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I can find this case that breaks down angle-angle-angle. If these work, just try to verify for yourself that they make logical sense why they would imply congruency. Now let's try angle-angle-side. Let's try angle-angle-side. So once again, let's have a triangle over here. It has some side. So this one's going to be a little bit more interesting.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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Let's try angle-angle-side. So once again, let's have a triangle over here. It has some side. So this one's going to be a little bit more interesting. So it has some side. That's the side right over there. And then it has two angles.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So this one's going to be a little bit more interesting. So it has some side. That's the side right over there. And then it has two angles. So let me draw the other sides of this triangle. I'll draw one in magenta and then one in green. And there's two angles and then the side.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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And then it has two angles. So let me draw the other sides of this triangle. I'll draw one in magenta and then one in green. And there's two angles and then the side. So let's say you have this angle. You have that angle right over there. Actually, I didn't have to put a double because that's the first angle that I'm.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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And there's two angles and then the side. So let's say you have this angle. You have that angle right over there. Actually, I didn't have to put a double because that's the first angle that I'm. So I have that angle, which we refer to as that first a. Then we have this angle, which is that second a. So if I know that there's another triangle that has one side having the same length, so let me draw it like that.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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Actually, I didn't have to put a double because that's the first angle that I'm. So I have that angle, which we refer to as that first a. Then we have this angle, which is that second a. So if I know that there's another triangle that has one side having the same length, so let me draw it like that. It has one side having the same length. It has one angle on that side that has the same measure. So it has a measure like that.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So if I know that there's another triangle that has one side having the same length, so let me draw it like that. It has one side having the same length. It has one angle on that side that has the same measure. So it has a measure like that. And so this side right over here could be of any length. We aren't constraining what the length of that side is. But whatever the angle is on the other side of that side is going to be the same as this green angle right over here.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So it has a measure like that. And so this side right over here could be of any length. We aren't constraining what the length of that side is. But whatever the angle is on the other side of that side is going to be the same as this green angle right over here. So for example, it could be like that. And then you could have a green side go like that. It could be like that and have the green side go like that.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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But whatever the angle is on the other side of that side is going to be the same as this green angle right over here. So for example, it could be like that. And then you could have a green side go like that. It could be like that and have the green side go like that. And if we have, so the only thing we're assuming is that this is the same length as this, and that this angle is the same measure as that angle, and that this measure is the same measure as that angle. And this magenta line can be of any length, and this green line can be of any length. We in no way have constrained that.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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It could be like that and have the green side go like that. And if we have, so the only thing we're assuming is that this is the same length as this, and that this angle is the same measure as that angle, and that this measure is the same measure as that angle. And this magenta line can be of any length, and this green line can be of any length. We in no way have constrained that. But can we form any triangle that is not congruent to this? Because the bottom line is this green line is going to have to touch this one right over there. And the only way it's going to touch that one right over there is if it starts right over here.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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We in no way have constrained that. But can we form any triangle that is not congruent to this? Because the bottom line is this green line is going to have to touch this one right over there. And the only way it's going to touch that one right over there is if it starts right over here. Because we're constraining this angle right over here. We're constraining that angle. And so it looks like angle-angle side does indeed imply congruency.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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And the only way it's going to touch that one right over there is if it starts right over here. Because we're constraining this angle right over here. We're constraining that angle. And so it looks like angle-angle side does indeed imply congruency. So that does imply congruency. So let's just do one more just to kind of try out all of the different situations. What if we have, and I'm running out of a little bit of real estate right over here at the bottom.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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And so it looks like angle-angle side does indeed imply congruency. So that does imply congruency. So let's just do one more just to kind of try out all of the different situations. What if we have, and I'm running out of a little bit of real estate right over here at the bottom. What if we tried out side-side angle? So once again, draw a triangle. So it has one side there.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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What if we have, and I'm running out of a little bit of real estate right over here at the bottom. What if we tried out side-side angle? So once again, draw a triangle. So it has one side there. It has another side there. And then I don't have to do those hash marks just yet. So one side, then another side, and then another side.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So it has one side there. It has another side there. And then I don't have to do those hash marks just yet. So one side, then another side, and then another side. And what happens if we know that there's another triangle that has two of the sides the same, and then the angle after it? So for example, it would have that side just like that. And then it has another side, but we're not constraining the angle.