### Angles and Arc

- Central angle: An angle whose vertex is at the center of a circle
- The sum of the measures of the central angles of a circle all adjacent to each other is 360 degrees
- Know the difference between a minor and major arc
- Arc measure: The measure of a minor arc is the measure of its central angle. The measure of a major arc is 360 degrees minus the measure of its central angle
- The measure of a semicircle is 180 degrees
- Arc Addition Postulate: The measures of an arc formed by two adjacent arcs is the sum of the measures of the two arcs
- Arc Length: A part of the circumference proportional to the measure of the arc compared to the entire circle
- Concentric Circles: Circles that lie in the same plane with the same center, but have different radii
- Concentric circles, along with all other circles, are similar
- Circles that have the same radii are congruent
- Congruent circles are also similar circles
- Two arcs of one circle with the same measure are congruent arcs
- Congruent arcs also have the same arc length

### Angles and Arc

∠BAC is the central angle.

The sum of the measures of the central angles of a circle all adjacent to each other can be more than 360

^{o}.

- Minor arc: ⁀BC
- Major arc: ⁀BEC

Major arc: ⁀BEC

- Minor arc: m⁀BC = 110
^{o} - Major arc: m⁀BEC = 250
^{o}

^{o}

Major arc: m⁀BEC = 250

^{o}

If O is a point on ⁀MN, then m⁀MO + m⁀ON = m⁀MN.

.

- [120/360] = [x/(2πr)]

Concentric circles are ______ congruent.

Congruent arcs have same length.

Circle A and circle C are congruent, AB = 12, find the length of ⁀DFE.

- The radius of circle C is 12
- m⁀DFE = 255
^{o} - [255/360] = [x/2p*12]

Circle A and circle C are congruent, determine whether ⁀BD is congruent to ⁀EF .

- m⁀BD = 360 - 85 - 150 = 125
- m⁀EF = 125
- The radius of the two circles are the same

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Angles and Arc

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Central Angle 0:06
- Definition of Central Angle
- Sum of Central Angles 1:17
- Sum of Central Angles
- Arcs 2:27
- Minor Arc
- Major Arc
- Arc Measure 5:24
- Measure of Minor Arc
- Measure of Major Arc
- Measure of a Semicircle
- Arc Addition Postulate 8:25
- Arc Addition Postulate
- Arc Length 9:43
- Arc Length and Example
- Concentric Circles 16:05
- Concentric Circles
- Congruent Circles and Arcs 17:50
- Congruent Circles
- Congruent Arcs
- Extra Example 1: Minor Arc, Major Arc, and Semicircle 20:14
- Extra Example 2: Measure and Length of Arc 22:52
- Extra Example 3: Congruent Arcs 25:48
- Extra Example 4: Angles and Arcs 30:33

### Geometry Online Course

### Transcription: Angles and Arc

*Welcome back to Educator.com.*0000

*For the next lesson, we are going to go over angles and arcs of circles.*0002

*The first angle we are going to go over is called the central angle.*0009

*A central angle is an angle within a circle whose vertex is at the center of the circle.*0013

*Here is an arrow to show you that this is a central angle.*0020

*This arc right here that the angle is hugging is called the intercepted arc, because it is intercepting the circle.*0026

*This right here...we are going to go over arcs in a little bit, but this is an arc--it is part of the circle.*0037

*The arc that the angle is essentially hugging is called the intercepted arc, so this is the intercepted arc.*0045

*Again, remember that this side of the central angle is a radius of the circle, and so is this side; both sides have to be radii of the circle.*0062

*The vertex is at the center; and this is called the central angle.*0072

*Here, we have a whole bunch of central angles; there is the center; and if we add up all of these central angles*0079

*(remember that no central angles can be overlapping with each other--they have to be adjacent;*0092

*they have to be side-by-side), all of these central angles together are going to add up to 360.*0098

*And we know that, if we go all the way around, a full circle is going to give us 360 degrees.*0109

*It is all of these central angles added up together--that forms a circle; all of these together are going to be 360 degrees.*0119

