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So how can we figure out this angle? This one's a little bit trickier. Well, the key here is to realize that this 93-degree angle, it is vertical to this whole angle right over here. And we know from geometry, which we're still learning as we do this example problem, that vertical angles are going to have the same measure. So if this one is 93 degrees, then this entire blue one right over here is also going to be, let me write it, this is also going to be 93 degrees. So 93 degrees, that's going to be made up of this red angle that we care about and the 38 degrees. So this red one, which is the measure of the central angle, it's also the arc measure of arc AB, is going to be 93 minus, 93 degrees minus 38 degrees.
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Finding arc measures Mathematics II High School Math Khan Academy.mp3
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And we know from geometry, which we're still learning as we do this example problem, that vertical angles are going to have the same measure. So if this one is 93 degrees, then this entire blue one right over here is also going to be, let me write it, this is also going to be 93 degrees. So 93 degrees, that's going to be made up of this red angle that we care about and the 38 degrees. So this red one, which is the measure of the central angle, it's also the arc measure of arc AB, is going to be 93 minus, 93 degrees minus 38 degrees. So what is that going to be? Let's see, 93, I can write degrees there, minus 38 degrees. That is going to be equal to, let's see, if it was 93 minus 40, it would be 53, and it's going to be two more, it's going to be 55 degrees.
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Finding arc measures Mathematics II High School Math Khan Academy.mp3
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So this red one, which is the measure of the central angle, it's also the arc measure of arc AB, is going to be 93 minus, 93 degrees minus 38 degrees. So what is that going to be? Let's see, 93, I can write degrees there, minus 38 degrees. That is going to be equal to, let's see, if it was 93 minus 40, it would be 53, and it's going to be two more, it's going to be 55 degrees. 55 degrees. And we are done. This angle right over here is 55 degrees.
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Finding arc measures Mathematics II High School Math Khan Academy.mp3
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That is going to be equal to, let's see, if it was 93 minus 40, it would be 53, and it's going to be two more, it's going to be 55 degrees. 55 degrees. And we are done. This angle right over here is 55 degrees. If you were to add this angle measure, plus 38 degrees, you'd get 93 degrees. And that has the same measure, because it's vertical, with this angle right over here, with angle DPE. All right, let's do one more of these.
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Finding arc measures Mathematics II High School Math Khan Academy.mp3
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This angle right over here is 55 degrees. If you were to add this angle measure, plus 38 degrees, you'd get 93 degrees. And that has the same measure, because it's vertical, with this angle right over here, with angle DPE. All right, let's do one more of these. So we have, in the figure below, and it doesn't quite fit on the page, but we'll scroll down in a second, AB is the diameter of circle P, is the diameter of circle P. All right, so AB is the diameter, let me label that. So AB is the diameter, so it's going straight across, straight across the circle. What is the arc measure of ABC in degrees?
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Finding arc measures Mathematics II High School Math Khan Academy.mp3
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All right, let's do one more of these. So we have, in the figure below, and it doesn't quite fit on the page, but we'll scroll down in a second, AB is the diameter of circle P, is the diameter of circle P. All right, so AB is the diameter, let me label that. So AB is the diameter, so it's going straight across, straight across the circle. What is the arc measure of ABC in degrees? So ABC, so they're making us go the long way around. This is a major arc they're talking about. Let me draw it.
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Finding arc measures Mathematics II High School Math Khan Academy.mp3
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What is the arc measure of ABC in degrees? So ABC, so they're making us go the long way around. This is a major arc they're talking about. Let me draw it. Arc A, what is the arc measure of arc ABC? So we're going the long way around, so it's a major arc. So what is that going to be?
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Finding arc measures Mathematics II High School Math Khan Academy.mp3
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Let me draw it. Arc A, what is the arc measure of arc ABC? So we're going the long way around, so it's a major arc. So what is that going to be? Well, it's going to be, in degrees, the same measure as the angle, as the central angle that intercepts it. So it's going to be the same thing as this central angle right over here. Well, what is that central angle going to be?
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Finding arc measures Mathematics II High School Math Khan Academy.mp3
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So what is that going to be? Well, it's going to be, in degrees, the same measure as the angle, as the central angle that intercepts it. So it's going to be the same thing as this central angle right over here. Well, what is that central angle going to be? Well, since we know that this is a diameter, since AB is a diameter, we know that this part of it, this part of it is going to be 180 degrees. We're going halfway around the circle, 180 degrees. And so if we wanted to look at this whole angle, the angle that intercepts the major arc ABC, it's going to be the 180 degrees plus 69 degrees.
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Finding arc measures Mathematics II High School Math Khan Academy.mp3
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Well, what is that central angle going to be? Well, since we know that this is a diameter, since AB is a diameter, we know that this part of it, this part of it is going to be 180 degrees. We're going halfway around the circle, 180 degrees. And so if we wanted to look at this whole angle, the angle that intercepts the major arc ABC, it's going to be the 180 degrees plus 69 degrees. So we're going to have 180 degrees plus 69 degrees, which is equal to, what is that? 249, 249 degrees. That's the arc measure of this major arc ABC.