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So one side, then another side, and then another side. And what happens if we know that there's another triangle that has two of the sides the same, and then the angle after it? So for example, it would have that side just like that. And then it has another side, but we're not constraining the angle. We aren't constraining this angle right over here, but we're constraining the length of that side. So let me color code it. So that blue side is that first side.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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And then it has another side, but we're not constraining the angle. We aren't constraining this angle right over here, but we're constraining the length of that side. So let me color code it. So that blue side is that first side. Then we have this magenta side right over there. So this is going to be the same length as this right over here. But let me make it at a different angle to see if I can disprove it.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So that blue side is that first side. Then we have this magenta side right over there. So this is going to be the same length as this right over here. But let me make it at a different angle to see if I can disprove it. So let's say it looks like that. Or actually, let me make it even more interesting. Let me try to make it like that.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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But let me make it at a different angle to see if I can disprove it. So let's say it looks like that. Or actually, let me make it even more interesting. Let me try to make it like that. So it's a very different angle. But now it has to have the same angle out here. It has to have that same angle out here.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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Let me try to make it like that. So it's a very different angle. But now it has to have the same angle out here. It has to have that same angle out here. So it has to be roughly that angle. So it actually looks like we can draw a triangle that is not congruent, that has the same two sides being the same length, and then an angle is different. For example, this is pretty much that.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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It has to have that same angle out here. So it has to be roughly that angle. So it actually looks like we can draw a triangle that is not congruent, that has the same two sides being the same length, and then an angle is different. For example, this is pretty much that. I made this angle smaller than this angle. These two sides are the same. This angle is the same now, but what the byproduct of that is that this green side is going to be shorter on this triangle right over here.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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For example, this is pretty much that. I made this angle smaller than this angle. These two sides are the same. This angle is the same now, but what the byproduct of that is that this green side is going to be shorter on this triangle right over here. So you don't necessarily have congruent triangles with side-side angle. So this is not necessarily congruent, or similar. It gives us neither congruency nor similarity.
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Other triangle congruence postulates Congruence Geometry Khan Academy.mp3
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So in algebra, when something is equal to another thing, it means that their quantities are the same. But if we're now all of a sudden talking about shapes, and we say that those shapes are the same, the shapes are the same size and shape, then we say that they're congruent. And just to see a simple example here, I have this triangle right over there. And let's say I have this triangle right over here. And if you are able to shift this triangle and rotate this triangle and flip this triangle, you can make it look exactly like this triangle, as long as you're not changing the lengths of any of the sides or the angles here. But you can flip it. You can shift it.
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Corresponding parts of congruent triangles are congruent.mp3
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And let's say I have this triangle right over here. And if you are able to shift this triangle and rotate this triangle and flip this triangle, you can make it look exactly like this triangle, as long as you're not changing the lengths of any of the sides or the angles here. But you can flip it. You can shift it. You can do any and rotate it. So you can shift. Let me write this.
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Corresponding parts of congruent triangles are congruent.mp3
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You can shift it. You can do any and rotate it. So you can shift. Let me write this. You can shift it. You can flip it. You can flip it.
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Corresponding parts of congruent triangles are congruent.mp3
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Let me write this. You can shift it. You can flip it. You can flip it. And you can rotate. If you can do those three procedures to make the exact same triangle, to make them look exactly the same, then they are congruent. And if you say that a triangle is congruent, and let me label these.
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Corresponding parts of congruent triangles are congruent.mp3
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You can flip it. And you can rotate. If you can do those three procedures to make the exact same triangle, to make them look exactly the same, then they are congruent. And if you say that a triangle is congruent, and let me label these. So let's call this triangle A, B, and C. And let's call this D. Let me call it X, Y, and Z. So if we were to say, if we make the claim that both of these triangles are congruent, so if we say triangle A, B, C is congruent, and the way you specified it, it looks almost like an equal sign. But it's an equal sign with this little curly thing on top.
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Corresponding parts of congruent triangles are congruent.mp3
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And if you say that a triangle is congruent, and let me label these. So let's call this triangle A, B, and C. And let's call this D. Let me call it X, Y, and Z. So if we were to say, if we make the claim that both of these triangles are congruent, so if we say triangle A, B, C is congruent, and the way you specified it, it looks almost like an equal sign. But it's an equal sign with this little curly thing on top. Let me write it a little bit neater. So we would write it like this. If we know that triangle A, B, C is congruent to triangle X, Y, Z, that means that their corresponding sides have the same length and their corresponding angles and their corresponding angles have the same measure.
