Angles in Standard Position - Video Tutorials & Practice Problems

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concept

Drawing Angles in Standard Position

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5m

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Everyone. So in a previous video where we discussed the basics of triangles, we said one way to classify a triangle was based on its angle and an angle is just where two of the sides of a triangle will meet. Well, a lot of times we're gonna be working with angles, we need to work with them, not in a shape or floating around in space, but instead on an X and Y coordinate system like this angles are one of the most important things that you'll need. Not just for this course, but you also be using them regularly in other math courses. I wanna make sure you have a great understanding of them. So I'm gonna introduce you to angles and show you how to draw them. There's some conceptual things you'll need to know and then we'll do some examples together. Let's get started here. So the first I want to talk about is the definition of an angle. Your textbooks won't have a great definition for this, but it's basically just the gap or the space that you create between two line segments or sides of a triangle. A lot of times when we work on the xy coordinate system, we don't even care about that third side of the triangle. But the angle is really just sort of this gap that you make here in between these two line segments. All right now, the bigger the gap, the bigger the angle. And we came up with a number system for this a long time ago, like thousands of years ago where basically you measure in degrees from zero all the way to 360. As you go around the XY coordinate system, the ancient Babylonians a long time ago, decided that the X axis over here would be zero. And then as you go around the angle gets bigger, if you go around a full circle over here, you've gone a full 360 degrees. That's probably I know that you're familiar with, which actually cuts up our X and Y axis into increments of 90 right? If a quarter of the circle that's 90 another quarter of the circle, that's another 90. So it'd be 180 another quarter of the circle just be 270. And then all the way back around will be 360 then you would just start all the way over again. All right. So then what would something like 60 degrees look like? Let's just get right into our first example. If these things are 0 91 82 70 what would 60 degrees look like? First of all, how do you even draw, what an angle is? How do I know which of the sides to draw? Can I draw an angle that the kind of look like this with those two sides? But what we do is a long time ago, we decided that the way we're gonna draw angles is from the initial side, the initial side is gonna be along the positive X axis. In other words, it's gonna be this side over here and you're gonna draw from that to the terminal side. The terminal side is usually gonna be the one that's not on the x axis. All right. So that is the terminal side whenever you have an angle that's drawn like this and all of your angles are gonna be drawn like this. Your, your textbooks refer to that as an angle drawn in standard position. In fact, most of the time you're not even gonna see this initial side, you'll just see something that looks like this. That's how your angles will be shown. All right. So that's how you draw an angle. So then how do you get, how do you get to 60 degrees? Well, basically, you can use your axes as guides for this, right? If I have a sort of angle that sweeps out like this and I go all the way around, oops, it looks like this, that would be 90 degrees, right? So I want something that's gonna be a little bit less than 90. All right. If I draw something that looks halfway between zero and 90 that's gonna be 45 degrees halfway between 0 90. If I need something that looks a little bit, sort of more steep than that. In fact, it's actually gonna look something like exactly what I've drawn over here with this line. If you do it. Exactly like this, this is about what 60 degrees is going to look like. Now again, this has to, you know, it doesn't have to be perfect because it's just a sketch, but I have a little protractor here. That kind of helps me sort of gauge these angles. And if you draw this, this is gonna be about, it's about like 50 degrees, which is close enough, right? That's what I've got there. All right. So that's how you draw 60 degrees. Let's move on to the next one, the next one I'm gonna draw in purple over here. How do you draw 100 and 50 degrees? Well, again, the initial side is gonna be the same. It's always gonna be along that positive X axis. And what I'm gonna do here is I'm gonna start to draw a little angle that goes all the way around in this direction. All right. So notice here how this goes from 0 to 90 100 and 50 is gonna be a little bit more than 90. But if you draw all the way to the left, that's gonna be 100 and 80 which is gonna be too much. So it's gonna have to be somewhere here in this quadrant and it's gonna be close to 180 not exactly close to 90. So it's gonna look something like this. All right, if you measure this angle all the way from the positive X axis like this, that's gonna be about 150 degrees. All right. So we've got those two that are done so pretty straightforward there. Now, one of the things you might have noticed here is that both of these examples have been positive angles. We've always been drawing them up like this from the positive X axis in the counterclockwise direction. This is just a convention that we came up a long time ago with which is that we're gonna draw positive angles in the counter clockwise direction against the direction of the clock. That just means that if you have a negative angle, like for example, negative 60 degrees, you're gonna go the other way, you're gonna draw those in the clockwise direction. All right. So how would I draw negative 60? Well, positive 60 would be if I went up from zero like this and drew an angle like that negative 60 would be in the opposite direction. I have to go down like this in this direction. And if I go clockwise down from zero, that's gonna be negative 60 degrees. That's one way I like to remember this positive angles go up from zero, clockwise, negative angles will go down from zero. That's one way I like to remember this. That's really all there is to it. That's how you sketch angles in standard position. All right. So one last thing I wanna mention here is you may be asked to sort of classify types of angles and it's actually very similar to how we did this for triangles. So the three words we use for this are gonna be acutes of T and rights exactly like what we use for triangles. And basically those things have to do with what they are relative to 90 acute angles are gonna be less than 90 degrees. So for example, this is an acute angle. That example A that we work with example, B which is 100 and 50 degrees. That's gonna be something that's bigger than 90. So that's an obtuse angle and a right angle is exactly equal to 90 degrees. All right. There's something really quick that you might be asked for. That's it for this one, folks. Thanks for watching and let me know if you have any questions.