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Finding arc measures Mathematics II High School Math Khan Academy.mp3
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So let's see, this is triangle ABC, and it looks like at first he rotates triangle ABC about point C to get it right over here. So that's what they're depicting in this diagram. And then they say Cassan concluded it is not possible to map triangle ABC onto triangle GFE using a sequence of rigid transformations. So the triangles are not congruent. So what I want you to do is pause this video and think about is Cassan correct that they are not congruent because you cannot map ABC, triangle ABC onto triangle GFE with rigid transformations? All right, so the way I think about it, he was able to do the rotation that got us right over here. So it was rotation about point C. And so this point right over here, let me make sure I get this right, this would have become B prime, and then this is A prime, and then C is mapped to itself.
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Congruent shapes and transformations.mp3
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So the triangles are not congruent. So what I want you to do is pause this video and think about is Cassan correct that they are not congruent because you cannot map ABC, triangle ABC onto triangle GFE with rigid transformations? All right, so the way I think about it, he was able to do the rotation that got us right over here. So it was rotation about point C. And so this point right over here, let me make sure I get this right, this would have become B prime, and then this is A prime, and then C is mapped to itself. So C is equal to C prime. But he's not done. There's another rigid transformation he could do, and that would be a reflection about the line FG.
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Congruent shapes and transformations.mp3
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So it was rotation about point C. And so this point right over here, let me make sure I get this right, this would have become B prime, and then this is A prime, and then C is mapped to itself. So C is equal to C prime. But he's not done. There's another rigid transformation he could do, and that would be a reflection about the line FG. So if he reflects about the line FG, then this point is going to be mapped to point E, just like that. And then if you did that, you would see that there is a series of rigid transformations that maps triangle ABC onto triangle GFE. So Cassan is not correct.
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Congruent shapes and transformations.mp3
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Draw the image of ABCP under a dilation whose center is at P and a scale factor is 1 2 3rds. What are the lengths of the side AB and its image? So we're going to do a dilation centered at P. So if we're centering a dilation at P, that means that every other point is going, and its scale factor is 1 2 3rds. That means once we perform the dilation, every point is going to be 1 2 3rds times as far away from P. Well, P is zero away from P, so its image is still going to be at P. So let's put that point right over there. Now, point C is going to be 1 2 3rds times as far as it is right now. So let's see, right now it is six away. It's at negative three, and P is, or its X coordinate is the same, but in the Y direction, P is at three, C is at negative three, so it's six less.
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Thinking about dilations Transformations Geometry Khan Academy.mp3
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That means once we perform the dilation, every point is going to be 1 2 3rds times as far away from P. Well, P is zero away from P, so its image is still going to be at P. So let's put that point right over there. Now, point C is going to be 1 2 3rds times as far as it is right now. So let's see, right now it is six away. It's at negative three, and P is, or its X coordinate is the same, but in the Y direction, P is at three, C is at negative three, so it's six less. We want to be 1 2 3rds times as far away. So what's 1 2 3rds of six? Well, 2 3rds of six is four, so it's going to be, six plus four is going to be, you're going to be ten away.
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Thinking about dilations Transformations Geometry Khan Academy.mp3
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It's at negative three, and P is, or its X coordinate is the same, but in the Y direction, P is at three, C is at negative three, so it's six less. We want to be 1 2 3rds times as far away. So what's 1 2 3rds of six? Well, 2 3rds of six is four, so it's going to be, six plus four is going to be, you're going to be ten away. So three minus ten, that gets us to negative seven, so that gets us right over there. Now, point A, right now it is three more in the horizontal direction than point P's X coordinate, so we want to go 1 2 3rds as far. So what is 1 2 3rds times three?
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Thinking about dilations Transformations Geometry Khan Academy.mp3
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Well, 2 3rds of six is four, so it's going to be, six plus four is going to be, you're going to be ten away. So three minus ten, that gets us to negative seven, so that gets us right over there. Now, point A, right now it is three more in the horizontal direction than point P's X coordinate, so we want to go 1 2 3rds as far. So what is 1 2 3rds times three? Well, that's going to be three plus 2 3rds of three, which is another two, so that's going to be five. So we're going to get right over there, then we can complete the rectangle. And notice, point B is now 1 2 3rds times as far in the horizontal direction.
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Thinking about dilations Transformations Geometry Khan Academy.mp3
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So what is 1 2 3rds times three? Well, that's going to be three plus 2 3rds of three, which is another two, so that's going to be five. So we're going to get right over there, then we can complete the rectangle. And notice, point B is now 1 2 3rds times as far in the horizontal direction. It was three away in the horizontal direction, now it is five away from P's X coordinate. And in the vertical direction, in the vertical direction, in the Y direction, it was six below P's Y coordinate. Now it is 1 2 3rds times as far.
