle are going to be similar. We haven't proven it yet. But how will that help us get something about BC up here? But we just showed that BC and FC are the same thing. So this is going to be the same thing.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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We haven't proven it yet. But how will that help us get something about BC up here? But we just showed that BC and FC are the same thing. So this is going to be the same thing. If we want to prove it, if we can prove that FC, the ratio of AB to AD is the same thing as the ratio of FC to CD, we're going to be there. Because BC, we just showed, is equal to FC. But let's not start with the theorem.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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So this is going to be the same thing. If we want to prove it, if we can prove that FC, the ratio of AB to AD is the same thing as the ratio of FC to CD, we're going to be there. Because BC, we just showed, is equal to FC. But let's not start with the theorem. Let's actually get to the theorem. So FC is parallel to AB, able to set up this one isosceles triangle, show these sides are congruent. Now let's look at some of the other angles here and make ourselves feel good about it.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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But let's not start with the theorem. Let's actually get to the theorem. So FC is parallel to AB, able to set up this one isosceles triangle, show these sides are congruent. Now let's look at some of the other angles here and make ourselves feel good about it. Well, if we look at triangle ABD, so this triangle right over here, and triangle FDC, we already established that they have one set of angles that are the same. And then they also both, ABD has this angle right over here, which is a vertical angle with this one over here. So they're congruent.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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Now let's look at some of the other angles here and make ourselves feel good about it. Well, if we look at triangle ABD, so this triangle right over here, and triangle FDC, we already established that they have one set of angles that are the same. And then they also both, ABD has this angle right over here, which is a vertical angle with this one over here. So they're congruent. And we know if two triangles have two angles that are the same, actually, the third one's going to be the same as well. Or you could say by the angle-angle similarity postulate, these two triangles are similar. So let me write that down.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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So they're congruent. And we know if two triangles have two angles that are the same, actually, the third one's going to be the same as well. Or you could say by the angle-angle similarity postulate, these two triangles are similar. So let me write that down. You want to make sure you get the corresponding sides right. We now know by angle-angle, and I'm going to start at the green angle, that triangle B, and then the blue angle, BDA is similar to triangle, so once again, let's start with the green angle F, then you go to the blue angle FDC. And here, we want to eventually get to the angle bisector theorem, so we want to look at the ratio between AB and AD.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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So let me write that down. You want to make sure you get the corresponding sides right. We now know by angle-angle, and I'm going to start at the green angle, that triangle B, and then the blue angle, BDA is similar to triangle, so once again, let's start with the green angle F, then you go to the blue angle FDC. And here, we want to eventually get to the angle bisector theorem, so we want to look at the ratio between AB and AD. Similar triangles, either you could find the ratio between corresponding sides are going to be similar triangles, or you could find the ratio between two sides of a similar triangle and compare them to the ratio of the same two corresponding sides on the other similar triangle, and they should be the same. So by similar triangles, we know that the ratio of AB, and this, by the way, was by angle-angle similarity. Want to write that down.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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And here, we want to eventually get to the angle bisector theorem, so we want to look at the ratio between AB and AD. Similar triangles, either you could find the ratio between corresponding sides are going to be similar triangles, or you could find the ratio between two sides of a similar triangle and compare them to the ratio of the same two corresponding sides on the other similar triangle, and they should be the same. So by similar triangles, we know that the ratio of AB, and this, by the way, was by angle-angle similarity. Want to write that down. So now that we know they're similar, we know that the ratio of AB to AD is going to be equal to, and we could even look here for the corresponding sides, the ratio of AB, the corresponding side is going to be CF. It's going to equal CF over AD. AD is the same thing as CD.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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Want to write that down. So now that we know they're similar, we know that the ratio of AB to AD is going to be equal to, and we could even look here for the corresponding sides, the ratio of AB, the corresponding side is going to be CF. It's going to equal CF over AD. AD is the same thing as CD. And so we know the ratio of AB to AD is equal to CF over CD. But we just proved to ourselves, because this is an isosceles triangle, that CF is the same thing as BC right over here. So CF is the same thing as BC.
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Angle bisector theorem proof Special properties and parts of triangles Geometry Khan Academy.mp3
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