Geometry 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 angles0079

(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