Measuring angles in degrees

now that we know what an angle is let's think about how we can measure them and we already hinted it one way to think about the measure of an angle in the last video where we said look this angle XYZ seems more open than angle BAC so maybe the measure of angle XYZ should be larger than the angle of BAC and that is exactly the way we think about the measures of angles but what I want to do in this video is come up with an exact way to measure an angle so what I've drawn over here is a little bit of a of a half circle and it looks very similar to a tool that you can by your local school supply store to measure angles so this is actually a little bit of a drawing of a protractor of a protractor and what we do in something like a protractor and you could even construct one with a piece of paper is we've taken a half circle right here and we've divided it into 180 sections and each of these marks marks 10 of those sections and what you do for any given angle is you put one of the sides of the angle so each of the Rays of an angle are considered one of its sides so you put the vertex you put the vertex of the angle at the center at the center of this half circle or if you're dealing with an actual protractor at the center of that protractor and then you put one side along the zero mark you put one side along the zero mark so I'm going to redo this I'm going to redraw this angle right over here at the center of this protractor so if we said this is y then Z goes right over here and then the other ray the other ray ray Y Z and ray Y X in this circumstance will go roughly in that direction and so it is pointing it is pointing on the protractor to the let's see this looks like this is the seventieth section this is the idiot section so maybe this is I would say I would guess the 77th section so this is pointing to 77 right over here and so when we measure an angle so we could say the measure of angle X Y is the assuming that I drew it the right way right over here we could say the measure of the measure of angle X Y Z sometimes they'll just say angle XYZ is equal to but this is a bit more formal the measure of angle XYZ is equal to 77 and what we do is we call each of these each of these little sections we call them degrees so it's equal to 77 77 sometimes it's written like that a little a little and the same way you would write degrees for the temperature outside so you could write 77 degrees like that or you could actually write out the word right over there so each of these sections are a degree so we're measuring in degrees and I want to be clear degrees aren't the only way to measure angles really anything that measures the openness so when you go into trigonometry you'll learn that you can measure angles not only in degrees but also using something called radians but I'll leave that to another day so let's measure this other angle angle B AC so once again I'll put a at the center and then AC I'll put along I'll put along the zero Degree edge of this half circle or of this protractor and then I'll point a B in the dwell assuming that I'm drawing it exactly the way that it's over there normally instead of moving the angle you could actually move the protractor to the angle so you it looks something like that and you could see that it's pointing to right about well let's just say right about the 30 degree the 30 degree mark so we could say that the measure of angle BAC is equal to 30 degrees and so you can look just straight up from evaluating this these numbers that 77 degrees is clearly larger than 30 degrees and so it is a larger angle which makes sense because it is a more it is a more open angle and in general there's a couple of interesting angles to think about if you have zero degree angle you actually have something that's just a closed angled it really is just array at that point as you get larger and larger or as you get more and more open you eventually get to a point where one of the Rays is completely straight up and down while the other one is left right so you could imagine an angle that looks like this you could imagine an angle that looks like this where one ray go straight up down like that and the other ray goes straight right left or you could imagine something like an angle that looks like this where at least the way you're looking at it doesn't one doesn't look straight up down or one does it look straight right left but if you rotate it if you rotate it it would look just like this thing right over here where one is going straight up and down and one is going straight right and left and you can see from this from our measure right over here that that gives us a 90-degree angle which is very very it's a very interesting angle it shows up many many times in geometry and trigonometry and there's a special word for a 90-degree angle it is called a right angle so this right over here assuming if you rotate it around would look just like this we would call this a right angle right right angle and there is a notation to show that it's a right angle you draw a little kind of part of a box right over there and that tells us that this is going exact if you were to rotate it exactly up and down while this is going exactly right and left if you were to rotate it properly or vice versa and then as you go even wider you get wider and wider and wider and wider until you get all the way to a to an angle that looks like this so you could imagine an angle where where the two rays in that angle form a line the two rays so let's say this is point X this is point Y and this is point Z you could call this angle Z X Y but it's really so open that it's formed an actual line here Z X and y are collinear and what we have right over here this is a hundred and eighty degree angle or we should say the measure of angle Z X Y measure of angle Z X Y is 180 degrees and you can actually go beyond that so if you were to go all the way around the circle so that you would get back to 360 degrees and then you could keep going round and round and round and you see you'll start to see a lot more of that when you when you enter a trigonometry class now there's one last or two last things that I want to introduce in the in this video is there are special words and I'll talk about more types of angles in the next video but if an angle is less than 90 degrees if an angle is less than 90 degrees so for example both of these Aang that we started our discussion with our less than 90 degrees we call them acute angles acute acute so anything so this is acute so that is an acute angle and that is an acute angle right over here they are less than 90 degrees what does a non acute angle look like and there's a word for it other than non acute well it would be more than 90 degrees so for example let me do this in a color I haven't used an angle that looks like this and let me draw it a little bit better than that an angle that looks like this so that's one of that's one side of the angle or one of the Rays and then that I'll put the other one along the baseline right over here clearly this is larger than 90 degrees if I were to approximate let's see that's 100 110 120 almost 130 so let's call that maybe 128 degree angle we call this an obtuse angle we call this an obtuse angle the way I remember it is acute it's kind of a cute angle it's nice and small if if I believe acute in either Latin or Greek or maybe both means something like pin or sharp so that's one way to think about it the the an acute angle seems much sharper obtuse I kind of imagined something that's that's kind of lumbering and large and so that's how I you know or you could think it's not acute it's not nice and small and pointy so that's one way to think about it but these are just general this is just general terminology for different types of angles less than 90 degrees you have an acute angle at 90 degrees you have a right angle larger than 90 degrees you have an obtuse angle and then if you get all the way to 180 degrees your angle actually forms a line