Geometry (all content)
More examples of the Angle Game. Created by Sal Khan.
Welcome back. Let's do a couple more angle game problems, and hopefully this will make you an angle game expert. So let's start, I have the star drawn again, and let's say we know the following angles. We know this angle right here is 41 degrees. We know this angle here is 113 degrees. We know this angle here is 101 degrees. And what we have to figure out -- this is the goal of this angle game -- we want to figure out what this angle is. And like always, I encourage you to try it on your own. Pause the video and then just try to work it through. If you get stuck, then play the video again and hopefully I'll have a solution for you. So pause right now, but otherwise let me explain how to do this. So let's see, we know this, this and this, and we're going to figure out this angle. So how can we figure out this angle? What are the possible strategies? Well, if we knew this angle here, we could say they're supplementary. But that angle seems like a hard angle to figure out too, because it's not a part of any triangles. But this angle is a part of this triangle right here, right? So if we were able to figure out this angle and this angle, these green angles, if we're able to figure out these green angles, then we could figure out this brown angle, which is the goal of this angle game. So, this could also be a good time to pause because I just gave you a hint. This green angle, well it's supplementary to this angle right here, so that means it adds up to 180 degrees, and that's clear because it's on kind of the same line. So this is 101 degrees and this is going to be 79 degrees, right? So it adds up to 180 degrees. That's 79. Now how can we figure out this angle? Well, it's kind of left by itself out in the corner of some place, so we could see if it's part of any triangles. But we already said it's part of this triangle. But that doesn't help us because we don't know this angle and that's actually our goal. What other triangles is it a part of? Well, it's a part of this triangle right here. That's why I like the star problem because it has all these triangles in it that might not be obvious to you the first time you look at it. But the more you look at you see all these triangles. So it's part of this triangle, and it's also part of this triangle. I'm going to draw this triangle another color because I think it'll be clear to you that this is a useful triangle to see that's it's a part of. So we have that triangle. So do we know two of the angles of that triangle? Well sure. We know this angle and we know this angle. So we know that this angle plus 113 plus 41 is going to equal 180 degrees because of the three angles of a triangle. So let me call this, I don't know, g for green. Let's call this g for green. So we know g plus 113 degrees, that's this one right here, plus 41 -- remember, we're looking at this triangle; that's the hardest part just keeping track of which triangle we're looking at -- is going to equal 180 degrees. g plus, what is this, 154? Right? 40, 50, 154 equals 180 degrees. That's always where I mess up on the addition. And so g is equal to, what is this, 26 degrees, right, because I just subtract 154 from both sides. So we're almost there. So we figured out g, we know this green angle. We just have to figure out this, and they're all part of this triangle, this small one right here. This small triangle. So our goal, which is let's call this x. x plus g, which is 26 degrees -- we just figured that out. 26 plus this angle, 79 -- and we figured that out because it was supplementary to this angle -- is going to equal 180 degrees. So x plus, what is this, 105 equal to 180. So x is equal to 75 degrees, if I did my addition and subtraction correctly. So x is equal to 75 degrees. And then we are done. Let's do another one of these problems. These problems are all generated on the [? Card ?] Academy website dynamically by the computer. Whoever wrote this software must be a genius. But anyway, back to the problem. Let me draw some more. So this is going to be a pretty straightforward drawing. It's pretty much just two triangles next to each other. Like that and then let me draw another line that goes like that, and then we draw a line that goes like that, and I think I have done my drawing. There you go. I'm have done my drawing. So let's see. What do we know about this triangle and what do we need to figure out? I'm going to tell you that this angle here, this big angle here, is 86 degrees. We also know that this angle here is 28 degrees. And we also know that this angle here is 122 degrees. And our goal, our mission in this round is to figure out what this angle is. And maybe we can do it, we can do it in a good color. Maybe we can do it in a couple of different ways. So one thing we could do is we could figure out what this angle is, so we could just subtract this green angle from 86 and we would get our answer. Well, this angle's easy, right, because we know two angles of this triangle, so we could figure that out. Let's just call this, I don't know, let's call this y. So y plus 122 plus 28 degrees is going to equal 180. So y plus 150 is equal to 180. So y is equal to 30 degrees, right? So this is equal to 30 degrees. So this is 30 degrees, and this big angle here is 86. So our goal, let's call that x, so x is going to just be equal to the big angle, 86 minus this angle we just figured out, minus 30. So x is going to be equal to 50 degrees. Done. That was a pretty straightforward problem. Let's see if we could figure that out any other way. Well, we could say instead of doing it that way -- let's forget we just solved it that way. We could say this angle here is supplementary to this 122 degree angle, right, so it has to add up to 180. So this plus 122 is 180, so what does that make this? It makes this 58 degrees, right? This plus this is going to be 180. So we figured out this. If we could figure out this, then we could use this triangle. How do we figure out this angle? Well, we could look at this big triangle here, and we know this side, right, and we could figure out this. Let's call this z. So we know that z plus this angle, plus 28, plus this big angle, plus 86 is equal to 180. So z plus, what is this, 106, 114 is equal to 180. So z is equal to, what is this, 66 degrees. I don't know if I'm doing any of my math correctly, but let's just hope. z equals 66. So z is 66, this angle is 58, and now we can use this triangle here to figure out what this angle is, our x. So x plus 66 plus 58 is equal to 180. I already think I might have made a mistake some place in the addition. So this time around I get x is equal to -- let's see, 66 plus 58 is 110 plus 14. So 180 minus 124. So now I got it, x is equal to 56 degrees. Oh great, I actually got the right answer. I was looking at this, I thought it was 50, but this was 56, right -- 86 minus 30. So x is equal to 56 degrees again. So we did it two different ways. That's what I wanted to show you. There's actually not a right answer, as long as you kind of get there eventually. We solved it two different ways and I did all my addition and subtraction correctly, and you get the exact same answer. So hopefully you find the angle game fun and you'll be playing this with your friends. I'll see you later.