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Now you can subtract 30 from both sides, and we are left with 2x is equal to 150. Divide both sides by 2, you get x is equal to 75. So we've figured out that x is equal to 75, and now we're at the home stretch. We need to figure out angle BED. Well, that's going to be angle C, so x is equal to the measure of angle CED. So BED is CED plus BEC. So the 60 degrees plus 75 degrees.
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Problem involving angle derived from square and circle Congruence Geometry Khan Academy.mp3
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We're told that a certain mapping in the xy-plane has the following two properties. Each point on the line y is equal to three x minus two maps to itself. Any point p not on the line maps to a new point p prime in such a way that the perpendicular bisector of the segment p, p prime is the line y is equal to three x minus two. Which of the following statements is true? So is this describing a reflection, a rotation, or a translation? So pause this video and see if you can work through it on your own. All right, so let me just try to visualize this.
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Identifying transformation described with other algebra and geometry concepts.mp3
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Which of the following statements is true? So is this describing a reflection, a rotation, or a translation? So pause this video and see if you can work through it on your own. All right, so let me just try to visualize this. So, and I'll just do a very quick, so if that's my y-axis, and that this right over here is my x-axis, three x minus two might look something like this. The line three x minus two would look something like that. And so what we're saying is, or what they're telling us, is any point on this after the transformation maps to itself.
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Identifying transformation described with other algebra and geometry concepts.mp3
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All right, so let me just try to visualize this. So, and I'll just do a very quick, so if that's my y-axis, and that this right over here is my x-axis, three x minus two might look something like this. The line three x minus two would look something like that. And so what we're saying is, or what they're telling us, is any point on this after the transformation maps to itself. Now that by itself is a pretty good clue that we're likely dealing with a reflection. Because remember, with a reflection, you reflect over a line. But if a point sits on the line, well it's just gonna continue to sit on the line.
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Identifying transformation described with other algebra and geometry concepts.mp3
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And so what we're saying is, or what they're telling us, is any point on this after the transformation maps to itself. Now that by itself is a pretty good clue that we're likely dealing with a reflection. Because remember, with a reflection, you reflect over a line. But if a point sits on the line, well it's just gonna continue to sit on the line. But let's just make sure that the second point is consistent with it being a reflection. So any point p not on the line, so let's see, point p right over here, it maps to a new point p prime in such a way that the perpendicular bisector of p p prime is the line y equals three x minus two. So I need to connect, so the line three x minus two, y is equal to three x minus two, would be the perpendicular bisector of the segment between p and what?
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Identifying transformation described with other algebra and geometry concepts.mp3
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But if a point sits on the line, well it's just gonna continue to sit on the line. But let's just make sure that the second point is consistent with it being a reflection. So any point p not on the line, so let's see, point p right over here, it maps to a new point p prime in such a way that the perpendicular bisector of p p prime is the line y equals three x minus two. So I need to connect, so the line three x minus two, y is equal to three x minus two, would be the perpendicular bisector of the segment between p and what? Well let's see, I'd have to draw a perpendicular line. It would have to have the same length on both sides of the line y equals three x minus two. So p prime would have to be right over there.
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Identifying transformation described with other algebra and geometry concepts.mp3
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So I need to connect, so the line three x minus two, y is equal to three x minus two, would be the perpendicular bisector of the segment between p and what? Well let's see, I'd have to draw a perpendicular line. It would have to have the same length on both sides of the line y equals three x minus two. So p prime would have to be right over there. So once again, this is consistent with being a reflection. P prime is equidistant on the other side of the line as p. So I definitely feel good that this is going to be a reflection right over here. Let's do another example.
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Identifying transformation described with other algebra and geometry concepts.mp3
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So p prime would have to be right over there. So once again, this is consistent with being a reflection. P prime is equidistant on the other side of the line as p. So I definitely feel good that this is going to be a reflection right over here. Let's do another example. So here we are told, and I'll switch my colors up, a certain mapping in the plane has the following two properties. Point O maps to itself. Every point V on a circle C centered at O, maps to a new point W on a circle, on circle C, so that the counterclockwise angle from segment OV to OW measures 137 degrees.
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Identifying transformation described with other algebra and geometry concepts.mp3
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Let's do another example. So here we are told, and I'll switch my colors up, a certain mapping in the plane has the following two properties. Point O maps to itself. Every point V on a circle C centered at O, maps to a new point W on a circle, on circle C, so that the counterclockwise angle from segment OV to OW measures 137 degrees. So is this a reflection, rotation, or translation? Pause this video and try to figure it out on your own. All right, so let's see.
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Identifying transformation described with other algebra and geometry concepts.mp3
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Every point V on a circle C centered at O, maps to a new point W on a circle, on circle C, so that the counterclockwise angle from segment OV to OW measures 137 degrees. So is this a reflection, rotation, or translation? Pause this video and try to figure it out on your own. All right, so let's see. We're talking about circle centered at O. So let's see, let me just say, so I have this point O. It maps to itself on its transformation.
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Identifying transformation described with other algebra and geometry concepts.mp3
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All right, so let's see. We're talking about circle centered at O. So let's see, let me just say, so I have this point O. It maps to itself on its transformation. Now every point V on circle C centered at O, so let's see, let's say this is circle C centered at point O. So I'm gonna try to draw a decent-looking circle here. You get the idea.