*Remember that, if you are missing one of the angles from here, but you are given three out of these four different central angles,*0129

*then you just have to add them up and subtract that from 360--that is the sum of central angles.*0137

*So now, arcs: remember: arcs are part of the circle--they are on the circle.*0149

*And these central angles--we have the same central angle for this circle and this circle--they are the same central angle.*0159

*But there are two different arcs that we can talk about.*0166

*Now, for this one, if we just look at this intercepted arc from here all the way to here, the arc right here, this is called the minor arc.*0190

*It is less than 180 degrees, so it is less than half; and it is the intercepted arc from the central angle.*0191

*So then, here we can call this arc AB; and the way you do that--you write AB, and you have to write a little arc above it.*0202

*We know that, if it is a segment, then we do a little flat line above it.*0212

*But because it is an arc, it has to be a little curve.*0218

*Now, that is a minor arc; now, for this one, the other arc, right here, from here all the way around,*0225

*like this, it is called the major arc, because it is bigger; it is greater than 180 degrees.*0237

*This is the major arc; the same central angle divides the circle up into two different arcs, the minor arc and the major arc.*0251

*Now, the major arc is a little bit different, because, if I say arc AB, then whoever I am talking to has to know which arc I am talking about.*0261

*If I just say arc AB, how are they going to know if I am talking about this arc or this big arc, the major arc?*0273

*When it comes to the major arc, there has to be another point here, like this; and you have to give it three variables.*0282

*You have to say arc ACB, with the same little arc over it.*0292

*If you just say arc AB, using two points, then you are talking about the minor arc.*0302

*If you are going to talk about the major arc, you have to name it with three variables: arc ACB.*0308

*And that is how you know if you are talking about the minor arc or the major arc.*0317

*Arc measure: the measure of a minor arc is the measure of its central angle.*0327

*So again, we know that the central angle is the angle whose vertex is on the center.*0335

*This central angle, this angle right here, and the intercepted arc have the same exact measure.*0344

*And it is important to know the difference between measure and length.*0352

*Now, we are going to go over length in a little bit; but the measure is the angle measure.*0357

*Here, whatever angle measure this is--let's say this is 100--then the measure of the intercepted arc is also going to be 100 degrees.*0365

*If I label this as AB, and this as D, then I can say that the measure of angle ADB is 100, and the measure of arc AB is also 100.*0379

*Keep in mind: the central angle and the intercepted arc have the same exact measure.*0405

*Now, that is the minor arc; if you want to look for the major arc, which is all this right here...*0410

*well, we know that all of the angles within the circle have to add up to 360.*0418

*So, you can just do 360 minus this 100, and then that will give you the measure of the major arc.*0423

*And we know that the measure of a semicircle (oh, that is a bad circle; I will do that again...let's say a circle like that,*0432

*but a little bit darker)...there is the center, and I am going to draw a segment (chord) through the center, which makes it a diameter;*0452

*the measure of this angle right here--even though it doesn't look like an angle, it is still an angle;*0465

*this still has a measure of 180 degrees--that is what it is saying that the measure of a semicircle is.*0477

*If this angle is 180 degrees, then all of this right here, the arc--that would be the intercepted arc of this angle, so this is also 180.*0485

*So, that is what it is saying: a semicircle, half the circle, has the measure of 180 degrees.*0498

*Arc Addition Postulate: The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.*0507

*If you remember, from the beginning of the course, you went over the Segment Addition Postulate and the Angle Addition Postulate.*0517

*This is the exact same thing; we are just using arcs instead of segments and angles.*0528

*Arc Addition Postulate: If you have this arc right here, PR, and Q is between (it doesn't have to be the midpoint;*0536

*just anywhere between) P and R, then we know that the measure of arc PQ plus the measure of arc QR is going to be the measure of arc PR.*0544

*If Q is a point on PR, then the measure of PQ, plus the measure of QR, is going to equal the measure of arc PR.*0563

*That is the same exact thing as the Angle Addition Postulate and the Segment Addition Postulate.*0573

*Arc length: now, this is different from arc measure.*0586

*We went over arc measure already; we know that arc measure is the same measure as the central angle.*0591

