Geometry Angles

Welcome back to Educator.com.0000

This next lesson is on angles.0002

OK, first, let's go over what an angle is: an angle is a figure formed by two non-collinear rays with a common endpoint.0007

Let's go over what a ray is: a ray is a segment like this, with one endpoint and one end extending indefinitely.0021

It is like part of it is a line, and part of it is a segment--one endpoint and one end going continuously.0035

An angle is a figure when we have two rays together, like this here, with a common endpoint.0048

Here is one ray, and here is another ray.0059

Now, just to go over rays a little bit: if you have a ray like this, then to write it...0063

now, we know that, if we have a line, we write it like this, using symbols...0072

if we have a ray, then you are going to write it like this, with one arrow.0078

Now, be careful: you cannot write it like this--you can't do that, because the endpoint is at A.0082

So then, the arrow has to be pointing to the point that is closer to the arrow.0095

So, if it is AB, it is going this way; so then you have to point the arrow that way: AB.0101

It can't be BA, because that is showing you that that is going the other way.0106

Now, also, don't write it like that, either--the direction has to be going to the right.0110

This is the right way; not like that, and not like that.0120

Opposite rays are when you have two rays that extend in opposite directions.0130

If they go in opposite directions...they have a common endpoint; one ray is going this way; the other ray is going this way.0137

Those are opposite rays: you have two rays: one is going to the right, and one is going to the left.0147

They form a line, because they are going in exactly opposite directions.0153

So then, opposite rays are two rays that extend in opposite directions to form a line.0159

Here is a diagram of an angle; now, this angle right here, where the two rays meet, the common endpoint is called the vertex.0166

And each of these rays in the angle are called sides.0187

This is a side, and then this is a side; this is a vertex, and these are sides, of an angle.0193

List all of the possible names for the angles: here, this is actually made up of three angles.0204

We have this angle right here, and I can just label that angle 1; this is another angle right here, and then this big angle.0214

To list all of the names, you are going to have to look at the different angles.0229

Depending on what angle you are looking at, you are going to call it by a different name.0236

This first one right here is going to be angle ABD; make sure that the middle letter is the vertex.0240

This has to be angle ABD; if you do angle ADB, that is going to look like that; ADB is like this, and that is not the angle, so it has to be angle ABD.0253

You can also say angle DBA, as long as the vertex is in the middle.0267

This can also be angle 1, because this whole angle is labeled as angle 1.0277

That is the first one; and then the next one can be angle DBC (again, with the vertex in the middle), angle CBD, or angle 2.0290

And then, the big one...now, if I just had an angle like this, just a single angle, and the vertex...0307

let's say this was different...EFG: if I had an angle like that, this can be angle angle EFG, angle GFE, or angle F.0314

It could be angle F, even though F is just the vertex; as long as you have only one angle--0333

this vertex is only for a single angle--then you can name the angle by its vertex, so this can be angle F.0341

In this case, I have three different angles here, so I can't label this whole...0351

even if I am talking about the big one, the whole thing, I can't label it as angle B, because angle B...this is a vertex for three different angles.0358

So, I cannot label it angle B; so instead, you have to just list it all out.0369

You are going to just say angle...the big one is label ABC; angle CBA; and that is it--those are the only two names for that big one.0375

It is not by 1 and 2; the 1 is for this angle, and then the 2 is for this angle.0390

An angle separates a plane into three parts; if I have an angle (actually, let me just draw it out--0402

there is my angle), then it separates it into its interior (the interior is the inside--that is all of this right here, on the inside),0414

and then the exterior (which is all of this, the outside of the angle), and then on the angle--the angle itself.0428

So, when you have an angle, there are three different parts: the inside, on the angle, and outside the angle.0441

And just so you are more familiar with these words: interior/exterior is inside/outside.0451

Angles are measured in units called degrees: so if you have this angle right here, this angle could be 110 degrees.0458

The number of degrees in an angle is the measure.0475

If I want to say that this angle is angle ABC, then I can say the measure (and m is for the measure) of angle ABC is 110.0480