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Identifying transformation described with other algebra and geometry concepts.mp3
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It maps to itself on its transformation. Now every point V on circle C centered at O, so let's see, let's say this is circle C centered at point O. So I'm gonna try to draw a decent-looking circle here. You get the idea. This is not the best hand-drawn circle ever, all right? So every point, let's just pick a point V here. So let's say that that is the point V on a circle centered at O, maps to a new point W on the circle C. So maybe it maps to a new point W on, actually, let me keep reading, W on circle C, so that the counterclockwise angle from OV to OW measures 137 degrees.
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Identifying transformation described with other algebra and geometry concepts.mp3
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You get the idea. This is not the best hand-drawn circle ever, all right? So every point, let's just pick a point V here. So let's say that that is the point V on a circle centered at O, maps to a new point W on the circle C. So maybe it maps to a new point W on, actually, let me keep reading, W on circle C, so that the counterclockwise angle from OV to OW measures 137 degrees. Okay, so we wanna know the angle from OV to OW, going counterclockwise, is 137 degrees. So this right over here is 137 degrees. And so this would be the segment OW.
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Identifying transformation described with other algebra and geometry concepts.mp3
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So let's say that that is the point V on a circle centered at O, maps to a new point W on the circle C. So maybe it maps to a new point W on, actually, let me keep reading, W on circle C, so that the counterclockwise angle from OV to OW measures 137 degrees. Okay, so we wanna know the angle from OV to OW, going counterclockwise, is 137 degrees. So this right over here is 137 degrees. And so this would be the segment OW. W would go right over there. And so what this looks like is, well, if we're talking about angles and we are rotating something, this point corresponds to this point. It's essentially the point has been rotated by 137 degrees around point O.
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Identifying transformation described with other algebra and geometry concepts.mp3
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And so this would be the segment OW. W would go right over there. And so what this looks like is, well, if we're talking about angles and we are rotating something, this point corresponds to this point. It's essentially the point has been rotated by 137 degrees around point O. So this right over here is clearly a rotation. This is a rotation. Sometimes reading this language at first is a little bit daunting.
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Identifying transformation described with other algebra and geometry concepts.mp3
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It's essentially the point has been rotated by 137 degrees around point O. So this right over here is clearly a rotation. This is a rotation. Sometimes reading this language at first is a little bit daunting. It was a little bit daunting to me when I first read it. But when you actually just break it down and you actually try to visualize what's going on, you'll say, okay, well, look, they're just taking point V and they're rotating it by 137 degrees around point O. And so this would be a rotation.
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Identifying transformation described with other algebra and geometry concepts.mp3
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Sometimes reading this language at first is a little bit daunting. It was a little bit daunting to me when I first read it. But when you actually just break it down and you actually try to visualize what's going on, you'll say, okay, well, look, they're just taking point V and they're rotating it by 137 degrees around point O. And so this would be a rotation. Let's do one more example. So here we are told, so they're talking about, again, a certain mapping in the XY plane. Each circle O with radius R and centered at X, Y is mapped to a circle O prime with radius R and centered at X plus 11 and then Y minus seven.
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Identifying transformation described with other algebra and geometry concepts.mp3
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And so this would be a rotation. Let's do one more example. So here we are told, so they're talking about, again, a certain mapping in the XY plane. Each circle O with radius R and centered at X, Y is mapped to a circle O prime with radius R and centered at X plus 11 and then Y minus seven. So once again, pause this video. What is this, reflection, rotation, or translation? All right, so you might be tempted, if they're talking about circles like we did in the last example, maybe they're talking about a rotation.
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Identifying transformation described with other algebra and geometry concepts.mp3
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Each circle O with radius R and centered at X, Y is mapped to a circle O prime with radius R and centered at X plus 11 and then Y minus seven. So once again, pause this video. What is this, reflection, rotation, or translation? All right, so you might be tempted, if they're talking about circles like we did in the last example, maybe they're talking about a rotation. But look, what they're really saying is is that if I have a circle, let's say I have a circle right over here, centered right over here, after, so this is X comma Y, centered at X comma Y, it's mapped to a new circle O prime with the same radius. So if this is the radius, it's mapped to a new circle with the same radius, but now it is centered at, now it is centered at X plus 11, so our new X coordinate is gonna be 11 larger, X plus 11, and our Y coordinate is gonna be seven less. But we have the exact same radius.
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Identifying transformation described with other algebra and geometry concepts.mp3
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All right, so you might be tempted, if they're talking about circles like we did in the last example, maybe they're talking about a rotation. But look, what they're really saying is is that if I have a circle, let's say I have a circle right over here, centered right over here, after, so this is X comma Y, centered at X comma Y, it's mapped to a new circle O prime with the same radius. So if this is the radius, it's mapped to a new circle with the same radius, but now it is centered at, now it is centered at X plus 11, so our new X coordinate is gonna be 11 larger, X plus 11, and our Y coordinate is gonna be seven less. But we have the exact same radius. We have the exact same radius. So our circle would still, so we have the exact same radius right over here. So what just happened to this circle?