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Thinking about dilations Transformations Geometry Khan Academy.mp3
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And notice, point B is now 1 2 3rds times as far in the horizontal direction. It was three away in the horizontal direction, now it is five away from P's X coordinate. And in the vertical direction, in the vertical direction, in the Y direction, it was six below P's Y coordinate. Now it is 1 2 3rds times as far. It is ten below P's Y coordinate. So let's answer these questions. The length of segment AB, well, we already saw that, that is, we're going from three to negative three, that is six units long.
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Thinking about dilations Transformations Geometry Khan Academy.mp3
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Now it is 1 2 3rds times as far. It is ten below P's Y coordinate. So let's answer these questions. The length of segment AB, well, we already saw that, that is, we're going from three to negative three, that is six units long. And its image, well, it's 1 2 3rds as long. We see it over here, we're going from three to negative seven. Three minus negative seven is ten.
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Thinking about dilations Transformations Geometry Khan Academy.mp3
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Let's say given this diagram right over here, we know that the length of segment AB is equal to the length of AC. So AB, which is this whole side right over here, the length of this entire side as a given is equal to the length of this entire side right over here. So that's the entire side right over there. And then we also know that angle ABF is equal to angle AC. You can see their measures are equal, or this implies that they're congruent or they have the same measures. It's equal to angle ACE. So this angle right over here is congruent to that angle right over there, or you could say that they have the same measure.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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And then we also know that angle ABF is equal to angle AC. You can see their measures are equal, or this implies that they're congruent or they have the same measures. It's equal to angle ACE. So this angle right over here is congruent to that angle right over there, or you could say that they have the same measure. Now the first thing that I want to attempt to prove in this video is whether BF has the same length as CE. So let's try to do that. So we already know a few things.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So this angle right over here is congruent to that angle right over there, or you could say that they have the same measure. Now the first thing that I want to attempt to prove in this video is whether BF has the same length as CE. So let's try to do that. So we already know a few things. I could do a two column proof. Actually, let me just do it just so that in case you have to do two column proofs in your class, you can kind of see how to do it more formally. So let's write our statements.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So we already know a few things. I could do a two column proof. Actually, let me just do it just so that in case you have to do two column proofs in your class, you can kind of see how to do it more formally. So let's write our statements. And then over here I'm going to write my reason for the statement. So let me just rewrite this. Just have a formal two column proof.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So let's write our statements. And then over here I'm going to write my reason for the statement. So let me just rewrite this. Just have a formal two column proof. So we know AB is equal to AC. So this is statement one, and this is given. We know statement two, that angle ABF is equal to angle ACE.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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Just have a formal two column proof. So we know AB is equal to AC. So this is statement one, and this is given. We know statement two, that angle ABF is equal to angle ACE. Once again, that was given. Now the other interesting thing, we have an angle and we have a side on each of these triangles. And then what you can see is both of the triangles, and when I say both of the triangles, I'm talking about triangle ABF and triangle ACE.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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We know statement two, that angle ABF is equal to angle ACE. Once again, that was given. Now the other interesting thing, we have an angle and we have a side on each of these triangles. And then what you can see is both of the triangles, and when I say both of the triangles, I'm talking about triangle ABF and triangle ACE. And they both share this vertex at A. At point A is a vertex for both of these. So we could say angle AF, we could say angle BAF, we could say is equal to angle CAE.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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And then what you can see is both of the triangles, and when I say both of the triangles, I'm talking about triangle ABF and triangle ACE. And they both share this vertex at A. At point A is a vertex for both of these. So we could say angle AF, we could say angle BAF, we could say is equal to angle CAE. That makes it a little bit clearer that we're dealing with two different triangles right here, but it really is the exact same angle. It's equal to itself right there. That's our third statement.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So we could say angle AF, we could say angle BAF, we could say is equal to angle CAE. That makes it a little bit clearer that we're dealing with two different triangles right here, but it really is the exact same angle. It's equal to itself right there. That's our third statement. And we could say it's obvious, some people would call this the reflexive property, that it's obvious that an angle is equal to itself. And so we could say it's obvious, or we could call it maybe the reflexive property, that an angle is clearly reflexive. Obviously equal to itself, even if we label it different ways, this angle is going to be the same measure.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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That's our third statement. And we could say it's obvious, some people would call this the reflexive property, that it's obvious that an angle is equal to itself. And so we could say it's obvious, or we could call it maybe the reflexive property, that an angle is clearly reflexive. Obviously equal to itself, even if we label it different ways, this angle is going to be the same measure. And now we have something interesting going on. We have an angle, a side, and an angle. So what we end up having is that triangle, so by angle side angle, we have the triangle BAF.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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Obviously equal to itself, even if we label it different ways, this angle is going to be the same measure. And now we have something interesting going on. We have an angle, a side, and an angle. So what we end up having is that triangle, so by angle side angle, we have the triangle BAF. So our statement number four, I'm running out of space right here, statement number, I'll go down here, statement number here is triangle BAF. Let me highlight it in a little blue right here. BAF.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So what we end up having is that triangle, so by angle side angle, we have the triangle BAF. So our statement number four, I'm running out of space right here, statement number, I'll go down here, statement number here is triangle BAF. Let me highlight it in a little blue right here. BAF. So that's this entire triangle right over here. Half of the trick of some of these problems is just seeing the right triangle. So we started with this wide angle, we went through the side that we knew, and we went to this orange angle right over here.