*So then, if we are looking for the measure of this arc right here, it is the intercepted arc of the central angle, so they are going to have the same measure.*0602

*Length, however, is different: length is talking about the part of the circumference.*0613

*It is like if this whole thing is 2πr; then what is the length of just this part right here, this piece?*0619

*So, it is kind of like perimeter; you are looking for the part of the perimeter.*0629

*But you know that the perimeter for a circle is the circumference; it measures the outside, the length of the whole circle.*0634

*But the arc length...we are just trying to figure out the length of this small portion of it, so it is different than measure.*0643

*The way you do this: now, if you have the measure of the central angle here, θ,*0652

*and this is the radius, r, we are going to make this into proportions.*0663

*To find the length of arc AB, we are going to look for the measure of arc AB; that is what we are going to look for.*0669

*We are going to create a proportion, since we know that it is proportional; the arc with all of the measure is going to be proportional to the arc length.*0680

*We are going to create a proportion to help us solve for the arc length.*0692

*We are going to have to create a ratio for the measure, and then we are going to create a ratio for length.*0700

*And then, we are just going to make them equal to each other.*0713

*To find the ratio of the measure...here, from arc AB, this is the measure.*0718

*If this is θ, if the central angle has a measure of θ, then so will arc AB.*0731

*So then, that will be θ over the whole; it is kind of like part over whole, θ over...what is the measure of the whole thing? 360.*0739

*So, when it comes to measure, this is going to be our ratio.*0756

*And then, the length: for the length, the part is the one that we are looking for.*0763

*So then, this is going to be arc (I shouldn't put "measure"--just "the length of AB")...I am going to write x;*0771

*let's say that this is x right here; that is what we are looking for--that is the part.*0782

*And then, the whole thing: well, if you are trying to find the length of the whole circle, that is known as circumference,*0789

*so it would just be circumference, or you could write 2πr, which is the formula for circumference.*0796

*This is essentially the formula to figure out the length of the arc.*0802

*And all it is: it is just part and whole for measure, and the part and the whole for the length.*0810

*We know that the measure has to do with angles and degrees: the central angle is the part for the measure,*0817

*and then it is over the whole circle, which is 360; then with length, it is the length of the arc AB,*0829

*which is what we are trying to find, over the whole thing, 2πr.*0837

*Let's say, now, that θ (we are doing an example) is 50 degrees, and the radius is 10.*0844

*Let's solve for it: this is 50, so arc AB has an angle measure of 50, over 360, which is equal to x,*0860

*which is what we are looking for (we are looking for this, the arc length), over the whole length, which is 2π times 10.*0878

*So then, here we have a proportion; and we are going to just cross-multiply and solve for it.*0890

*If you have a calculator (I have a calculator here on my screen), you are going to do 2 times π times 10 times 50, which is going to equal 360x.*0896

*That is 2 times 10, times 50, times π; and then, from here, you can just divide the 360; and we get x as 8.73.*0918

*That is the arc length--that is the part of the circumference of the circle.*0954

*And we will do another example later on in this section.*0962

* Concentric circles are two or more circles that lie on the same plane (it is flat), and they share the same center.*0967

*This circle right here and this big circle right here--they are known as concentric circles, because they have the same center.*0983

*But concentric circles have to have a different radius, because the radius, obviously, for this one is going to be shorter than the radius for this big one.*0992

*They have different radii; and we know that they can't be congruent.*1009

*Concentric circles cannot be congruent, because they do have different radii.*1019

*And so, instead, they are going to be similar; and what makes them similar?*1027

*Remember: anything with the same shape, but a different size, is similar; so these two circles...*1033

*And all circles are actually similar; concentric circles are going to be similar...it doesn't matter if I draw another circle here;*1041

*I can draw a tiny little circle; it is still going to be similar to both of those, because it is the same shape.*1049

*All circles have the same shape; they just can be different sizes--some could be big; some could be small.*1054

*But since they do have the same size, they are all going to be similar; so just keep that in mind.*1061