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Corresponding parts of congruent triangles are congruent.mp3
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But it's an equal sign with this little curly thing on top. Let me write it a little bit neater. So we would write it like this. If we know that triangle A, B, C is congruent to triangle X, Y, Z, that means that their corresponding sides have the same length and their corresponding angles and their corresponding angles have the same measure. So if we make this assumption, or if someone tells us that this is true, then we know, for example, that AB is going to be equal to XY. The length of segment AB is going to be equal to the length of segment XY. And we could denote it like this.
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Corresponding parts of congruent triangles are congruent.mp3
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If we know that triangle A, B, C is congruent to triangle X, Y, Z, that means that their corresponding sides have the same length and their corresponding angles and their corresponding angles have the same measure. So if we make this assumption, or if someone tells us that this is true, then we know, for example, that AB is going to be equal to XY. The length of segment AB is going to be equal to the length of segment XY. And we could denote it like this. And I'm assuming that these are the corresponding sides. And you can see it, actually, by the way we've defined these triangles. A corresponds to X, B corresponds to Y, and then C corresponds to Z right over there.
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Corresponding parts of congruent triangles are congruent.mp3
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And we could denote it like this. And I'm assuming that these are the corresponding sides. And you can see it, actually, by the way we've defined these triangles. A corresponds to X, B corresponds to Y, and then C corresponds to Z right over there. So AB, side AB, is going to have the same length as side XY. And you can sometimes, if you don't have the colors, you would denote it just like that. These two lengths are, or these two line segments have the same length.
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Corresponding parts of congruent triangles are congruent.mp3
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A corresponds to X, B corresponds to Y, and then C corresponds to Z right over there. So AB, side AB, is going to have the same length as side XY. And you can sometimes, if you don't have the colors, you would denote it just like that. These two lengths are, or these two line segments have the same length. And you can actually say this, and you don't always see it written this way. You could also make the statement that line segment AB is congruent to line segment XY. But congruence of line segments really just means that their lengths are equivalent.
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Corresponding parts of congruent triangles are congruent.mp3
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These two lengths are, or these two line segments have the same length. And you can actually say this, and you don't always see it written this way. You could also make the statement that line segment AB is congruent to line segment XY. But congruence of line segments really just means that their lengths are equivalent. So these two things mean the same thing. If one line segment is congruent to another line segment, that just means the measure of one line segment is equal to the measure of the other line segment. And so we can go through all the corresponding sides.
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Corresponding parts of congruent triangles are congruent.mp3
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But congruence of line segments really just means that their lengths are equivalent. So these two things mean the same thing. If one line segment is congruent to another line segment, that just means the measure of one line segment is equal to the measure of the other line segment. And so we can go through all the corresponding sides. If these two characters are congruent, we also know that BC is going to be the length of YZ, assuming that those are the corresponding sides. And we could put these double hash marks right over here to show that these two lengths are the same. And then if we go to the third side, we also know that these are going to have the same length, or the line segments themselves are going to be congruent.
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Corresponding parts of congruent triangles are congruent.mp3
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And so we can go through all the corresponding sides. If these two characters are congruent, we also know that BC is going to be the length of YZ, assuming that those are the corresponding sides. And we could put these double hash marks right over here to show that these two lengths are the same. And then if we go to the third side, we also know that these are going to have the same length, or the line segments themselves are going to be congruent. So we also know that the length of AC is going to be equal to the length of XZ. Not only do we know that all of the corresponding sides are going to have the same length, if someone tells us that a triangle is congruent, we also know that all the corresponding angles are going to have the same measure. So for example, we also know that this angle's measure is going to be the same as the corresponding angle's measure.
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Corresponding parts of congruent triangles are congruent.mp3
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And then if we go to the third side, we also know that these are going to have the same length, or the line segments themselves are going to be congruent. So we also know that the length of AC is going to be equal to the length of XZ. Not only do we know that all of the corresponding sides are going to have the same length, if someone tells us that a triangle is congruent, we also know that all the corresponding angles are going to have the same measure. So for example, we also know that this angle's measure is going to be the same as the corresponding angle's measure. And the corresponding angle is right over here. It's between this orange side and this blue side, or this orange side and this purple side, I should say. And between the orange side and this purple side.
