Understanding Central and Inscribed Angles in Circles: Key Concepts Explained

let's look at uh central angles and inscribed angles so I have a couple pictures here a couple circles that show

what a central angle and inscribed angle are um so central angle just as a reminder is an angle where the vertex is

at the center of the circle the very center inscribed angle is where the vertex touches the circles on the circle

so what so so what about them what's special about them well one special thing about a central angle is if I know

let's say this Arc here is 98° great boy bands of all time so this

central angle would also be 98 degrees this angle is the same as the intercepted Arc for an inscribed angle

let's say this is 100 and looks like it's about 164° is my guess the inscribed

angle is half it's always half of the intercepted Arc so this would be 82° right there okay so what I did was I

figured let's look at a few examples and make sure we you know we get it so I have six examples here if I have this

Arc AB is 90 Dees question that would most likely come up is what is angle M well we just talked about this the arc

is the same as the central angle so this would be 90 and vice versa if they give you the central angle you can assume

that the arc is the same okay uh if this is 160 degrees this Arc from C to D then half of 160 would be 80 so if you're

watching this for notes purposes what I would encourage I probably should have said this before I went over these two

is maybe pause it I'll scroll down slightly so you can see the others copy them down and see if you can figure out

like the missing angles and arcs on these shap shapes okay this angle is 70 so what

would this Arc be right here well uh if this is 70 we just talked about this the angle is half of the arc so here we

divide it by two but if we know the angle we're going to multiply by two so 70 * 2 is 140 be about

140 all right here the bottom one we have an arc and and we got some funky looking quadrilateral happening here so

one question would be what's the central angle well it's the same as the outside Arc so 100

degrees another question would be what is the inscribed angle well if this is 100 this would

have to be half so 50° other question would be well what is this angle

and this angle and the answer is I don't know there's really no way to tell without

more information um they look like they're about the same but they're not necessarily the same so we can only find

these two um a third question or I guess fourth question might be uh if we want

jkl if I asked you for this J KL that would be how much well remember Circle 360 so if this is 100 that means

from J to K to L would have to be 260 because 360 minus 100 is 260 okay so this time you're given the

inscribed angle and if I extend out these lines n to p and N to Q if this is 40° that means this intercepted Arc here

has to be twice that so this would be 80° and then this central angle would be

80° obviously it's uptu so it's not a great drawing but this is how the math would shake out okay so what we're going

to find on this one is we're going to figure out what this angle is we're going to figure out what this angle is

we're going to figure out what this Arc is and what this Arc is and we could probably also figure out

these angles here we'll do that so let's start with this Arc SX if we start at

V and we extend out our lines here for angle V I guess this would be angle

xvs this is 20° and that means that this Arc SX is the intercepted Arc for this 20°

inscribed angle so this would have to be 40° because it's twice as big as the inscribed

angle sorry about the Bell uh so let's clear this

out and then if we want to do the same thing for this if this is 30 Dees we extend that out this would have

to be 60° because it's double that okay so I'm going to clean this up again so then a question might come up

with what is this angle right here at T well if you look this angle at

T the intercepted Arc is 40° so the inscribed angle would have to be half of that which is

20 and then likewise with this angle if you're asked to find whoops I got a little bit excited

there let me clean this up if you're asked to find this angle here well the intercepted Arc would be

60 degrees so this would be half which would be 30 degrees and then like I mentioned

earlier if we wanted to find this angle we have a triangle here so if this is 30 and 20 that's 50 we know these

three angles in a triangle have to add up to 180 so 180 minus 50 would be 130° and then you could figure out the

other three because we have some vertical angles here so this is 130 and we'd have 50 and 50 here and here

because 130 and 50 are a linear pair so they have to be supplementary okay so there's a few

examples of central angles and inscribed angles