*A few more things to go over: circles that have the same radii are congruent.*1074

*So, if you have a circle here, and that is r, and a circle here, and that is also r--the same variable--then they are going to be congruent.*1081

*And if you were to take one of these circles and place it on top of this one, then they would be exactly the same, because they have the same radius.*1096

*That is why they would be congruent.*1106

*Congruent circles are also similar circles; they are the same size and the same shape.*1110

*We know that, any time they have the same shape, whether or not they have the same size, they are similar.*1118

*So, congruent circles are also similar.*1124

*Two arcs of one circle with the same measure are congruent arcs.*1128

*Now, they have to be from the same circle, because we know that angle measures don't have to do with size.*1135

*We can have something really big, and we can have something really small, but they will have the same angle measure, or the same measure.*1144

*Within the same circle, two arcs with the same measure will be congruent.*1153

*If you have, let's say, from here to here, and from here to here--let's say that this arc right here is 90 degrees,*1164

*and this arc right here is also 90 degrees--then they will be congruent arcs.*1178

*Congruent arcs also have the same arc length: well, if they have the same measure (not angle measure, because they are arcs),*1185

*then they will also have the same length, because they are the same proportion of the whole circumference, and that is what length is.*1198

*So, that means that this is as long as this will be, because they have the same measure.*1207

*Going over examples: Name a minor arc, a major arc, and a semicircle.*1216

*A minor arc (and we have a few, so you can just name whichever one) can be arc AB, arc CD, arc DC, arc DA or AD.*1223

*Any of those can be minor arcs; so we can just say that a minor arc is arc AB.*1241

*A major arc: make sure that the major arc is greater than 180--that is what makes it major.*1250

*And you have to use three variables to name a major arc.*1260

*So, if we want to talk about this all the way to D, that would be a major arc.*1265

*And we have A, B, C, D, but you are not going to use all four; you can just use A...*1276

*you have to start with A, and you can name either B or C, just one of those; and then D would be the last one.*1282

*Or if you want to go the other way, then it would be D, then CA; or you could do DBA.*1290

*It doesn't matter, as long as you name one of the points that is in between those two.*1298

*A major arc would be arc ABD; if you said arc ACD, that would be the exact same thing.*1306

*And a semicircle: we know that a semicircle is exactly half, so this is a semicircle.*1317

*And for this one, you don't have to name three; you can just name two, because,*1328

*whether you go this way or whether you go that way, it will be the same measure, so I can just say arc AC.*1334

*You can also, if you want to specify which semicircle you are talking about, name the variable in the middle.*1342

*For our semicircle, we can say arc ABC, because AC can be this one or this one.*1349

*Sometimes, it doesn't matter; they have the same measure, and we know that a semicircle has a measure of 180.*1359

*But if I want to talk about this one, then it would be arc ABC.*1366

*The next one: Find the measure and the length of arc AB.*1372

*Here, that is arc AB; when it comes to measure, we know that the measure of arc AB (I can write it like this--*1379

*the measure of arc AB) is going to be the same as the central angle; this is the central angle; and that will be 96 degrees.*1395

*Now, for arc length, remember: we have to create that proportion.*1408

*I am going to create a proportion comparing the measure and the length.*1414

*For the measure, the part and the whole right here are going to be from the arc and from the whole circle.*1426

*From the arc, the measure is 96, over the whole circle, 360.*1435

*If you want, I can just write arc and circle.*1444

*And then, the arc length is what I am looking for; that is x, over the length of the circle (is the circumference, so)--that will be 2π times the radius, 2πr.*1453

*And that is 8; so then, from there, you are just going to solve this proportion.*1469

*This will be 16π times 96 equals 360x; and then, we divide the 360, so use your calculator; and x should be...*1476

*And your calculator should have a π button, but if it doesn't, then you could just use 3.14...and we get 13.40.*1507

*This is the arc length; now, if this was in, let's say, inches, then this will also be in inches; that is arc length.*1529

*The measure is this, and the length is that.*1541

*The next one: Tell whether the given arcs are congruent.*1550

*Here are the arcs: arc AB and arc EF; we are trying to see if they are congruent; and for this one, it is arc AC and arc BD.*1555