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Corresponding parts of congruent triangles are congruent.mp3
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So for example, we also know that this angle's measure is going to be the same as the corresponding angle's measure. And the corresponding angle is right over here. It's between this orange side and this blue side, or this orange side and this purple side, I should say. And between the orange side and this purple side. And so it also tells us that the measure of angle is this BAC is equal to the measure of angle of angle YXZ. Let me write that angle symbol a little less. Like a measure of angle YXZ.
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Corresponding parts of congruent triangles are congruent.mp3
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And between the orange side and this purple side. And so it also tells us that the measure of angle is this BAC is equal to the measure of angle of angle YXZ. Let me write that angle symbol a little less. Like a measure of angle YXZ. We could also write that as angle BAC is congruent to angle YXZ. And once again, like line segments, if one line segment is congruent to another line segment, it just means that their lengths are equal. And if one angle is congruent to another angle, it just means that their measures are equal.
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Corresponding parts of congruent triangles are congruent.mp3
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Like a measure of angle YXZ. We could also write that as angle BAC is congruent to angle YXZ. And once again, like line segments, if one line segment is congruent to another line segment, it just means that their lengths are equal. And if one angle is congruent to another angle, it just means that their measures are equal. So we know that those two corresponding angles have the same measure. They're congruent. We also know that these two corresponding angles have the same measure.
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Corresponding parts of congruent triangles are congruent.mp3
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And if one angle is congruent to another angle, it just means that their measures are equal. So we know that those two corresponding angles have the same measure. They're congruent. We also know that these two corresponding angles have the same measure. And I'll use a double arc to specify that this has the same measure as that. So we also know that the measure of angle ABC is equal to the measure of angle XYZ. And then finally, we know that this angle, if we know that these two characters are congruent, that this angle is going to have the same measure as this angle as its corresponding angle.
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Corresponding parts of congruent triangles are congruent.mp3
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We also know that these two corresponding angles have the same measure. And I'll use a double arc to specify that this has the same measure as that. So we also know that the measure of angle ABC is equal to the measure of angle XYZ. And then finally, we know that this angle, if we know that these two characters are congruent, that this angle is going to have the same measure as this angle as its corresponding angle. So we know that the measure of angle ACB is going to be equal to the measure of angle XZY. Now what we're going to concern ourselves a lot with is how do we prove congruence? Because it's cool.
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Corresponding parts of congruent triangles are congruent.mp3
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So the first question I'll ask you, if you do one revolution, if you have an angle that went all the way around once, how many radians is that? Well, we know that that is 2 pi radians. Now, that exact same angle, if we were to measure it in degrees, how many degrees is that? Well, you've heard of people doing a 360, doing one full revolution. That is equal to 360 degrees. Now, can we simplify this? It's important to write this little superscript circle.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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Well, you've heard of people doing a 360, doing one full revolution. That is equal to 360 degrees. Now, can we simplify this? It's important to write this little superscript circle. That's literally the units under question. Sometimes it doesn't look like a unit, but it is a unit. You could literally write degrees instead of that little symbol.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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It's important to write this little superscript circle. That's literally the units under question. Sometimes it doesn't look like a unit, but it is a unit. You could literally write degrees instead of that little symbol. And the units right here, of course, are the word radians. Now, can we simplify this a little bit? Well, sure.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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You could literally write degrees instead of that little symbol. And the units right here, of course, are the word radians. Now, can we simplify this a little bit? Well, sure. Both 2 pi and 360 are divisible by 2, so let's divide things by 2. And if we do that, what do we get for what pi radians are equal to? Well, on the left-hand side here, we're just left with pi radians.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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Well, sure. Both 2 pi and 360 are divisible by 2, so let's divide things by 2. And if we do that, what do we get for what pi radians are equal to? Well, on the left-hand side here, we're just left with pi radians. And on the right-hand side here, 360 divided by 2 is 180, and we have still the units, which are degrees. So we get pi radians are equal to 180 degrees, which actually answered the first part of our question. We wanted to convert pi radians.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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Well, on the left-hand side here, we're just left with pi radians. And on the right-hand side here, 360 divided by 2 is 180, and we have still the units, which are degrees. So we get pi radians are equal to 180 degrees, which actually answered the first part of our question. We wanted to convert pi radians. Well, we just figured out pi radians are equal to 100, 180 degrees. Pi radians are equal to 180 degrees. And if you want to think about it, we know pi radians are halfway around a circle like that, and that's the same thing as 180 degrees.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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We wanted to convert pi radians. Well, we just figured out pi radians are equal to 100, 180 degrees. Pi radians are equal to 180 degrees. And if you want to think about it, we know pi radians are halfway around a circle like that, and that's the same thing as 180 degrees. So now let's think about the second part of it. We want to convert negative pi over 3 radians. Let me do this in a new color.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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And if you want to think about it, we know pi radians are halfway around a circle like that, and that's the same thing as 180 degrees. So now let's think about the second part of it. We want to convert negative pi over 3 radians. Let me do this in a new color. Negative pi over 3. So negative pi over 3 radians. How can we convert that to degrees?