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Identifying transformation described with other algebra and geometry concepts.mp3
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But we have the exact same radius. We have the exact same radius. So our circle would still, so we have the exact same radius right over here. So what just happened to this circle? Well, we kept the radius the same and we just shifted, we just shifted our center to the right by 11, plus 11, and we shifted it down by seven. We shifted it down by seven. So this is clearly a translation.
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Identifying transformation described with other algebra and geometry concepts.mp3
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We've got two right triangles here. Let's say we also know that they both have an angle whose measure is equal to theta. So angle A is congruent to angle D. What do we now know about these two triangles? Well, for any right, or any triangle, if you know two of the angles, you're going to know the third angle because the sum of the angles of a triangle add up to 180 degrees. So if you have two angles in common, that means you're going to have three angles in common. And if you have three angles in common, you are dealing with similar triangles. Let me make that a little bit clearer.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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Well, for any right, or any triangle, if you know two of the angles, you're going to know the third angle because the sum of the angles of a triangle add up to 180 degrees. So if you have two angles in common, that means you're going to have three angles in common. And if you have three angles in common, you are dealing with similar triangles. Let me make that a little bit clearer. So if this angle is theta, this is 90, they all have to add up to 180 degrees. That means that this angle plus this angle up here have to add up to 90. We've already used up 90 right over here.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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Let me make that a little bit clearer. So if this angle is theta, this is 90, they all have to add up to 180 degrees. That means that this angle plus this angle up here have to add up to 90. We've already used up 90 right over here. So this angle, angle A and angle B need to be complements. So this angle right over here needs to be 90 minus theta. Well, we can use the same logic over here.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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We've already used up 90 right over here. So this angle, angle A and angle B need to be complements. So this angle right over here needs to be 90 minus theta. Well, we can use the same logic over here. We already used up 90 degrees over here. So we have a remaining 90 degrees between theta and that angle. So this angle is going to be 90 degrees minus theta.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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Well, we can use the same logic over here. We already used up 90 degrees over here. So we have a remaining 90 degrees between theta and that angle. So this angle is going to be 90 degrees minus theta. 90 degrees minus theta. You have three congruent, three corresponding angles being congruent, you are dealing with similar triangles. Now, why is that interesting?
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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So this angle is going to be 90 degrees minus theta. 90 degrees minus theta. You have three congruent, three corresponding angles being congruent, you are dealing with similar triangles. Now, why is that interesting? Well, we know from geometry that the ratio of corresponding sides of a similar triangles are always going to be the same. So let's explore the corresponding sides here. Well, the side that jumps out when you're dealing with the right triangles the most is always the hypotenuse.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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Now, why is that interesting? Well, we know from geometry that the ratio of corresponding sides of a similar triangles are always going to be the same. So let's explore the corresponding sides here. Well, the side that jumps out when you're dealing with the right triangles the most is always the hypotenuse. So this right over here is the hypotenuse. This hypotenuse is going to correspond to this hypotenuse right over here. And we could write that down.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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Well, the side that jumps out when you're dealing with the right triangles the most is always the hypotenuse. So this right over here is the hypotenuse. This hypotenuse is going to correspond to this hypotenuse right over here. And we could write that down. This is the hypotenuse of this triangle. This is the hypotenuse of that triangle. Now, this side right over here, side BC, what side does that correspond to?
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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And we could write that down. This is the hypotenuse of this triangle. This is the hypotenuse of that triangle. Now, this side right over here, side BC, what side does that correspond to? Well, if you look at this triangle, you can kind of view it as the side that is opposite this angle theta. So it's opposite. If you go across a triangle, you get there.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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Now, this side right over here, side BC, what side does that correspond to? Well, if you look at this triangle, you can kind of view it as the side that is opposite this angle theta. So it's opposite. If you go across a triangle, you get there. So let's go opposite angle D. If you go opposite angle A, you get to BC. Opposite angle D, you get to EF. You get EF, so it corresponds to this side right over here.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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If you go across a triangle, you get there. So let's go opposite angle D. If you go opposite angle A, you get to BC. Opposite angle D, you get to EF. You get EF, so it corresponds to this side right over here. And then finally, side AC is the one remaining one. We could view it as, well, there's two sides that make up this angle A. One of them is a hypotenuse.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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You get EF, so it corresponds to this side right over here. And then finally, side AC is the one remaining one. We could view it as, well, there's two sides that make up this angle A. One of them is a hypotenuse. We could call this maybe the adjacent side to it. And so D corresponds to A. And so this would be the side that corresponds.