Finding arc measures

- So I have some example questions here from Khan Academy on arc measure. And like always, I encourage you to pause the video after you see each of these questions, and try to solve them before I do. So this first question says what is the arc measure, in degrees, of arc AC on circle P below. So this is point A, that is point C, and when they're talking about arc AC, since they only have two letters here, we can assume that it's going to be the minor arc. When we talk about the minor arc. There's two potential arcs that connect point A and point C. There's the one here on the left, and then there's the one, there is the one on the right. And since C isn't exactly straight down from A, it's a little bit to the right, the shorter arc, the arc with the smaller length, or the minor arc is going to be this one that I'm depicting here right on the right. So what is this arc measure going to be? Well, the measure of this arc is going to be exactly the same thing as, in degrees, as the measure of the central angle that intercepts the arc. So that central angle, let me do it in a different color, I'll do it in this blue color, that central angle is angle C, P, A. Angle C, P, A, and the measure of that central angle is going to be 70 degrees plus 104 degrees. It's going to be this whole thing right over there. So it's going to be 174 degrees. One hundred and seventy four degrees, that's the arc measure, in degrees, of arc AC. Let's keep doing these. So let me do another one. So, this next one asks us, in the figure below, in the figure below, segment AD-- so this is point A, this is point D, so segment AD is this one right over here. Let me see if I can draw that. That's AD right over there, AD and CE are diameters of the circle. So let me draw CE, so CE is, we're going to connect point C and E. These are diameters. So, let me, so they go straight. Whoops, I'm using the wrong tool, let me... So those are, somehow I should, alright. So, those are di-- whoops, how did that happen? So let me, somehow my pen got really big, alright. That'll be almost there, ok. So CE, there you go. So those are both diameters of the circle P. What is the arc measure of AB, of arc AB in degrees? So arc AB, once again there's two potential arcs that connect point A and B. There's the minor arc, and since this only has two letters we'll assume it's the minor arc. It's going to be this one over here. There's a major arc, but to not the major arc they would've said something like A, E, B or A, D, B or arc A, C, B to make us go this kind of, this long way around. But this is arc AB, so we, in order to find the arc measure, we just really have to find the measure of this central angle. This is the central angle that intercepts that arc, or you can even say it defines that arc in some way. So how can we figure out this angle? And this one's a little bit trickier. Well, the key to, the key here is to realize that this 93 degree angle, it is vertical to this whole angle right over here. And we know from geometry, which we're still learning as we do this example problem, that vertical angles are going to have the same measure. So if this one on, this one is 93 degrees, then this entire blue one right over here is also gonna be, let me write it, this is also gonna be 93 degrees. So 93 degrees, that's gonna be made up of this red angle, that we care about, and the 38 degrees. So this red one, which is the measure of the central angle, it's also the arc measure of arc AB, is going to be 93 minus, 93 degrees minus 38 degrees. So what is that going to be? Let's see, 93, I can write degrees there, minus 38 degrees, that is going to be equal to, let's see if it was 93 minus 40 it would be, it would be 53, it's gonna be two more, it's gonna be 55 degrees. Fifty five degrees, and we are done. This angle right here is 55 degrees. If you were to add this angle meausre, plus 38 degrees, you would get 93 degrees, and that has the same measure because it's vertical with this angle right over here, with angle D, P, E. Alright, let's do one more of these. So, we have in the figure below, and it doesn't quite fit on the page, but we'll scroll down in a second, AB is the diameter of circle P, is the diameter of circle P. Alright, so AB is the diameter, let me label that. So AB is the diameter. It's going straight across, straight across the circle. What is the arc measure of A, B, C in degrees? So A, B, C. So they're making us go the long way around. This is a major arc they're talking about. Let me draw it. Arc A, what is the arc measure of arc A, B, C. So we're going the long way around. So it's a major arc. So what is that going to be. Well it's going to be in degrees, the same measure as the angle, as the central angle that intercepts it. So it's going to be the same thing as this central angle right over here. Well, what is that central angle going to be? Well, since we know that this is the diameter, since AB is the diameter, we know that this part of it is going to 180 degrees. We're going halfway around the circle. One hundred eighty degrees. And so if we wanna look at this whole angle, the angle that intercepts the major arc A, B, C, is going to be 180 degrees plus 69 degrees. So we're going to have 180 degrees, plus 69 degrees which is equal to, what is that, 249, 249 degrees. That's the arc measure of this major arc A, B, C.