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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Let me do this in a new color. Negative pi over 3. So negative pi over 3 radians. How can we convert that to degrees? What do we get based on this information right over here? Well, to figure this out, we need to know how many degrees there are per radian. If we need to multiply this times degrees, and I'm going to write the word out because if I just wrote a little circle here, it would be hard to visualize that as a unit.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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How can we convert that to degrees? What do we get based on this information right over here? Well, to figure this out, we need to know how many degrees there are per radian. If we need to multiply this times degrees, and I'm going to write the word out because if I just wrote a little circle here, it would be hard to visualize that as a unit. Degrees per radian. So how many degrees are there per radian? Well, we know that for every 180 degrees, we have pi radians.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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If we need to multiply this times degrees, and I'm going to write the word out because if I just wrote a little circle here, it would be hard to visualize that as a unit. Degrees per radian. So how many degrees are there per radian? Well, we know that for every 180 degrees, we have pi radians. Or you could say that there are 180 over pi degrees per radian. And this is going to work out. We have however many radians we have times the number of degrees per radian.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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Well, we know that for every 180 degrees, we have pi radians. Or you could say that there are 180 over pi degrees per radian. And this is going to work out. We have however many radians we have times the number of degrees per radian. So of course, the units are going to work out. Radians cancel out. The pi also cancels out.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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We have however many radians we have times the number of degrees per radian. So of course, the units are going to work out. Radians cancel out. The pi also cancels out. So you're left with negative 180 divided by 3, leaving us with negative 60. And we don't want to forget the units. We could write them out.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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The pi also cancels out. So you're left with negative 180 divided by 3, leaving us with negative 60. And we don't want to forget the units. We could write them out. That's the only units that's left. Degrees, which we could write out. We could write out the word degrees or we could just put that little symbol there.
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Example Converting radians to degrees Trigonometry Khan Academy.mp3
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And to graph a circle, you have to know where its center is, and you have to know what its radius is. So let me see if I can change that. And you have to know what its radius is. So what we need to do is put this in some form where we can pick out its center and its radius. Let me get my little scratch pad out and see if we can do that. So this is that same equation. And what I essentially want to do is I want to complete the square in terms of x and complete the square in terms of y to put it into a form that we can recognize.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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So what we need to do is put this in some form where we can pick out its center and its radius. Let me get my little scratch pad out and see if we can do that. So this is that same equation. And what I essentially want to do is I want to complete the square in terms of x and complete the square in terms of y to put it into a form that we can recognize. So first, let's take all of the x terms. So let's take all of the x terms. So you have x squared and 4x on the left-hand side.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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And what I essentially want to do is I want to complete the square in terms of x and complete the square in terms of y to put it into a form that we can recognize. So first, let's take all of the x terms. So let's take all of the x terms. So you have x squared and 4x on the left-hand side. So I could rewrite this as x squared plus 4x. And I'm going to put some parentheses around here because I want to complete the square. And then I have my y terms.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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So you have x squared and 4x on the left-hand side. So I could rewrite this as x squared plus 4x. And I'm going to put some parentheses around here because I want to complete the square. And then I have my y terms. I'll circle those in. Well, the red looks too much like the purple. I'll circle those in blue.