Angle measurement & circle arcs

we already know that an angle is formed when two rays share common end points so for example let's say that this is one ray right over here and then this is one another ray right over here and then they would form an angle and this point right over here their common end point is called the vertex of that angle now we also know that not all angles seem the same for example this is one angle here and then we could have another angle we could have another angle that looks something that looks something like something like this that looks something like this and viewed this way it looks like this one is much more this is much more open so I'll say more open more open and this one right over here seems less open so to avoid having to just say oh more open and less open and actually becoming a little bit more exact about it we actually want to measure how open an angle is or we want to have a measure of the angle now the most typical way that angles are measured there's actually two major ways that they're measured the one the most typical unit is in degrees but later on in high school you will also see the unit of radians being used especially when you learn trigonometry but the degrees convention really comes from a circle so let's draw ourselves a circle right over here so that's a circle and the convention is is that and when I say a convention is just kind of what everyone has been doing the convention is is that you have 360 degrees in a circle so let me explain that so if that's the center of the circle and if we make this ray R are kind of our starting point or our one side of our angle if you go all the way around the circle if you go all the way around the circle that represents 360 degrees 360 degrees and the notation is 360 and then this little superscript circle represents degrees this could be read as 360 degrees now you might be saying where did this 360 number come from and no one knows for sure but there's some type there's hints in history and there's hints in in the way that the the universe works or at least the Earth's rotation around the Sun you might recognize or you might already realize that there are 365 days in a non-leap year 366 in a leap year and so you could imagine ancient astronomers might have said well you know that's pretty close that's pretty close to that's pretty close to 360 that's pretty close to 360 and in fact several ancient calendars including the Persians and the Mayans had 360 days in their year and 360 is also a much neater number than 365 it has many many more factors as to visible oh it's another way of saying it's divisible by a bunch of things but anyway this has just been the convention once again what history has handed us that a circle work is viewed to have 360 degrees and so the one way we could measure an angle is you could put one of the Rays of an angle right over here at this part of the circle and then the other array of the angle the other array of the angle will look something like this and then the fraction of the circle that's intersected by these two rays the fraction of the circle circumference that is intersected by these two arrays the measure of this angle would be that fraction of degrees so for example so for example let's say that this right over here this right over here this length right over here is 1/6 is 1/6 of the circles circumference circumference circumference so it's 1/6 of the way around the circle then this angle right over here is going to be 1/6 of 360 degrees so in this case this would be 60 degrees I could do another example so let's say I had a circle like this and I'll draw an angle up with the vertex at the center of the angle I'll put one of the Rays right over here you could consider that to be zero degrees or the right side of the other angle if the other array was also here would be zero degrees and then I'll make the other array of this angle let's say it went straight up let's say it went straight up like this well in this situation the arc the arc that connects these two end points just like this this represents 1/4 of the circumference of the circle this is right over here 1/4 of the circumference circumference circumference so this angle right over here is going to be 1/4 of 360 degrees 360 degrees divided by 4 is going to be 90 degrees an angle like this one where one ray is straight up and down and the other one is goes to the right left direction we would call this a perpendicular angle we would say these two rays are perpendicular or we would call this a right angle and the way that we oftentimes will denote that is by a symbol like this but this literally means a 90-degree angle let's do one more example let's do one more example of this just to make sure that we understand what's going on actually let's lay out at least one more example maybe one more if we have time so let's say that we have an angle that looks like this one one swore I'm going to put its Center I'm going to put its vertex at the center of the circle that's one ray of the angle and let's say that this is the other Ray this right over here is the other ray of the angle I encourage you to pause this video and tell and try to figure out what the measure what the measure of this angle right over here is well let's think about where the Rays intersect the circle they intersect there and they're the arc that connects them on the circle is that arc right over there that is literally half of the circumference of the circle that is half of the circumference half of the way around of the circle circumference of the circle so this angle is going to be half of 360 degrees and half of 360 is 108 degrees and when you view it this way these two rays share a common endpoint and together they're really forming a line here and let's just do one more example because I said I would let me let me paste another circle let me draw another angle let me draw another angle so let's say that's one ray of the angle and this is the other ray this is the other ray of the angle right over here and we care there's actually two angles or forms there's actually two angles formed in all of these there's one angle that's formed right over here and you might recognize that to be a 90-degree angle but what would we really care about in this example is this angle right over here so once again where does it intersect the circle we care about this arc right over here because that's the arc that is that corresponds to this angle right over here and it looks like we've gone three-fourths around the circle so this angle is going to be 3/4 of 6 of 360 degrees 1/4 of 360 degrees is 90 so three of those is going to be 270 degrees