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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BAF. So that's this entire triangle right over here. Half of the trick of some of these problems is just seeing the right triangle. So we started with this wide angle, we went through the side that we knew, and we went to this orange angle right over here. We started at this angle, then we went to this orange angle across the side E that we know is congruent to that side over there, and then we went to the side, the angle, the vertex that's not labeled. So triangle BAF we now know is going to be congruent to triangle. We start at the wide angle, go to the orange angle, and then go to the unlabeled angle.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So we started with this wide angle, we went through the side that we knew, and we went to this orange angle right over here. We started at this angle, then we went to this orange angle across the side E that we know is congruent to that side over there, and then we went to the side, the angle, the vertex that's not labeled. So triangle BAF we now know is going to be congruent to triangle. We start at the wide angle, go to the orange angle, and then go to the unlabeled angle. It's going to be congruent to angle to triangle CAF. So this is kind of a messily drawn version, but I think you get the idea. These two triangles are going to be congruent.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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We start at the wide angle, go to the orange angle, and then go to the unlabeled angle. It's going to be congruent to angle to triangle CAF. So this is kind of a messily drawn version, but I think you get the idea. These two triangles are going to be congruent. CAE, I should say, is congruent to triangle CAE. Wide angle, orange angle, and then the unlabeled angle in that triangle right over there. This comes straight out of angle, side, angle.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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These two triangles are going to be congruent. CAE, I should say, is congruent to triangle CAE. Wide angle, orange angle, and then the unlabeled angle in that triangle right over there. This comes straight out of angle, side, angle. This comes straight out of ASA. This is one angle, this is the side in between, and these are the two angles. It comes out of statements 1, 2, and 3.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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This comes straight out of angle, side, angle. This comes straight out of ASA. This is one angle, this is the side in between, and these are the two angles. It comes out of statements 1, 2, and 3. So they are congruent. We know that corresponding sides are going to be congruent. So we know our statement 5, we now know that BF is equal to CAE.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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It comes out of statements 1, 2, and 3. So they are congruent. We know that corresponding sides are going to be congruent. So we know our statement 5, we now know that BF is equal to CAE. This comes straight out of statement 4. We could say corresponding sides are congruent. Now let's take it up another notch.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So we know our statement 5, we now know that BF is equal to CAE. This comes straight out of statement 4. We could say corresponding sides are congruent. Now let's take it up another notch. Let's see if we can prove whether ED is equal to EF. Let's just keep going down this and see if we can prove whether ED is equal to EF. I put a question mark there because we haven't necessarily proven it yet.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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Now let's take it up another notch. Let's see if we can prove whether ED is equal to EF. Let's just keep going down this and see if we can prove whether ED is equal to EF. I put a question mark there because we haven't necessarily proven it yet. I want to prove that this little short line segment ED is equal to DF. The interesting thing that we might, at first it might not be so obvious, how do we figure out some type of congruency over that, but we do already have some information here. We know that BAF is congruent to CAE.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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I put a question mark there because we haven't necessarily proven it yet. I want to prove that this little short line segment ED is equal to DF. The interesting thing that we might, at first it might not be so obvious, how do we figure out some type of congruency over that, but we do already have some information here. We know that BAF is congruent to CAE. We also know that this side, right over here, let me do it in a color that I haven't used yet. Let me see, I have not, I've been using a lot of the colors in my palettes. So we know that from these two congruent triangles, that side AE, which is part of CAE, we know that AE is going to be equal to AF, that these two sides are congruent.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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We know that BAF is congruent to CAE. We also know that this side, right over here, let me do it in a color that I haven't used yet. Let me see, I have not, I've been using a lot of the colors in my palettes. So we know that from these two congruent triangles, that side AE, which is part of CAE, we know that AE is going to be equal to AF, that these two sides are congruent. The reason why is because they're corresponding sides of congruent triangles. AF is the side opposite the white angle on BAF, triangle BAF, and AE is the side opposite the white angle on triangle CAE, which we know are congruent. So we know that AE is equal to AF.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So we know that from these two congruent triangles, that side AE, which is part of CAE, we know that AE is going to be equal to AF, that these two sides are congruent. The reason why is because they're corresponding sides of congruent triangles. AF is the side opposite the white angle on BAF, triangle BAF, and AE is the side opposite the white angle on triangle CAE, which we know are congruent. So we know that AE is equal to AF. Once again, this comes from statement four, and we can even say corresponding sides congruent. Same reason as we gave right up here. Now what's interesting here is, this isn't even a triangle that we're seeing up here, but this information that these two characters are congruent, help us with this part over here.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So we know that AE is equal to AF. Once again, this comes from statement four, and we can even say corresponding sides congruent. Same reason as we gave right up here. Now what's interesting here is, this isn't even a triangle that we're seeing up here, but this information that these two characters are congruent, help us with this part over here. Because we know that AB is equal to AC, that was given. And so we know that EB, let me write it over here, and I'll make it a little bit messy right over here. Statement seven will give us some space.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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Now what's interesting here is, this isn't even a triangle that we're seeing up here, but this information that these two characters are congruent, help us with this part over here. Because we know that AB is equal to AC, that was given. And so we know that EB, let me write it over here, and I'll make it a little bit messy right over here. Statement seven will give us some space. We know that BE is going to be equal to CF. Let me write that down. We know that BE is equal to CF.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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Statement seven will give us some space. We know that BE is going to be equal to CF. Let me write that down. We know that BE is equal to CF. And why do we know that? Let me put the reason right over here. Let me try to clean up my work a little bit.