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Sec 10.2 Notes AP

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10.2 is about central angles in the arcs that they make no you know what an angle is an arc is a curve it is specifically part of a circle is an arc any part of that Circle that you cut off is considered an art so the first thing we're going to talk about is a central angle in a central angle is its vertex on the center so this is a central angle now we can name this angle by the three letters that is dhx  so angle dhx names this or 8X HD now this Google which is right here that's your angle makes two arcs it cuts off one on the inside right here and then it cuts off and separate one on the outside and one is definitely much bigger than the other now this red one that we have cut off this is a minor Arc right here now to name an arc you name it by its two endpoints and minor Arc you can just Name by the two endpoints this is DMX and what you would do is that you would do DX with an arc on top of this other one right here on the backside is a major arc and you have to use three letters to do my major art because the minor Arc is from d2x but the major started D and goes all the way around X but you got to give it directions Saudi RX and so we would name this one D x with an arc on top so two letters for minor you have to have three for a major now you can use three for a minor it's okay it'll show the direction that it's going a minor now there's a couple things about these two arcs that you need to know okay so the minor Arc right here  okay always on the inside of the angle  and it is always less than 180 degrees okay the major arc is on the outside  of the central angle and it is always greater than 100 need to grease know they have measurements just like angles have measurements of degree and here's the thing you know that there's 300 60 degrees in a circle will there's 360° on all the angles in the center and on the outside of the circle so this angle whatever it's measure is cuts off the same part of the circle that makes the arc so the actual measurement of this angle equals the measurement of the minor so are central angle  equals the minor Arc and now we're talking about measurements in these are always in degrees so that minor are the major arc  will full 360 degrees minus the central angle or the minor Arc because they're the same amount so that's how you're going to find their measurement so if we know that this is 100° this angle then this Arc is going to be 100° and the major arc is going to be too hun 60 degrees 360 - the 100K so we have our central angle are minor Arc enter major arc okay and the central angle equals the measurement minor Arc in the major arc is 360 - the central angle or minus the minor Arc  okay the other thing that you need to know here is minor arcs are less than 180 major Arts are greater than 180 what if it's exactly 180 will think about this a semicircle you kind of know what a semicircle is it's half of a circle right well half of 360 is exactly 180 so if we have a diameter anywhere in the circle it cuts the circle in half so you have half of a circle here which is called a semicircle so if an arc is exactly 180 it's called a semi circle and a diameter cuts that semi circle  are cuts a circle into 2 cm Necco soap you have 180 degrees on this side and you have 190° on this side okay so it's made buy a diameter  and it has two equal exactly a hundred ninety degrees so you're going to have a minor and major arc or a semicircle now this page is one postulate and three things they're just some rules about circles and a couple of them are very basic so this one partial it 10-1 arc addition postulate the measure of an arc formed by two adjacent arcs is the sum of the measure of the two Arts soap BC that art is a B+ BC you can add to arts and make the bigger art that's all it's an congruent circles how do you have congruent circles same shape same size all circles are the same shape so they just have to be the same size and we is that by their radii circles are congruent circles if and only if they have the same radius so any two circles that have the same radius or diameter have to Beacon net  10-4 congruent central angles in the same Circle or in congruent circles two minor arcs are congruent if and only if their corresponding central angles are congruent so here we have a pair of vertical angles in a circle we still have vertical angles if these two angles are equal they're in the same Circle then these two arcs are equal which if the angles are equal they angle equals Arc tangle equals Arts Archer equal and if you were to have two circles to another Circle over here if it had the same radius so they're congruent circles if the angles are equal the Arts equal awesome and then the last one there in 10.5 similar circles will think about what similar means similar means same shape different size all circles are the same shape than all circles are similar just basic knowledge  how much for size 1 through identify the given Arts as major minor or semi-circles then find the measure of each Arc okay so we have this the first thing I see is purple where to eat a c and b your perpendicular what is that do well that gives us 90 degree angle so we know that's a right angle and we know that's a writing what are thing you always want to look for his look for diameters those are straight lines and they cut off semi-circles also so here's a straight line if this is 40 can you find this one since that's a so 180 - 40 and we get 140 degrees for this Angle now we know all arson each one of these is a central angle because they all have their Center on the circle I mean vertex on the center so now central angles you know the minor arcs they make so this Arc is 90° this Arc is 90° this Arc is 140° and this Arc is 40 degrees if you add all of that up you  you get a 360° total on the whole thing and that's a circle art let's look at ABS two letters it has to be a minor Arc cuz that's the only one you can name with two letters major and semi-circles you have to name the three so baby  is this Arc it's a minor Arc  and it's 90 degrees  ABC started a go to be go to see that is a semicircle that's half of it and that is 180 grease from A to B all the way to D you need to add 1990 and 40 you get 200 20 degrees so that is a major arc K this is from B to C B to C minor Arc 90°  BAC so start it be go to a and go all the way around to see so we've got 91 4 M40 and you get 270 degrees which is a major arc d a b from D go to to be so 140 + 90 you get 230 degrees which again is a major Park in from a 2-D 2-D that's 140 degrees so that play minor it's less than 180 and then see daddy is 40 degrees and that is a minor Arc because it is less than hundred and eighty degrees  number 9 in circle e above till weather ABC is congruent to ATC Arc explain why or not ABC that's 190 it's a semicircle a DC that's a semicircle its 182 so yes they are congruent because they have the same measure in the same Circle and it's because they're in the same Circle and they have the same measurement they have to be congruent okay in number 10 in circle play find the measure of De now want you to pause and I want you to try and work this out and see if you can figure it out remember linear vertical all those kind of angles still apply then started up and see if you got it right okay you should have done that D was 125 degrees now the way you would have got what is these are vertical angles so if this is 55 this is 55 this is a diameter so it makes a straight line so 180 - 55 is 120 innocence this angle is 125 this art is 125  11 read it see if you can figure it out and then Paws and come back and see if you got it right couple of hints okay first of all it's as fun the value of x then find the measure of angle a be so we have to find FedEx's to plug it back in think about you have every Ark around that Circle what is that have to add up to work on it  okay you should that in the measure of angle of not angle of rkd to be 125 degrees now X was 25 you added 5x + 40 + 70 75 + 2x equals 360 collect like terms your ex's is 7 x 40 7075 is 185 360 then subtract the 185 over here and you get 175 so 7x equals 175 / 7 x 25 they want the measure of Arc AB which is 5X so 5 * X is 25 * 25 is 125 major assignment don't forget that number Section 1 and 2 is due as soon as you walk in the door on Thursday you'll have Tuesday in class to ask me questions and work on

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Sec 10.2 Notes AP

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