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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One of them is a hypotenuse. We could call this maybe the adjacent side to it. And so D corresponds to A. And so this would be the side that corresponds. Now, the whole reason I did that is to leverage that corresponding sides, the ratio between corresponding sides of similar triangles is always going to be the same. So for example, the ratio between BC and the hypotenuse BA, so let me write that down. BC over BA is going to be equal to EF over ED.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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And so this would be the side that corresponds. Now, the whole reason I did that is to leverage that corresponding sides, the ratio between corresponding sides of similar triangles is always going to be the same. So for example, the ratio between BC and the hypotenuse BA, so let me write that down. BC over BA is going to be equal to EF over ED. EF, the length of segment EF over the length of segment ED. Over the length of segment ED. Or we could also write that the length of segment AC, so AC over the hypotenuse, over this triangle's hypotenuse, over AB is equal to DF over DE.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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BC over BA is going to be equal to EF over ED. EF, the length of segment EF over the length of segment ED. Over the length of segment ED. Or we could also write that the length of segment AC, so AC over the hypotenuse, over this triangle's hypotenuse, over AB is equal to DF over DE. Once again, this green side over the orange side, these are similar triangles. They're corresponding to each other. So this is equal to, this is equal to DF over DE.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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Or we could also write that the length of segment AC, so AC over the hypotenuse, over this triangle's hypotenuse, over AB is equal to DF over DE. Once again, this green side over the orange side, these are similar triangles. They're corresponding to each other. So this is equal to, this is equal to DF over DE. Over DE. Or we could also say, we could keep going, but I'll just do another one. Or we could say that the ratio of this side, right over this blue side to the green side, so of this triangle, BC, the length of BC over CA, over CA is going to be the same as the blue, the ratio between these two corresponding sides, the blue over the green.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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So this is equal to, this is equal to DF over DE. Over DE. Or we could also say, we could keep going, but I'll just do another one. Or we could say that the ratio of this side, right over this blue side to the green side, so of this triangle, BC, the length of BC over CA, over CA is going to be the same as the blue, the ratio between these two corresponding sides, the blue over the green. EF over DF. Over DF. And we got all of this from the fact that these are similar triangles.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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Or we could say that the ratio of this side, right over this blue side to the green side, so of this triangle, BC, the length of BC over CA, over CA is going to be the same as the blue, the ratio between these two corresponding sides, the blue over the green. EF over DF. Over DF. And we got all of this from the fact that these are similar triangles. So if this is true for all similar triangles, or this is true for any right triangle that has an angle theta, then those two triangles are going to be similar, and all of these ratios are going to be the same. Well, maybe we can give names to these ratios relative to the angle theta. So from angle theta's point of view, from theta's point of view, I'll write theta right over here, or we can just remember that.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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And we got all of this from the fact that these are similar triangles. So if this is true for all similar triangles, or this is true for any right triangle that has an angle theta, then those two triangles are going to be similar, and all of these ratios are going to be the same. Well, maybe we can give names to these ratios relative to the angle theta. So from angle theta's point of view, from theta's point of view, I'll write theta right over here, or we can just remember that. What are these, what is the ratio of these two sides? Well, from theta's point of view, that blue side is the opposite side, it's opposite, so the opposite side of the right triangle. And then the orange side, we've already labeled the hypotenuse.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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So from angle theta's point of view, from theta's point of view, I'll write theta right over here, or we can just remember that. What are these, what is the ratio of these two sides? Well, from theta's point of view, that blue side is the opposite side, it's opposite, so the opposite side of the right triangle. And then the orange side, we've already labeled the hypotenuse. So from theta's point of view, this is the opposite side over the hypotenuse. And I keep stating from theta's point of view, because that wouldn't be the case for this other angle, for angle B. From angle B's point of view, this is the adjacent side over the hypotenuse.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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And then the orange side, we've already labeled the hypotenuse. So from theta's point of view, this is the opposite side over the hypotenuse. And I keep stating from theta's point of view, because that wouldn't be the case for this other angle, for angle B. From angle B's point of view, this is the adjacent side over the hypotenuse. And we'll think about that relationship later on. But let's just all think of it from theta's point of view right over here. So from theta's point of view, what is this?
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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From angle B's point of view, this is the adjacent side over the hypotenuse. And we'll think about that relationship later on. But let's just all think of it from theta's point of view right over here. So from theta's point of view, what is this? Well, theta's right over here. Clearly, A, B, and D, E, A, B, and D, E are still the hypotenuses, hypotenai, I don't know how to say that in plural again. And what is AC?