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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We know that BE is equal to CF. And why do we know that? Let me put the reason right over here. Let me try to clean up my work a little bit. This column has been slowly drifting to the left. But how do we know that BE is equal to CF? We know that the length of BE is equal to the length of BA minus AE.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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Let me try to clean up my work a little bit. This column has been slowly drifting to the left. But how do we know that BE is equal to CF? We know that the length of BE is equal to the length of BA minus AE. I should say AB, that's how I call it up here. So it's equal to AB minus AE, which is the same thing based on these last few things that we saw, as saying AC minus AF. Because AB is equal to AC, so that's equal to AC.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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We know that the length of BE is equal to the length of BA minus AE. I should say AB, that's how I call it up here. So it's equal to AB minus AE, which is the same thing based on these last few things that we saw, as saying AC minus AF. Because AB is equal to AC, so that's equal to AC. And AE, we already showed, is the same thing as AF. So AC minus AF, and AC minus AF is the same thing as CF right over here. It's equal to CF right over there.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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Because AB is equal to AC, so that's equal to AC. And AE, we already showed, is the same thing as AF. So AC minus AF, and AC minus AF is the same thing as CF right over here. It's equal to CF right over there. And we know that because we know this from statement one, we know it from statement five, and we know it from statement six. Actually, we didn't need statement five there. We just need one and six.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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It's equal to CF right over there. And we know that because we know this from statement one, we know it from statement five, and we know it from statement six. Actually, we didn't need statement five there. We just need one and six. Let's say we need this from one and six is what we had to do there. So we just know that, look, this side is equal to that side. This little part is equal to that part.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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We just need one and six. Let's say we need this from one and six is what we had to do there. So we just know that, look, this side is equal to that side. This little part is equal to that part. So if you subtract the big part minus the little part, this right over here is going to be equal to this right over here. That's all we're showing. So this yellow side is equal to this yellow side right over here.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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This little part is equal to that part. So if you subtract the big part minus the little part, this right over here is going to be equal to this right over here. That's all we're showing. So this yellow side is equal to this yellow side right over here. Now, the other thing that we know, and this is straight out of vertical angles, is that this angle, EDB, is going to be congruent to angle FDC. So let me write that down again. Eight, we know that angle EDB is going to be equal to angle FDC.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So this yellow side is equal to this yellow side right over here. Now, the other thing that we know, and this is straight out of vertical angles, is that this angle, EDB, is going to be congruent to angle FDC. So let me write that down again. Eight, we know that angle EDB is going to be equal to angle FDC. That comes straight out of vertical angles. Vertical angles are congruent or their measures are equal. And now all of a sudden we have something interesting again.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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Eight, we know that angle EDB is going to be equal to angle FDC. That comes straight out of vertical angles. Vertical angles are congruent or their measures are equal. And now all of a sudden we have something interesting again. We have orange angle, white angle, side. So we know that these two smaller triangles are congruent. So now we know, and I don't want to lose my diagram, we know that triangle BED, so statement number nine, we know that triangle BED is congruent to triangle, same sides, white angle, yellow side, then orange angle.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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And now all of a sudden we have something interesting again. We have orange angle, white angle, side. So we know that these two smaller triangles are congruent. So now we know, and I don't want to lose my diagram, we know that triangle BED, so statement number nine, we know that triangle BED is congruent to triangle, same sides, white angle, yellow side, then orange angle. White angle, white angle, let me be careful here. White angle, so B is white angle, E is the unlabeled angle, and then D is the orange labeled angle. So we want to start at C, unlabeled angle, orange angle.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So now we know, and I don't want to lose my diagram, we know that triangle BED, so statement number nine, we know that triangle BED is congruent to triangle, same sides, white angle, yellow side, then orange angle. White angle, white angle, let me be careful here. White angle, so B is white angle, E is the unlabeled angle, and then D is the orange labeled angle. So we want to start at C, unlabeled angle, orange angle. So CFD. So triangle CFD. And this comes straight from, once again, orange angle, white angle, side.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So we want to start at C, unlabeled angle, orange angle. So CFD. So triangle CFD. And this comes straight from, once again, orange angle, white angle, side. So angle, angle, side. Orange angle, white angle, side. So this comes straight out of angle, angle, side congruency.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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And this comes straight from, once again, orange angle, white angle, side. So angle, angle, side. Orange angle, white angle, side. So this comes straight out of angle, angle, side congruency. And since we've now shown that this triangle is equal to that triangle, we know that their corresponding sides are equal. And then this is our home stretch. We now know, since these two triangles are congruent, we now know that ED is equal to DF, because their corresponding sides.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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So this comes straight out of angle, angle, side congruency. And since we've now shown that this triangle is equal to that triangle, we know that their corresponding sides are equal. And then this is our home stretch. We now know, since these two triangles are congruent, we now know that ED is equal to DF, because their corresponding sides. And I could write that right over here. ED is equal to DF. And once again, the reason here is the same thing up here.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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We now know, since these two triangles are congruent, we now know that ED is equal to DF, because their corresponding sides. And I could write that right over here. ED is equal to DF. And once again, the reason here is the same thing up here. Corresponding, so we know our statement 9, which means they're congruent. And corresponding sides, congruent. And we are done.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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And once again, the reason here is the same thing up here. Corresponding, so we know our statement 9, which means they're congruent. And corresponding sides, congruent. And we are done. So that was a pretty involved problem. But you see, once again, you go step by step. Just try to figure out whatever you can about each triangle, and you eventually get it.