That is how you would write it: the measure...and you write the m in front of the angle, angle ABC (what you name it) only when you are giving the measure.0499

The Protractor Postulate: from the last lesson (the last lesson was on segments), remember: we went over the Ruler Postulate.0519

This one is the Protractor Postulate; it is the same concept, but then, because it is an angle, you are not using a ruler; you are using a protractor.0529

And again, a postulate is any statement that is assumed to be true.0537

This is the protractor postulate--this whole thing right here, the statement.0542

Once we go over it, we can assume that it is true, because it is a postulate.0546

Given AB and a number r between 0 and 180, there is exactly one ray with endpoint A extending on either side of ray AB0552

such that the measure of the angle formed is r; all that that is saying is that,0567

if you have this ray right here (this is A; this is B), now, if you have a ray AB, there is only one ray0575

or a single ray that you can draw to get a certain angle measure.0597

If I want an angle measure of 80 from AB, then there is only one ray that I can draw that is going to give you an angle measure of 80.0607

You can't draw two different angles; but it is going to be on both sides--so it can be this side,0619

or I can have an angle on...this is AB...then it can go on the other side, too; so this also can be AB.0624

That is all that they are saying: just for this, if you draw a ray like this to make it 80, you can't draw a different type of ray to also make it 80.0634

It is going to be something else.0645

So, for a number r between 0 and 180, there is only one ray that you can extend from this ray right here to give you that angle measure.0647

And the Protractor Postulate is as if you put this at 0 on your protractor, if you have your protractor like that;0661

and then here is your protractor, and then you have all of your angle measures; that is how you would read it.0677

And then, whatever this says right here--this number--that is going to be your angle measure of this.0688

Make sure that this endpoint is at the 0 of the protractor.0694

Angle Addition Postulate: from the last lesson, the segments lesson, we went over the Segment Addition Postulate.0702

The Segment Addition Postulate was when we had a segment that was broken down into its parts.0711

For the Segment Addition Postulate, remember, you had...this is AB...B is just anywhere in between A and C;0720

then we can say that AB + BC equals AC; so this plus this equals AC.0730

In the same way, we have the Angle Addition Postulate.0741

And all that it is saying is that, if R is in the interior (remember, interior is inside) of angle PQS, the measure of angle PQR0748

plus the measure of angle RQS equals the measure of angle PQS.0764

Measure means that you are talking about degrees, the number of degrees in the angle.0776

If the measure of angle PQR, let's say, is 30 (let's say this angle measure is 30),0785

and the measure of angle RQS is, say, 50; then the measure of angle PQS:0794

you just add them up, and then it is going to be...the whole thing is 80 degrees.0805

That is just the Angle Addition Postulate; in the same way, remember, if this was 3, and this was 5, then AC...you add them up, and you get 8.0812

It is the same thing--the Angle Addition Postulate.0824

You can also say the other way around: if the measure of this angle, plus the measure of this angle,0828

equals the measure of the whole thing, then R is in the interior of angle PQS.0837

Classifying angles: let's go over the different types of angles.0852

An acute angle is any angle that is less than 90 degrees; less than 90 looks like that--an acute angle.0857

A right angle is any angle that measures perfectly 90 degrees, like that.0871

And then, an obtuse angle is any angle that is greater than 90, like that.0882

This is a small angle, a right angle, and a big angle--an obtuse angle.0893

Angle bisector: we also went over a segment bisector.0905

A segment bisector was when you have a segment (or...it could be a segment; it could be a line;0909

it could be a ray; it could be a plane)...anything that cuts the segment in half; it intersects the segment.0919

Let's draw a line: this line intersects this segment at its midpoint.0929

That means that this line is called the segment bisector, because it is bisecting this segment; it is cutting it in half.0936

The same thing happens with an angle bisector: it could be a segment, ray, or line that divides the angle ABC into two congruent angles.0944

This ray, ray BD, is an angle bisector; BD, that ray, is the angle bisector of angle ABC.0955

Again, the bisector is whatever is doing the cutting, the dividing; it has to divide it into two congruent parts; and the same here.0971