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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So from theta's point of view, what is this? Well, theta's right over here. Clearly, A, B, and D, E, A, B, and D, E are still the hypotenuses, hypotenai, I don't know how to say that in plural again. And what is AC? Or what is AC and what are DF? Well, these are adjacent to it. They're one of the sides, two sides that make up this angle that is not the hypotenuse.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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And what is AC? Or what is AC and what are DF? Well, these are adjacent to it. They're one of the sides, two sides that make up this angle that is not the hypotenuse. So this we can view as the ratio in either of these triangles between the adjacent side. So this is relative, once again, this is opposite angle B, but we're only thinking about angle A right here, or the angle that measures theta, or angle D right over here. Relative to angle A, AC is adjacent.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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They're one of the sides, two sides that make up this angle that is not the hypotenuse. So this we can view as the ratio in either of these triangles between the adjacent side. So this is relative, once again, this is opposite angle B, but we're only thinking about angle A right here, or the angle that measures theta, or angle D right over here. Relative to angle A, AC is adjacent. Relative to angle D, DF is adjacent. So this ratio right over here is the adjacent over the hypotenuse. And it's going to be the same for any right triangle that has an angle theta in it.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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Relative to angle A, AC is adjacent. Relative to angle D, DF is adjacent. So this ratio right over here is the adjacent over the hypotenuse. And it's going to be the same for any right triangle that has an angle theta in it. And then finally, this over here, this is going to be the opposite side. Once again, this was the opposite side over here. This ratio for either right triangle is going to be the opposite side over the adjacent side.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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And it's going to be the same for any right triangle that has an angle theta in it. And then finally, this over here, this is going to be the opposite side. Once again, this was the opposite side over here. This ratio for either right triangle is going to be the opposite side over the adjacent side. Over the adjacent side. And I really wanna stress the importance, and we're gonna do many, many more examples of this to make this very concrete. But for any right triangle that has an angle theta, the ratio between its opposite side and its hypotenuse is going to be the same.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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This ratio for either right triangle is going to be the opposite side over the adjacent side. Over the adjacent side. And I really wanna stress the importance, and we're gonna do many, many more examples of this to make this very concrete. But for any right triangle that has an angle theta, the ratio between its opposite side and its hypotenuse is going to be the same. That comes out of similar triangles, we just explored that. The ratio between the adjacent side to that angle that is theta and the hypotenuse is going to be the same for any of these triangles as long as it has that angle theta in it. And the ratio relative to the angle theta between the opposite side and the adjacent side, between the blue side and the green side, is always going to be the same.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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But for any right triangle that has an angle theta, the ratio between its opposite side and its hypotenuse is going to be the same. That comes out of similar triangles, we just explored that. The ratio between the adjacent side to that angle that is theta and the hypotenuse is going to be the same for any of these triangles as long as it has that angle theta in it. And the ratio relative to the angle theta between the opposite side and the adjacent side, between the blue side and the green side, is always going to be the same. These are similar triangles. So given that, mathematicians decided to give these things names. Relative to the angle theta, this ratio is always going to be the same.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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And the ratio relative to the angle theta between the opposite side and the adjacent side, between the blue side and the green side, is always going to be the same. These are similar triangles. So given that, mathematicians decided to give these things names. Relative to the angle theta, this ratio is always going to be the same. So they call this, the opposite of our hypotenuse, they call this the sine of the angle theta. So this is the sine, let me do this in a new color. This is by definition, and we're going to extend this definition in the future, this is sine of theta.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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Relative to the angle theta, this ratio is always going to be the same. So they call this, the opposite of our hypotenuse, they call this the sine of the angle theta. So this is the sine, let me do this in a new color. This is by definition, and we're going to extend this definition in the future, this is sine of theta. This right over here, by definition, is the cosine of theta. And this right over here, by definition, is the tangent. That, by definition, is the tangent of theta.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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This is by definition, and we're going to extend this definition in the future, this is sine of theta. This right over here, by definition, is the cosine of theta. And this right over here, by definition, is the tangent. That, by definition, is the tangent of theta. And a mnemonic that will help you remember this, and these really are just definitions. People realize, wow, by similar triangles, for any angle theta, this ratio is always going to be the same. Because of similar triangles, for any angle theta, this ratio is always going to be same.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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That, by definition, is the tangent of theta. And a mnemonic that will help you remember this, and these really are just definitions. People realize, wow, by similar triangles, for any angle theta, this ratio is always going to be the same. Because of similar triangles, for any angle theta, this ratio is always going to be same. This ratio is always going to be same, so let's make these definitions. And to help us remember it, there's the mnemonic SOH CAH TOA. So I'll write it like this.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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Because of similar triangles, for any angle theta, this ratio is always going to be same. This ratio is always going to be same, so let's make these definitions. And to help us remember it, there's the mnemonic SOH CAH TOA. So I'll write it like this. SOH, SOH, SOH is sine is opposite of our hypotenuse, CAH, cosine, cosine is adjacent over hypotenuse, cosine is adjacent over hypotenuse. And then finally, tangent, tangent is opposite over adjacent. Tangent is opposite, opposite over, opposite over adjacent.