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Congruent triangle example 2 Congruence Geometry Khan Academy.mp3
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What is this distance right over here between point A and point C? And I encourage you now to pause this video and try this out on your own. So I'm assuming you've given a go at it. So the key thing to realize here, if since AC is tangent to the circle at point C, that means it's going to be perpendicular to the radius between the center of the circle and point C. So this right over here is a right angle. And the reason why that is useful is now we know that triangle AOC is a right triangle. So we could use, if we know two of its sides, we could use the Pythagorean theorem to figure out the third. Now we clearly know OC.
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Example with tangent and radius Circles Geometry Khan Academy.mp3
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So the key thing to realize here, if since AC is tangent to the circle at point C, that means it's going to be perpendicular to the radius between the center of the circle and point C. So this right over here is a right angle. And the reason why that is useful is now we know that triangle AOC is a right triangle. So we could use, if we know two of its sides, we could use the Pythagorean theorem to figure out the third. Now we clearly know OC. Now OA, we don't know the entire side. They only give us that AB is equal to two. But the thing that might jump out in your mind is OB is a radius.
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Example with tangent and radius Circles Geometry Khan Academy.mp3
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Now we clearly know OC. Now OA, we don't know the entire side. They only give us that AB is equal to two. But the thing that might jump out in your mind is OB is a radius. OB is a radius. It's going to be the same length as any radius. So this is going to be three as well.
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Example with tangent and radius Circles Geometry Khan Academy.mp3
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But the thing that might jump out in your mind is OB is a radius. OB is a radius. It's going to be the same length as any radius. So this is going to be three as well. It's the distance between the center of the circle and a point on the circle, just like the distance between O and C. So this is going to be three as well. And so now we are able to figure out that the hypotenuse of this triangle has length five. And so we need to figure out what side, the length of segment AC is.
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Example with tangent and radius Circles Geometry Khan Academy.mp3
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So this is going to be three as well. It's the distance between the center of the circle and a point on the circle, just like the distance between O and C. So this is going to be three as well. And so now we are able to figure out that the hypotenuse of this triangle has length five. And so we need to figure out what side, the length of segment AC is. So let's just call that, I don't know, I'll call that, I will call that X. X. And so we know that X squared plus three squared, I'm just applying the Pythagorean theorem here, so plus three squared is going to be equal to the length of the hypotenuse squared, is going to be equal to five squared. And I know this is the hypotenuse.
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Example with tangent and radius Circles Geometry Khan Academy.mp3
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And so we need to figure out what side, the length of segment AC is. So let's just call that, I don't know, I'll call that, I will call that X. X. And so we know that X squared plus three squared, I'm just applying the Pythagorean theorem here, so plus three squared is going to be equal to the length of the hypotenuse squared, is going to be equal to five squared. And I know this is the hypotenuse. It's the side opposite the 90 degree angle. It's the longest side of the right triangle. So X squared, X squared plus nine, plus nine is equal to 25, is equal to 25.
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Example with tangent and radius Circles Geometry Khan Academy.mp3
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And I know this is the hypotenuse. It's the side opposite the 90 degree angle. It's the longest side of the right triangle. So X squared, X squared plus nine, plus nine is equal to 25, is equal to 25. Subtract nine from both sides, and you get, you get X squared is equal to 16. And so it should jump out at you that X is going to be equal to, X is equal to four. So X is equal to four.
