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### Course: Class 5>Unit 9

Lesson 1: Angle introduction

# Angles: introduction

Two rays that share the same endpoint form an angle. Learn about angles and the parts of an angle, like the vertex. Created by Sal Khan.

## Want to join the conversation?

• How are Angles used in everyday life?
• Angles are used to design the most basic and the most complex of polygons (shapes). If you look at your clothing, there are many different angles that are used to design skirts and dresses. Look at the collars of your shirts. Diverse angles used in any sort of art design draws the eye. How about stained glass windows?
• how many angles can you think of?
• Besides the 4 types of angles tabron517 mentioned, there are also full and reflex angles. A full angle is a full 360 degrees and a reflex angle is between 180 degrees and 360 degrees. A full angle looks like this: https://upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Full_angle.svg/380px-Full_angle.svg.png An obtuse angle is between 90 and 180 degrees and a straight angle is 180 degrees.

Hopefully you know the rest: acute --> less than 90, right ---> 90. In between, there's an infinite number of possible angles within the possible degrees an angle can have (you can have a 40.34 degree angle for example). If you actually draw angles out you'll find that you can really define an angle to be at either side of a vertex (the point where the two rays meet) and what you would probably consider the "other side" would usually be a reflex angle, which would look something like this: http://www.mathsisfun.com/images/reflexangles.gif
• Is a vertex the meeting point between two rays or two line segments or two lines, or does it not matter?
• Well, if you look at a angle, it has no arrows, but is still is a ray, and still goes on forever, the angle just doesn't have the arrows. I believe that angle are all rays, but I am not sure. It would be good question to google. Hope this was helpful to you!
• At , Sal says you can't call that angle A, then demonstrates another diagram. I know on the 2nd diagram that you can't call it H, but couldn't you call it A in the first one?
• Even Though it is an angle all by itself, the rule still applies.
• how do you tell a 360° angle from a 720° angle or a 45° angle from a 405° angle?
• If you have 720 degrees, you would have two rounds of 360 degrees instead of one. If you 405 degrees, you would have one round 180 degrees and 45 degrees.

Usually, we use degrees under 180 degrees and above 0 degrees.

Hope this helps.
• say that i wanted to name the angle BAC but i wanted name the reflex angle, how would i do it?
• you subtract 360 by the inside angel
• At , Sal says that the point is called AB. Can you call it ray BA?
• Yes! It always stays the same, even if you do switch it from AB to BA.
• why a right angle is ninety degrees?
• Each full circle is 360 degrees. A quarter of that would be a right angle, or 90 degrees.
• What is an angle? We jump directly into line arrays and segments and where angles are found but we have no clue of what an angle is at it's most basic explanation.
• An angle is a combination of two rays (half-lines) with a common endpoint. The latter is known as the vertex of the angle and the rays as the sides, sometimes as the legs and sometimes the arms of the angle.
• And pretend this isn't a greater than sign, but is this an angle? >
• Typically, we use the less than sign < to represent an angle. You can tell the difference between an angle and less than sign because angles will be indicated with letters (ex. <A).

## Video transcript

Let's say we have one ray over here that starts at point A and then goes through point B. And so we could call this ray-- let me draw that a little bit straighter-- we could call this ray AB. Ray AB starts at A, or has a vertex at A. And let's say that there's also a ray AC. So let's say that C is sitting right over there. And then I can draw another ray that goes through C. So this is ray AC. And what's interesting about these two rays is that they have the exact same vertex. They have the exact same vertex at A. And in general, what we have when we have two rays that have the exact same vertex, you have an angle. And you're probably already reasonably familiar with the concept of an angle, which I believe comes from the Latin for corner, which makes sense. This looks like a little bit of a corner right over here that we see at point A. But the geometric definition, or the one that you're most likely to see, is when two rays share a common vertex. And that common vertex is actually called the vertex of the angle. So A is the vertex. Not only is it the vertex of each of these rays, ray AB and ray AC, it is also the vertex of the angle. So the next thing I want to think about is how do we label an angle. You might be tempted to just label it angle A. But I'll show you in a second why that's not going to be so clear to someone, based on where our angle is actually sitting. So the way that you specify an angle-- and hopefully this will make sense in a second-- is that you say angle-- this is the symbol for angle, and it actually looks strangely similar to this angle right over here. But this little pointy thing, or it almost looks like a less-than sign. But it's not quite. It's flat on the bottom right over here. This is the symbol for angle. You'd say angle BAC. Or you could say angle CAB. In either case, they're kind of specifying this corner. Or sometimes you could view it as this opening right over here. And the important thing to realize is that you have the vertex in the middle of the letters. Now you might be saying, wait, why go through the trouble of listing all three of these letters. Why can't I just call this angle A? And to see that, let me show you another diagram. And although the geometric definition of an angle involves two rays that have the same vertex, in practice, you're going to see many angles that are made up of lines and line segments. And you could imagine that you could continue those line segments on and on in one direction. And then they would become rays. So in that way, they're consistent with this definition. But let's say I have one line segment that looks like that. Let me label some points here. So we've already used ABC. So I'm going to call this D and E, points D and E. So this is line segment DE. And let's say I also have a line segment FG. And let's say this point where these two line segments intersect, let's call that point point H. Now how could we specify this angle right over here? Can we just call that angle H? Well, no. Because if we just said angle H, the angle that has a vertex H, it could be this angle right over here. Or it could be this angle right over here. Let me draw it this way. You could view it that way. Or it could be that angle over there. It could be this angle over here. It could be this angle over here. Or it could be that angle over there. And so the only way to really specify which angle you're talking about well, is to give three letters. So if you really did want to talk about that angle right over there, you would call that angle EHG. So that is angle EHG. Or you could actually call that angle GHE. If you wanted to specify this angle right over here, the one made up of, if you imagine that ray and that ray, if you were to keep on going past those points, then you could call that angle DHG, or angle GHD. I think you get the point. This angle up here could be FHD or EHF. And this one could be FHD or DHF. And when you do it this way, it's very clear what angle you are referring to. So now that we have a general idea of what an angle is, and kind of how do we denote it with symbols, the next thing you might be curious about is, it doesn't look like all angles are kind of the same. It seems like some angles open up or are more open than others. And some are a little bit more closed in than others. And that actually is the case. So for example, let's take two angles here. So let's say I have one angle that looks like that. So I'll started reusing letters. So let's say that this is A, B and C. I could make these rays. I could keep on going and make them rays if I like. Or I could just keep them as line segments. So right over here, I have angle BAC. And let's say over here, I have angle-- so let me draw another one-- and let's say this is angle XYZ. And once again, I could draw them as rays if I like, to go on and on and on. So it's angle XY and Z. And so when you just look at these, you just eyeball these two angles, it looks like this one is more open. So this one looks more open. While this one over here looks more closed, at least relative to this one. So maybe when we measure angles, we should measure it based on how open or closed they are. And that actually is the case. And so without even telling you how we measure an angle, you could say that the measure of angle XYZ, the measure of this angle, is greater than the measure of this angle right over here. And any convention we use for measuring angles is essentially going to be a measure of how open or how closed an angle actually is. And I'll take that up in the next video where we'll see how to actually measure an angle.