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Similarity to define sine, cosine, and tangent Basic trigonometry Trigonometry Khan Academy.mp3
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What we have drawn over here is five different triangles. What I want to do in this video is figure out which of these triangles are congruent to which other of these triangles. And to figure that out, I'm just over here going to write our triangle congruency postulate. So we know that two triangles are congruent if all of their sides are the same, so side, side, side. We also know they are congruent if we have a side and then an angle between the sides, and then another side that is congruent, so side, angle, side. If we reverse the angles on the sides, we know that's also a congruence postulate. So if we have an angle and then another angle, and then the side in between them is congruent, then we also have two congruent triangles.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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So we know that two triangles are congruent if all of their sides are the same, so side, side, side. We also know they are congruent if we have a side and then an angle between the sides, and then another side that is congruent, so side, angle, side. If we reverse the angles on the sides, we know that's also a congruence postulate. So if we have an angle and then another angle, and then the side in between them is congruent, then we also have two congruent triangles. And then finally, if we have an angle and then another angle and then a side, then that is also. Any of these imply congruency. So let's see our congruent triangles.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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So if we have an angle and then another angle, and then the side in between them is congruent, then we also have two congruent triangles. And then finally, if we have an angle and then another angle and then a side, then that is also. Any of these imply congruency. So let's see our congruent triangles. So let's see what we can figure out right over here for these triangles. So right in this triangle ABC over here, we're given this length 7, then 60 degrees, and then 40 degrees. Or another way to think about it, we're given an angle, an angle, and a side.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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So let's see our congruent triangles. So let's see what we can figure out right over here for these triangles. So right in this triangle ABC over here, we're given this length 7, then 60 degrees, and then 40 degrees. Or another way to think about it, we're given an angle, an angle, and a side. 40 degrees, then 60 degrees, then 7. And in order for something to be congruent here, they would have to have an angle, angle, side given, at least unless maybe we have to figure it out some other way. But I'm guessing for this problem, they'll just already give us the angle.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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Or another way to think about it, we're given an angle, an angle, and a side. 40 degrees, then 60 degrees, then 7. And in order for something to be congruent here, they would have to have an angle, angle, side given, at least unless maybe we have to figure it out some other way. But I'm guessing for this problem, they'll just already give us the angle. So they'll have to have an angle, an angle, side. And it can't just be any angle, angle, and side. It has to be 40, 60, and 7.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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But I'm guessing for this problem, they'll just already give us the angle. So they'll have to have an angle, an angle, side. And it can't just be any angle, angle, and side. It has to be 40, 60, and 7. And it has to be in the same order. It can't be 60 and then 40 and then 7. If the 40 degree side has, if one of its side has the length 7, then that is not the same thing here.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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It has to be 40, 60, and 7. And it has to be in the same order. It can't be 60 and then 40 and then 7. If the 40 degree side has, if one of its side has the length 7, then that is not the same thing here. Here the 60 degree side has length 7. So let's see if any of these other triangles have this kind of 40, 60 degrees, and then the 7 right over here. So this has the 40 degrees and the 60 degrees, but the 7 is in between them.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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If the 40 degree side has, if one of its side has the length 7, then that is not the same thing here. Here the 60 degree side has length 7. So let's see if any of these other triangles have this kind of 40, 60 degrees, and then the 7 right over here. So this has the 40 degrees and the 60 degrees, but the 7 is in between them. So this looks like it might be congruent to some other triangle, maybe closer to something like angle, side, angle, because they have an angle, side, angle. So it wouldn't be that one. This one looks interesting.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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So this has the 40 degrees and the 60 degrees, but the 7 is in between them. So this looks like it might be congruent to some other triangle, maybe closer to something like angle, side, angle, because they have an angle, side, angle. So it wouldn't be that one. This one looks interesting. This is also angle, side, angle. So maybe these are congruent, but we'll check back on that. We're still focused on this one right over here.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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This one looks interesting. This is also angle, side, angle. So maybe these are congruent, but we'll check back on that. We're still focused on this one right over here. This one we have a 60 degrees, then a 40 degrees, and a 7. This is tempting. We have an angle, an angle, and a side.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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We're still focused on this one right over here. This one we have a 60 degrees, then a 40 degrees, and a 7. This is tempting. We have an angle, an angle, and a side. But the angles are in a different order. Here it's 40, 60, 7. Here it's 60, 40, 7.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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We have an angle, an angle, and a side. But the angles are in a different order. Here it's 40, 60, 7. Here it's 60, 40, 7. So it's an angle, an angle, and side, but the side is not on the 60 degree angle. It's on the 40 degree angle over here. So this doesn't look right either.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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Here it's 60, 40, 7. So it's an angle, an angle, and side, but the side is not on the 60 degree angle. It's on the 40 degree angle over here. So this doesn't look right either. Here we have 40 degrees, 60 degrees, and then 7. So this is looking pretty good. We have this side right over here is congruent to this side right over here.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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So this doesn't look right either. Here we have 40 degrees, 60 degrees, and then 7. So this is looking pretty good. We have this side right over here is congruent to this side right over here. Then you have your 60 degree angle right over here. 60 degree angle over here. It might not be obvious, because it's flipped, and they're drawn a little bit different.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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We have this side right over here is congruent to this side right over here. Then you have your 60 degree angle right over here. 60 degree angle over here. It might not be obvious, because it's flipped, and they're drawn a little bit different. But you should never assume that just the drawing tells you what's going on. And then finally, you have your 40 degree angle here, which is your 40 degree angle here. So we can say, we can write down that, and I'll do it.