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Example with tangent and radius Circles Geometry Khan Academy.mp3
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So we're told circle P is below. This is circle P right over here. What is the arc measure of arc BC in degrees? So this is point B, this is point C. Let me pick a different color so you can see the arc. And since they only gave us two letters, we really want to find the minor arc. So we want to find the shorter arc between B and C. So the major arc would be the long way around. And if they wanted to specify the major arc, they would have had to give us three letters to force us to go the long way around.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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So this is point B, this is point C. Let me pick a different color so you can see the arc. And since they only gave us two letters, we really want to find the minor arc. So we want to find the shorter arc between B and C. So the major arc would be the long way around. And if they wanted to specify the major arc, they would have had to give us three letters to force us to go the long way around. So if they said arc BAC or BDC, that would go the long way around. But since they just gave us just B and C, we assume it's going to be the minor arc. So we want to find that arc measure right over there.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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And if they wanted to specify the major arc, they would have had to give us three letters to force us to go the long way around. So if they said arc BAC or BDC, that would go the long way around. But since they just gave us just B and C, we assume it's going to be the minor arc. So we want to find that arc measure right over there. Now the arc measure is going to be the exact same measure in degrees as the measure of the central angle that intercepts that arc. So it's going to be the same thing as the measure of this central angle, which is 4k plus 159 degrees. So if we can figure out what k is, we're going to know what this central angle measure is.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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So we want to find that arc measure right over there. Now the arc measure is going to be the exact same measure in degrees as the measure of the central angle that intercepts that arc. So it's going to be the same thing as the measure of this central angle, which is 4k plus 159 degrees. So if we can figure out what k is, we're going to know what this central angle measure is. And then that's going to be the same thing as this arc measure. So how do we figure that out? Well, what might jump out at you is that this angle, angle BPC that we care about, is vertical to angle APD.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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So if we can figure out what k is, we're going to know what this central angle measure is. And then that's going to be the same thing as this arc measure. So how do we figure that out? Well, what might jump out at you is that this angle, angle BPC that we care about, is vertical to angle APD. These are vertical angles. And so vertical angles are going to have the same measure. They're going to be congruent.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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Well, what might jump out at you is that this angle, angle BPC that we care about, is vertical to angle APD. These are vertical angles. And so vertical angles are going to have the same measure. They're going to be congruent. So let's set these two measures equal to each other. So we know that 4k plus 159 is going to be equal to 2k plus 153. So let's get all of our k terms on the left-hand side and all of the non-k terms on the right-hand side.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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They're going to be congruent. So let's set these two measures equal to each other. So we know that 4k plus 159 is going to be equal to 2k plus 153. So let's get all of our k terms on the left-hand side and all of the non-k terms on the right-hand side. So let's subtract 2k from both sides. So we can subtract 2k from both sides. And let's subtract.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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So let's get all of our k terms on the left-hand side and all of the non-k terms on the right-hand side. So let's subtract 2k from both sides. So we can subtract 2k from both sides. And let's subtract. Well, let me just do that first. I don't want to skip steps. And so I got rid of the k's on the right-hand side.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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And let's subtract. Well, let me just do that first. I don't want to skip steps. And so I got rid of the k's on the right-hand side. So it's just going to be left with a 153. And on the left-hand side, 4k minus 2k is 2k. And I still have plus 159.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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And so I got rid of the k's on the right-hand side. So it's just going to be left with a 153. And on the left-hand side, 4k minus 2k is 2k. And I still have plus 159. Let's get rid of this 159 on the left-hand side. So let's subtract it. But if I do it on the left-hand side, I need to do it on the right-hand side as well.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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And I still have plus 159. Let's get rid of this 159 on the left-hand side. So let's subtract it. But if I do it on the left-hand side, I need to do it on the right-hand side as well. So subtract 159 from both sides. And I'm left with 2k is equal to 153 minus 159 is negative 6. So k is equal to, just divide both sides by 2, k is going to be equal to negative 3.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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But if I do it on the left-hand side, I need to do it on the right-hand side as well. So subtract 159 from both sides. And I'm left with 2k is equal to 153 minus 159 is negative 6. So k is equal to, just divide both sides by 2, k is going to be equal to negative 3. Now, you might be tempted to say, oh, negative 3. But that's not what we're trying. We're not just trying to solve for k. We're trying to figure out this angle measure, which is going to be the same as the arc measure that we care about.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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So k is equal to, just divide both sides by 2, k is going to be equal to negative 3. Now, you might be tempted to say, oh, negative 3. But that's not what we're trying. We're not just trying to solve for k. We're trying to figure out this angle measure, which is going to be the same as the arc measure that we care about. And that's just expressed in terms of k. So it's 4 times k plus 159. So that's going to be 4 times negative 3 plus 159. Well, what's that going to be?