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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And then I have my y terms. I'll circle those in. Well, the red looks too much like the purple. I'll circle those in blue. y squared and negative 4y. So we have plus y squared minus 4y. And then we have a minus 17.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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I'll circle those in blue. y squared and negative 4y. So we have plus y squared minus 4y. And then we have a minus 17. And I'll just do that in a neutral color. So minus 17 is equal to 0. Now, what I want to do is make each of these purple expressions perfect squares.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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And then we have a minus 17. And I'll just do that in a neutral color. So minus 17 is equal to 0. Now, what I want to do is make each of these purple expressions perfect squares. So how could I do that here? Well, this would be a perfect square if I took half of this 4 and I squared it. So if I made this plus 4, then this entire expression would be x plus 2 squared.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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Now, what I want to do is make each of these purple expressions perfect squares. So how could I do that here? Well, this would be a perfect square if I took half of this 4 and I squared it. So if I made this plus 4, then this entire expression would be x plus 2 squared. And you can verify that if you like. If you need review on completing the square, there's plenty of videos on Khan Academy on that. All we did is we took half of this coefficient and then squared it to get 4.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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So if I made this plus 4, then this entire expression would be x plus 2 squared. And you can verify that if you like. If you need review on completing the square, there's plenty of videos on Khan Academy on that. All we did is we took half of this coefficient and then squared it to get 4. Half of 4 is 2 squared to get 4. And that comes straight out of the idea. If you take x plus 2 and square it, it's going to be x squared plus twice the product of 2 and x plus 2 squared.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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All we did is we took half of this coefficient and then squared it to get 4. Half of 4 is 2 squared to get 4. And that comes straight out of the idea. If you take x plus 2 and square it, it's going to be x squared plus twice the product of 2 and x plus 2 squared. Now, we can't just willy-nilly add a 4 here. We had an equality before. Just adding a 4, it wouldn't be equal anymore.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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If you take x plus 2 and square it, it's going to be x squared plus twice the product of 2 and x plus 2 squared. Now, we can't just willy-nilly add a 4 here. We had an equality before. Just adding a 4, it wouldn't be equal anymore. So if we want to maintain the equality, we have to add 4 on the right-hand side as well. Now, let's do the same thing for the y's. Half of this coefficient right over here is a negative 2.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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Just adding a 4, it wouldn't be equal anymore. So if we want to maintain the equality, we have to add 4 on the right-hand side as well. Now, let's do the same thing for the y's. Half of this coefficient right over here is a negative 2. If we square a negative 2, it becomes a positive 4. We can't just do that on the left-hand side. We have to do that on the right-hand side as well.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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Half of this coefficient right over here is a negative 2. If we square a negative 2, it becomes a positive 4. We can't just do that on the left-hand side. We have to do that on the right-hand side as well. Now, what we have in blue becomes y minus 2 squared. And of course, we have the minus 17. But why don't we add 17 to both sides as well to get rid of this minus 17 here?
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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We have to do that on the right-hand side as well. Now, what we have in blue becomes y minus 2 squared. And of course, we have the minus 17. But why don't we add 17 to both sides as well to get rid of this minus 17 here? So let's add 17 on the left and add 17 on the right. So on the left, we're just left with these two expressions. And on the right, we have 4 plus 4 plus 17.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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But why don't we add 17 to both sides as well to get rid of this minus 17 here? So let's add 17 on the left and add 17 on the right. So on the left, we're just left with these two expressions. And on the right, we have 4 plus 4 plus 17. Well, that's 8 plus 17, which is equal to 25. Now, this is a form that we recognize. If you have the form x minus a squared plus y minus b squared is equal to r squared, we know that the center is at the point a comma b, essentially the point that makes both of these equal to 0, and that the radius is going to be r. So if we look over here, what is our a?