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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It might not be obvious, because it's flipped, and they're drawn a little bit different. But you should never assume that just the drawing tells you what's going on. And then finally, you have your 40 degree angle here, which is your 40 degree angle here. So we can say, we can write down that, and I'll do it. Let me think of a good place to do it. I'll write it right over here. We can write down that triangle ABC is congruent to triangle.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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So we can say, we can write down that, and I'll do it. Let me think of a good place to do it. I'll write it right over here. We can write down that triangle ABC is congruent to triangle. Now we have to be very careful with how we name this. We have to make sure that we have the corresponding vertices map up together. So for example, we started this triangle at vertex A.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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We can write down that triangle ABC is congruent to triangle. Now we have to be very careful with how we name this. We have to make sure that we have the corresponding vertices map up together. So for example, we started this triangle at vertex A. So point A right over here, that's where we have the 60 degree angle. That's the vertex of the 60 degree angle. So the vertex of the 60 degree angle over here is point N. So I'm going to go to N. And then we went from A to B.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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So for example, we started this triangle at vertex A. So point A right over here, that's where we have the 60 degree angle. That's the vertex of the 60 degree angle. So the vertex of the 60 degree angle over here is point N. So I'm going to go to N. And then we went from A to B. B was the side, was the vertex that we did not have any angle for. And we could figure it out. If these two guys add up to 100, then this is going to be the 80 degree angle.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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So the vertex of the 60 degree angle over here is point N. So I'm going to go to N. And then we went from A to B. B was the side, was the vertex that we did not have any angle for. And we could figure it out. If these two guys add up to 100, then this is going to be the 80 degree angle. So over here, the 80 degree angle is going to be M, the one that we don't have any label for. It's kind of the other side. It's the thing that shares the 7 length side right over here.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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If these two guys add up to 100, then this is going to be the 80 degree angle. So over here, the 80 degree angle is going to be M, the one that we don't have any label for. It's kind of the other side. It's the thing that shares the 7 length side right over here. So then we want to go to N, then M, and then finish up the triangle in O. And I want to really stress this, that we have to make sure we get the order of these right. Because then we're kind of referring to, we're not showing the corresponding vertices in each triangle.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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It's the thing that shares the 7 length side right over here. So then we want to go to N, then M, and then finish up the triangle in O. And I want to really stress this, that we have to make sure we get the order of these right. Because then we're kind of referring to, we're not showing the corresponding vertices in each triangle. So now we see vertex A, or point A, maps to point N on this congruent triangle. Vertex B maps to point M. And so you can say, look, AB, the length of AB, is congruent to NM. So it all matches up.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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Because then we're kind of referring to, we're not showing the corresponding vertices in each triangle. So now we see vertex A, or point A, maps to point N on this congruent triangle. Vertex B maps to point M. And so you can say, look, AB, the length of AB, is congruent to NM. So it all matches up. And we can say that these two are congruent by angle, angle side, by AAS. So we did this one is, this one right over here is congruent to this one right over there. Now let's look at these two characters.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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So it all matches up. And we can say that these two are congruent by angle, angle side, by AAS. So we did this one is, this one right over here is congruent to this one right over there. Now let's look at these two characters. So here we have an angle, 40 degrees, a side in between, and then another angle. So it looks like ASA is going to be involved. We look at this one right over here.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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Now let's look at these two characters. So here we have an angle, 40 degrees, a side in between, and then another angle. So it looks like ASA is going to be involved. We look at this one right over here. We have a 40 degrees, 40 degrees, 7, and then 60. And you might say, wait, here the 40 degrees is on the bottom, and here it's on the top. But remember, things can be congruent if you can flip them.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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We look at this one right over here. We have a 40 degrees, 40 degrees, 7, and then 60. And you might say, wait, here the 40 degrees is on the bottom, and here it's on the top. But remember, things can be congruent if you can flip them. If you can flip them, rotate them, shift them, whatever. So if you flip this guy over, you will get this one over here. And that would not have happened if you had flipped this one to get this one over here.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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But remember, things can be congruent if you can flip them. If you can flip them, rotate them, shift them, whatever. So if you flip this guy over, you will get this one over here. And that would not have happened if you had flipped this one to get this one over here. So you see, these two by, let me just make it clear. You have the 60 degree angle is congruent to this 60 degree angle. You have the side of length 7 is congruent to this side of length 7.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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And that would not have happened if you had flipped this one to get this one over here. So you see, these two by, let me just make it clear. You have the 60 degree angle is congruent to this 60 degree angle. You have the side of length 7 is congruent to this side of length 7. And then you have the 40 degree angle is congruent to this 40 degree angle. So once again, these two characters are congruent to each other. And we can write, I'll write it right over here, we can say triangle DEF is congruent to triangle.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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You have the side of length 7 is congruent to this side of length 7. And then you have the 40 degree angle is congruent to this 40 degree angle. So once again, these two characters are congruent to each other. And we can write, I'll write it right over here, we can say triangle DEF is congruent to triangle. And here we have to be careful again. Point D is the vertex for the 60 degree side. So I'm going to start at H, which is the vertex of the 60 degree side over here.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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And we can write, I'll write it right over here, we can say triangle DEF is congruent to triangle. And here we have to be careful again. Point D is the vertex for the 60 degree side. So I'm going to start at H, which is the vertex of the 60 degree side over here. It's congruent to triangle H. And then we went from D to E. E is the vertex on the 40 degree side, kind of the other vertex that shares the 7 length segment right over here. We want to go from H to G. HGI. And we know that from angle side angle.