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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We're not just trying to solve for k. We're trying to figure out this angle measure, which is going to be the same as the arc measure that we care about. And that's just expressed in terms of k. So it's 4 times k plus 159. So that's going to be 4 times negative 3 plus 159. Well, what's that going to be? 4 times negative 3 is negative 12 plus 159 is going to be 147. So this angle right here is a measure of 147 degrees. And you can calculate.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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Well, what's that going to be? 4 times negative 3 is negative 12 plus 159 is going to be 147. So this angle right here is a measure of 147 degrees. And you can calculate. That's the same thing as over here. 2 times negative 3 is negative 6 plus 153 is 147 degrees. These two are the same.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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And you can calculate. That's the same thing as over here. 2 times negative 3 is negative 6 plus 153 is 147 degrees. These two are the same. And so 147 degrees. This angle measure is the same as the measure of arc BC. Let's do one more of these.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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These two are the same. And so 147 degrees. This angle measure is the same as the measure of arc BC. Let's do one more of these. Circle P is below. What is the arc measure of BC in degrees? Now, since once again, they only gave us two letters, we can assume it is the minor arc.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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Let's do one more of these. Circle P is below. What is the arc measure of BC in degrees? Now, since once again, they only gave us two letters, we can assume it is the minor arc. So we care about BC. We care about this right over here. And so the measure of this arc is going to be the same thing as the measure of the central angle that intercepts that arc.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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Now, since once again, they only gave us two letters, we can assume it is the minor arc. So we care about BC. We care about this right over here. And so the measure of this arc is going to be the same thing as the measure of the central angle that intercepts that arc. And that measure is going to be the sum of these two angles. So it's going to be 4y plus 6 plus 7y minus 7. So what's that?
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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And so the measure of this arc is going to be the same thing as the measure of the central angle that intercepts that arc. And that measure is going to be the sum of these two angles. So it's going to be 4y plus 6 plus 7y minus 7. So what's that? 4y plus 7y. We can combine the y terms. It's going to be 11y.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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So what's that? 4y plus 7y. We can combine the y terms. It's going to be 11y. And then 6 minus 7 is going to be negative 1. So it's going to be 11y minus 1. And how do we figure that out?
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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It's going to be 11y. And then 6 minus 7 is going to be negative 1. So it's going to be 11y minus 1. And how do we figure that out? Well, how do we figure out what y is? We need to figure out what y is in order to figure out what 11y minus 1 is. Well, we know, let me write this down.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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And how do we figure that out? Well, how do we figure out what y is? We need to figure out what y is in order to figure out what 11y minus 1 is. Well, we know, let me write this down. So the angle that we care about is 11y minus 1. We know that that angle plus this big angle that I'm going to show in blue, that if we add them together, that it's going to be 360 degrees. Because we would have gone all the way around the circle.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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Well, we know, let me write this down. So the angle that we care about is 11y minus 1. We know that that angle plus this big angle that I'm going to show in blue, that if we add them together, that it's going to be 360 degrees. Because we would have gone all the way around the circle. So we know that 11y minus 1 plus 20y minus 11 is going to be equal to 360 degrees. And so now we can just solve for y. What is, let me get some new colors involved, what is 11y plus 20y?
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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Because we would have gone all the way around the circle. So we know that 11y minus 1 plus 20y minus 11 is going to be equal to 360 degrees. And so now we can just solve for y. What is, let me get some new colors involved, what is 11y plus 20y? Well, that's going to be 31y. And then if I have negative 1 and negative 11, that's going to be negative, let me do this in a different color. So that's going to be negative 1, negative 11.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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What is, let me get some new colors involved, what is 11y plus 20y? Well, that's going to be 31y. And then if I have negative 1 and negative 11, that's going to be negative, let me do this in a different color. So that's going to be negative 1, negative 11. That's negative 12. And that's going to be equal to 360 degrees. So let's see, we could add 12 to both sides to get rid of that negative 12 right over there.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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So that's going to be negative 1, negative 11. That's negative 12. And that's going to be equal to 360 degrees. So let's see, we could add 12 to both sides to get rid of that negative 12 right over there. And that's going to leave us with 31y is equal to 372. And so if we divide both sides by 31, it looks like 12. Yep, it'll go exactly 12 times.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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So let's see, we could add 12 to both sides to get rid of that negative 12 right over there. And that's going to leave us with 31y is equal to 372. And so if we divide both sides by 31, it looks like 12. Yep, it'll go exactly 12 times. So y is equal to 12. And remember, we weren't trying to solve for y. We were trying to solve for 11y minus 1.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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Yep, it'll go exactly 12 times. So y is equal to 12. And remember, we weren't trying to solve for y. We were trying to solve for 11y minus 1. So what is 11 times 12? We know that y is 12. 11 times 12 minus 1.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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We were trying to solve for 11y minus 1. So what is 11 times 12? We know that y is 12. 11 times 12 minus 1. Let's see, 11 times 12 is 121. And then 121 minus 1 is going to be, oh, sorry, no. It's going to be, my multiplication tables are off.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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11 times 12 minus 1. Let's see, 11 times 12 is 121. And then 121 minus 1 is going to be, oh, sorry, no. It's going to be, my multiplication tables are off. It's been a long day. 11 times 12 is going to be 132 minus 1 is going to be 131. And it's going to be in degrees.
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Finding arc measures with equations Mathematics II High School Math Khan Academy.mp3
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