*Here, we know that arc AB has a measure of 75; so the measure of arc AB is 75.*1573

*And because they are from two different circles, I am just going to say that they are congruent,*1592

*because these two circles have to be congruent for us to compare their arcs.*1599

*So then, these two circles are congruent; this is arc AB, 75; and now, we can look to see if this arc is going to be congruent to that arc,*1603

*because otherwise, I don't know the radius; and if they have the same radius, then I can say that they are congruent.*1614

*And then, I can compare their arcs; but since I don't know their radius, you just have to know that those circles are congruent.*1622

*Now, for this one, I don't know what the measure is for EF, so I want to find...*1629

*since I know that this central angle has the same measure as the arc, as long as I find this angle measure, I can find the measure of that arc.*1639

*So, how do you do it? Right here, this is 110, 165, and then that is unknown.*1648

*All three angles together are going to add up to 360; so this can be 165 + 110 is going to be 275;*1655

*I can subtract that from 360, and that is going to be 85 degrees right here.*1667

*If this is 85, then this will be 85 also; so this one is "no"; the measure of arc EF is 85 degrees.*1693

*This one is 75; this one is 85; so this one is "no"--those arcs are not congruent.*1711

*And this one: here, this is 55, so the measure of arc AC is 55; and then, for BD, we have to find that one.*1720

*Now, this is 55, and 70...we are missing this one and this one here.*1741

*But remember that this arc CD is a semicircle; so this one has a measure of 180.*1747

*So then, to find this arc measure, you can just do 360 - (70 + 55 + 180 + x).*1755

*Now, you can also, because this is 180, know that this is also 180; you can do that, also.*1778

*If you want to use the whole circle, you are just going to do it this way; or you can just do 55 + 70 + x and that will be also 180.*1789

*Here, we are going to get x as 55; so then, this arc BD is going to also be 55; the measure of arc BD is 55.*1798

*Therefore, these two arcs are congruent; so then, "yes," they are congruent.*1822

*And the last one: Use the circle to find the following.*1834

*Here, we are going to look for a few different things; the first one is going to be x.*1839

*Here, the center is P, so that is circle P; and then, we have BC, and we have CD being congruent; we have this thing, a semicircle...*1851

*Here, how do we find x? Well, this one right here, arc BAE, is 180 degrees, because it is a semicircle.*1870

*So, I can just take this x, add it to 5x, and make that equal to 180.*1881

*5x + x = 180; 6x = 180; divide the 6; x is equal to 30.*1888

*The next one: The measure of angle CPD: now, to find this one, we know that this is 30;*1907

*this is 150; so for the remaining angle measures, the central angles, we want to look for this one.*1925

*Now, since we know that this is 30, this one and this one are congruent, because the arcs are congruent.*1940

*Well, here, we know that this is also a semicircle; so if I label this, let's say (what variables shall I use?), as y, then this angle will also be y.*1948

*That means that this one, plus this, plus that, is going to equal 180; so I can do 180 = 30 + y + y.*1963

*180 = 30 + 2y; that is 150, since I subtract the 30; that equals 2y; I am going to divide the 2; I am going to get 75 = y.*1984

*y, we know, is the measure of angle CPD; so this will be 75 degrees; this is 75; this is also 75; and this one will be 30.*2001

*Now, this is 30, and I know that because these are vertical angles.*2023

*Since this is a diameter, and EB is a diameter, these will be vertical angles.*2031

*Or you can also say that this BCE, this whole thing right here, is a semicircle; it is going to add up to 180. So then, that is 30.*2037

*The measure of arc ED: well, the central angle right here is 30, so then this has to be 30 degrees.*2050

*And the measure of arc EAC is all of this together, this right here: 150 and 30 and 75.*2064

*So, this measure of arc EAC is going to equal 150 + 30 + 75.*2079

*This is 180 + 75, which is going to be 255; so that is the answer there: x is 30.*2093

*That is it for this lesson; thank you for watching Educator.com.*2120

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