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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So I'm going to start at H, which is the vertex of the 60 degree side over here. It's congruent to triangle H. And then we went from D to E. E is the vertex on the 40 degree side, kind of the other vertex that shares the 7 length segment right over here. We want to go from H to G. HGI. And we know that from angle side angle. By angle side angle. So that gives us that character right over there is congruent to this character right over here. And then finally, we're left with this poor chap.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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And we know that from angle side angle. By angle side angle. So that gives us that character right over there is congruent to this character right over here. And then finally, we're left with this poor chap. And it looks like it is not congruent to any of them. It is tempting to try to match it up to this one, especially because the angles here on the bottom, and you have the 7 side over here, angles here on the bottom, and you have the 7 side over here. But it doesn't match up.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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And then finally, we're left with this poor chap. And it looks like it is not congruent to any of them. It is tempting to try to match it up to this one, especially because the angles here on the bottom, and you have the 7 side over here, angles here on the bottom, and you have the 7 side over here. But it doesn't match up. Because the order of the angles aren't the same. You don't have the same corresponding angles. If you try to do this little exercise where you map everything to each other, you wouldn't be able to do it right over here.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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But it doesn't match up. Because the order of the angles aren't the same. You don't have the same corresponding angles. If you try to do this little exercise where you map everything to each other, you wouldn't be able to do it right over here. And this over here, it might have been a trick question where maybe if you did the math, if this was like a 40 or 60 degree angle, then maybe you could have matched this to some of the other triangles, or maybe even some of them to each other. But this last angle in all of these cases, 40 plus 60 is 100. This is going to be an 80 degree angle right over here.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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If you try to do this little exercise where you map everything to each other, you wouldn't be able to do it right over here. And this over here, it might have been a trick question where maybe if you did the math, if this was like a 40 or 60 degree angle, then maybe you could have matched this to some of the other triangles, or maybe even some of them to each other. But this last angle in all of these cases, 40 plus 60 is 100. This is going to be an 80 degree angle right over here. To add up to 180, this is an 80 degree angle. If this ended up by the math being a 40 or 60 degree angle, then it could have been a little bit more interesting. There might have been other congruent pairs.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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This is going to be an 80 degree angle right over here. To add up to 180, this is an 80 degree angle. If this ended up by the math being a 40 or 60 degree angle, then it could have been a little bit more interesting. There might have been other congruent pairs. But this is an 80 degree angle in every case. The other angle is 80 degrees. So this is just a loan, unfortunately, for him.
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Finding congruent triangles Congruence Geometry Khan Academy.mp3
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We're going to pick whether it's a square, rhombus, rectangle, parallelogram, trapezoid, none of the above. And I'm assuming we're going to pick the most specific one possible, because obviously all squares are rhombuses, or rhombi, I guess you'd say. Not all rhombi are squares. All squares are also rectangles. All squares, rhombi, and rectangles are parallelograms. So we want to be as specific as possible in picking this. So let's see, point A is at 1, 6.
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Classifying a quadrilateral on the coordinate plane Analytic geometry Geometry Khan Academy.mp3
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All squares are also rectangles. All squares, rhombi, and rectangles are parallelograms. So we want to be as specific as possible in picking this. So let's see, point A is at 1, 6. So 1, 6. I encourage you to pause this video and actually try this on your own before seeing how I do it, but I'll just proceed. So that's point A right over there.
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Classifying a quadrilateral on the coordinate plane Analytic geometry Geometry Khan Academy.mp3
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So let's see, point A is at 1, 6. So 1, 6. I encourage you to pause this video and actually try this on your own before seeing how I do it, but I'll just proceed. So that's point A right over there. Point B is at negative 5, 2. Negative 5, 2. That's point B.
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Classifying a quadrilateral on the coordinate plane Analytic geometry Geometry Khan Academy.mp3
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So that's point A right over there. Point B is at negative 5, 2. Negative 5, 2. That's point B. Point C is at some carbonated water, so some air is coming out. Point C is at negative 7, 8. Negative 7, 8.
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Classifying a quadrilateral on the coordinate plane Analytic geometry Geometry Khan Academy.mp3
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That's point B. Point C is at some carbonated water, so some air is coming out. Point C is at negative 7, 8. Negative 7, 8. So that is point C right over there. And then finally, point D is at 2, 11. 2, 11.
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Classifying a quadrilateral on the coordinate plane Analytic geometry Geometry Khan Academy.mp3
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Negative 7, 8. So that is point C right over there. And then finally, point D is at 2, 11. 2, 11. And actually, that kind of goes off the screen. This is 10, 11 would be right like this. So that would be 2, 11.
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Classifying a quadrilateral on the coordinate plane Analytic geometry Geometry Khan Academy.mp3
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2, 11. And actually, that kind of goes off the screen. This is 10, 11 would be right like this. So that would be 2, 11. If we were to extend this, this is 10, and this is 11 right up here. 2, 11. So let's see what this quadrilateral looks like.
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Classifying a quadrilateral on the coordinate plane Analytic geometry Geometry Khan Academy.mp3
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So that would be 2, 11. If we were to extend this, this is 10, and this is 11 right up here. 2, 11. So let's see what this quadrilateral looks like. You have this line right over here, this line right over there, that line right over there. And then you have this line like this, like this. And then you have this like this.
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Classifying a quadrilateral on the coordinate plane Analytic geometry Geometry Khan Academy.mp3
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So let's see what this quadrilateral looks like. You have this line right over here, this line right over there, that line right over there. And then you have this line like this, like this. And then you have this like this. So right off the bat, it's definitely a quadrilateral. I have four sides. But the key question, are any of these sides parallel to any of the other sides?
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Classifying a quadrilateral on the coordinate plane Analytic geometry Geometry Khan Academy.mp3
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