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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And on the right, we have 4 plus 4 plus 17. Well, that's 8 plus 17, which is equal to 25. Now, this is a form that we recognize. If you have the form x minus a squared plus y minus b squared is equal to r squared, we know that the center is at the point a comma b, essentially the point that makes both of these equal to 0, and that the radius is going to be r. So if we look over here, what is our a? We have to be careful here. Our a isn't 2. Our a is negative 2. x minus negative 2 is equal to 2.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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If you have the form x minus a squared plus y minus b squared is equal to r squared, we know that the center is at the point a comma b, essentially the point that makes both of these equal to 0, and that the radius is going to be r. So if we look over here, what is our a? We have to be careful here. Our a isn't 2. Our a is negative 2. x minus negative 2 is equal to 2. So the x-coordinate of our center is going to be negative 2. And the y-coordinate of our center is going to be 2. Remember, we care about the x value that makes this 0 and the y value that makes this 0.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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Our a is negative 2. x minus negative 2 is equal to 2. So the x-coordinate of our center is going to be negative 2. And the y-coordinate of our center is going to be 2. Remember, we care about the x value that makes this 0 and the y value that makes this 0. So the center is negative 2 comma 2. And this is the radius squared. So the radius is equal to 5.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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Remember, we care about the x value that makes this 0 and the y value that makes this 0. So the center is negative 2 comma 2. And this is the radius squared. So the radius is equal to 5. So let's go back to the exercise and actually plot this. So it's negative 2 comma 2. So our center is negative 2 comma 2.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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So the radius is equal to 5. So let's go back to the exercise and actually plot this. So it's negative 2 comma 2. So our center is negative 2 comma 2. So that's right over there. x is negative 2. y is positive 2. And the radius is 5.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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So our center is negative 2 comma 2. So that's right over there. x is negative 2. y is positive 2. And the radius is 5. So see, this would be 1, 2, 3, 4, 5. So we have to go a little bit wider than this. My pen is having trouble.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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And the radius is 5. So see, this would be 1, 2, 3, 4, 5. So we have to go a little bit wider than this. My pen is having trouble. There you go. 1, 2, 3, 4, 5. Let's check our answer.
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Completing the square to write equation in standard form of a circle Algebra II Khan Academy.mp3
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So parallel lines are lines that have the same slope, and they're different lines, so they never, ever intersect. So we need to look for different lines that have the exact same slope. And lucky for us, all of these lines are in y equals mx plus b, or slope intercept form. So you can really just look at these lines and figure out their slopes. The slope for line A, m is equal to 2. We see it right over there. For line B, our slope is equal to 3.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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So you can really just look at these lines and figure out their slopes. The slope for line A, m is equal to 2. We see it right over there. For line B, our slope is equal to 3. So these two guys are not parallel. And I'll graph it in a second, and you'll see that. And then finally, for line C, I'll do it in purple, the slope is 2.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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For line B, our slope is equal to 3. So these two guys are not parallel. And I'll graph it in a second, and you'll see that. And then finally, for line C, I'll do it in purple, the slope is 2. So m is equal to 2. I don't know if that purple is too dark for you. So line C and line A have the same slope, but they're different lines.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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And then finally, for line C, I'll do it in purple, the slope is 2. So m is equal to 2. I don't know if that purple is too dark for you. So line C and line A have the same slope, but they're different lines. They have different y-intercepts. So they're going to be parallel. And to see that, let's actually graph all of these characters.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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So line C and line A have the same slope, but they're different lines. They have different y-intercepts. So they're going to be parallel. And to see that, let's actually graph all of these characters. So line A, our y-intercept is negative 6, so the point 0, 1, 2, 3, 4, 5, 6. And our slope is 2. So if we move 1 in the positive x direction, we go up 2 in the positive y direction.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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And to see that, let's actually graph all of these characters. So line A, our y-intercept is negative 6, so the point 0, 1, 2, 3, 4, 5, 6. And our slope is 2. So if we move 1 in the positive x direction, we go up 2 in the positive y direction. 1 in x, up 2 in y. If we go 2 in x, we're going to go up 4 in y. We're going to go up 4 in y.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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So if we move 1 in the positive x direction, we go up 2 in the positive y direction. 1 in x, up 2 in y. If we go 2 in x, we're going to go up 4 in y. We're going to go up 4 in y. And I could just do up 2, then we're going to go 2, 4. And you're going to see it's all on the same line. So line A is going to look something like, do my best to draw it as straight as possible.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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We're going to go up 4 in y. And I could just do up 2, then we're going to go 2, 4. And you're going to see it's all on the same line. So line A is going to look something like, do my best to draw it as straight as possible. Line A, I could do a better version than that. Line A is going to look like, well, that's about just as good as what I just drew. That is line A.