2

Problem

Problem

What is the approximate measure of the angle shown below? Choose the most reasonable answer.

A

60°

B

150°

C

240°

D

300°

3

example

Example 1

Video duration:

3m

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All right, everyone. So let's use what we know about angles to go ahead and work out this example. We're gonna draw and sketch each of these angles in standard position. They're just sketches so they can kind of be rough. They don't have to be perfect. Let's go ahead and take a look at the first one, which is 45 degrees. What does that look like? Well, let's just use our axes as guides for this because we know this is zero. We know this is 90 this is 1 82 70. So 45 is gonna be somewhere between zero and 90. It's gonna be somewhere in this first quadrant over here. What is 45 look like? Well, if you actually think about it, 45 is, is exactly halfway between zero and 90 that's exactly what 90 divided by two is. So if I were to grab a line and start to sweep it out towards the 90 degree angle, I would stop basically at a halfway which is almost like a perfectly diagonal line like this, that's what 45 degrees looks like. So I'm just gonna take a line and I'm gonna sort of draw it so that I'm going to be like perfectly at like the northeast, you know, upright, uh, corner and I'm basically just gonna be cutting these two angles in half like this. So this would be 45 degrees. All right. That's about what that would look like. All right, let's look at the, the next one here, which is 210 degrees. All right. So 210 again, let's use our, our, our axis as guides. This is 90. Uh and this is gonna be 100 and 8210 is bigger than 180. So you're gonna start to go round in this direction over here into this third quadrant because this is gonna be 270 that's too big. So it's gonna be somewhere over here. All right. So if you look at sort of the halfway between, between 1 82 70 something like the halfway points like this, this is actually 225 you can figure that out just by taking the difference between those. All right. So that's actually too big as well. 2210 is gonna be somewhere inside of over here. It's gonna be somewhere in this area. So let's actually use sort of use what we know about zero and 30 to sort of draw this because if you look at this, the difference between 100 and 8210 is 30 degrees. So what does 30 look like in the first quadrant? Well, if you sort of draw this out, it's gonna look something that looks like this, right? That's about what a 30 degree angle looks like. So the difference between zero and 30 is this. So if you go halfway around the circle to 180 the difference between 1 82 10 is also basically what this is, but it's just in the third quadrant. So all that really means here is you can take this line at 30 you can kind of just imagine that it keeps extending out in this direction over here. I'm just gonna extend the line because now that angle is gonna be 30 degrees from 100 and 80 degrees, right? So this is gonna get me 210 over here. All right, it's 30 degrees from 180. But if you go all the way around the circle, that whole angle over there is 210 degrees. Hopefully, that made sense. All right. So now let's look at the last one over here, which is negative 100 degrees. All right. So remember these first two have been both positive numbers. So we've been going counterclockwise. Now we're actually gonna go in the clockwise direction the same way that the clock goes because it's a negative angle. All right. So in because it's going in the negative direction, our axes won't help us so much because we know this is 09 91 82 70 but we don't know what these negative angles are. Well, what happens is if you just go around in the other direction, you sort of just flip numbers, right? So this is 270 but this would be about what negative 90 degrees would look like. Remember these are right angles, right? This would be uh 100 and 80 but also it would be negative 100 and 80 if you go around in this direction and this would be negative 270. All right. So negative 100 would be going basically a full 90 plus a little bit extra. So if you were going in the cal in the clockwise direction, you would go past 90 then a little bit extra like 10 degrees. All right. So what happens is I'm actually gonna draw a line that looks something like this. All right. So this is a full nine degrees in the negative direction, but then it's a little bit extra. So this angle which we draw clockwise would be negative 100 degrees. All right. It's about what that would look like. Again, these are sketches, they don't have to be perfect, but hopefully you got something that looks like this. All right. Let me know if you have any questions. Thanks for watching.

4

Problem

Problem

Which angle is NOT a positive angle drawn in standard position?