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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So line A is going to look something like, do my best to draw it as straight as possible. Line A, I could do a better version than that. Line A is going to look like, well, that's about just as good as what I just drew. That is line A. Now let's do line B. Line B, the y-intercept is negative 6. So it has the same y-intercept, but its slope is 3.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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That is line A. Now let's do line B. Line B, the y-intercept is negative 6. So it has the same y-intercept, but its slope is 3. So if x goes up by 1, y will go up by 3. So x goes up by 1, y goes up by 3. If x goes up by 2, y is going to go up by 6.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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So it has the same y-intercept, but its slope is 3. So if x goes up by 1, y will go up by 3. So x goes up by 1, y goes up by 3. If x goes up by 2, y is going to go up by 6. 2, 4, 6. So 2, and then you go 2, 4, 6. So this line is going to look something like this.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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If x goes up by 2, y is going to go up by 6. 2, 4, 6. So 2, and then you go 2, 4, 6. So this line is going to look something like this. Trying my best to connect the dots. It has a steeper slope. And you see that when x increases, this blue line increases by more in the y direction.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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So this line is going to look something like this. Trying my best to connect the dots. It has a steeper slope. And you see that when x increases, this blue line increases by more in the y direction. So that is line B. And notice they do intersect. There's definitely not two parallel lines.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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And you see that when x increases, this blue line increases by more in the y direction. So that is line B. And notice they do intersect. There's definitely not two parallel lines. And then finally, let's look at line C. The y-intercept is 5. So 0, 1, 2, 3, 4, 5. The point 0, 5, it's y-intercept.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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There's definitely not two parallel lines. And then finally, let's look at line C. The y-intercept is 5. So 0, 1, 2, 3, 4, 5. The point 0, 5, it's y-intercept. And its slope is 2. So you increase by 1 in the x direction. You're going to go up by 2 in the y direction.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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The point 0, 5, it's y-intercept. And its slope is 2. So you increase by 1 in the x direction. You're going to go up by 2 in the y direction. If you decrease by 1, you're going to go down 2 in the y direction. If you increase by, well, you're going to go to that point. You're going to have a bunch of these points.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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You're going to go up by 2 in the y direction. If you decrease by 1, you're going to go down 2 in the y direction. If you increase by, well, you're going to go to that point. You're going to have a bunch of these points. And then if I were to graph the line, let me do it one more time. If I were to decrease by 2, I'm going to have to go down by 4. Negative 4 over negative 2 is still a slope of 2.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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You're going to have a bunch of these points. And then if I were to graph the line, let me do it one more time. If I were to decrease by 2, I'm going to have to go down by 4. Negative 4 over negative 2 is still a slope of 2. So 1, 2, 3, 4. And I can do that one more time. Get right over there.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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Negative 4 over negative 2 is still a slope of 2. So 1, 2, 3, 4. And I can do that one more time. Get right over there. And then you'll see the line. The line will look like that. It will look just like that.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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Get right over there. And then you'll see the line. The line will look like that. It will look just like that. And notice that line C and line A never intersect. They have the exact same slope. Different y-intercepts, same slope.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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It will look just like that. And notice that line C and line A never intersect. They have the exact same slope. Different y-intercepts, same slope. So they're increasing at the exact same rate. But they're never going to intersect each other. So line A and line C are parallel.
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Parallel lines from equation Mathematics I High School Math Khan Academy.mp3
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So if you're doing this on the Khan Academy exercise, you would actually just click on a point right over there, and it would show up, but this would be the reflection of point A across the line L. Let's do another example. So here we're asked, plot the image of point B under a reflection across the X axis. Alright, so this is point B, and we're going to reflect it across the X axis right over here. So to go from B to the X axis, it's exactly five units below the X axis, one, two, three, four, five. So if we were to reflect across the X axis, essentially create its mirror image, it's going to be five units above the X axis, one, two, three, four, five. So that's where the image would be. Maybe we could denote that with a B prime.
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Reflecting points across horizontal and vertical lines.mp3
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So to go from B to the X axis, it's exactly five units below the X axis, one, two, three, four, five. So if we were to reflect across the X axis, essentially create its mirror image, it's going to be five units above the X axis, one, two, three, four, five. So that's where the image would be. Maybe we could denote that with a B prime. We are reflecting across the X axis. Let's do another example. So here, they tell us point C prime is the image of C, which is at the coordinates negative four comma negative two, under a reflection across the Y axis.
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Reflecting points across horizontal and vertical lines.mp3
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