This is a segment bisector; this would be the angle bisector, because it is intersecting the angle at exactly its middle.0985

Then, if this is 30, then this has to be 30.0996

Angle relationships: adjacent angles are angles that are next to each other with a common vertex and a side.1006

Angles 1 and 2 are adjacent angles, because they share a side, which is this right here, and a vertex.1028

If I have an angle like this, and then an angle like that, 1 and 2, even though they share a common side,1037

these angles would not be adjacent, because they don't share a vertex.1049

It has to be a side and a vertex; so this is not adjacent angles; these would be adjacent angles.1056

Again, two angles that share a side and a vertex are adjacent angles.1063

Vertical angles: if you have two lines that are intersecting each other1070

("intersecting," meaning that they cross each other--they meet, right here), then they form vertical angles.1077

Vertical angles, when they cross, would be the opposite angle; so this angle, right here, and this angle are vertical angles.1087

Non-adjacent means that they are not next to each other; they are not sharing a side and a vertex.1096

Even though vertical angles share a vertex, they are not sharing a side.1102

The two sides of this angle are this and this; the two sides of this angle are this and this.1106

They are not adjacent; they are non-adjacent angles formed by intersecting lines.1117

In this one right here, 1 and 2 would be vertical angles.1124

Now, this and this are also vertical angles; so then, in this one, 3 and 4 are also vertical angles.1127

There are two pairs of vertical angles when two lines are intersected; always, there are always vertical angles involved.1140

If I have an angle that looks like that, these are not considered...1150

because this is not a line, these are not vertical angles, and these are not vertical angles.1161

They have to be straight lines that are intersecting.1175

Linear pair: a linear pair are adjacent angles that form a line by opposite rays.1181

Remember: opposite rays form a line, so adjacent angles are two angles, like this, angles 1 and 2.1191

Here is angle 1, and here is angle 2; see how they form a line?1205

They are also adjacent, because they are sharing a side and a vertex.1213

A linear pair is two angles that just form a line; a linear pair is the pair of angles that forms a line.1218

Right angles formed by perpendicular lines have perpendicular lines; they are perpendicular;1233

then we know that this is 90 degrees; then that means that this is 90 degrees,1245

because we know that a straight line measures 180 degrees.1253

This is also 90, and this is 90; we are going to go over this again later, but perpendicular lines form right angles.1262

Supplementary angles are two angles that add up to 180, two angles whose measures have a sum of 180.1276

Supplementary is 180; complementary angles are two angles that add up to 90.1290

It has to be two angles that add up to 90 and two angles that add up to 180; this is 90.1302

Now, if I have an angle like this (let's say that this is 120 degrees), then I can say...1310

OK, if I ask you for the angle that makes it add up to 180, that is called the supplement.1325

This is 60 degrees, because these two angles together...if they are supplementary angles, then they have to add up to 180.1338

Two angles (1,2) that add up to 180...1347

If I want to compare them to each other, then I can say, "This is the supplement of this, and this is the supplement of that."1357

So, if I ask you, "What is the supplement of 120?" the answer will be 60.1364

"What is the supplement of 60?" 120; but together, they are supplementary angles.1373

The same thing for complementary: I have two angles; let's say this is 50, and this is 40; they are complementary angles; together they add up to 90.1380

But I can say that this is the complement of 40, and 40 is the complement of 50; together, they are complementary angles.1400

So again, supplementary is 180, and complementary is 90.1411

Sometimes, just be careful not to get confused between complementary and supplementary--which one is 90 and which one is 180.1418

Complementary starts with a C; it comes before S; C is before S, and then 90 is before 180; that is one way you can remember it.1428

Complementary is 90; supplementary is 180; C before S, 90 before 180.1440

Let's do a few examples: the first one: State all possible names for angle 1.1450

Here is angle 1; all of the possible names are...(now, I am not going to write measure of angle,1457

because I am not dealing with its degree measure)...just naming them, you are going to write1467

angle AFB, angle BFA (make sure that the vertex is in the middle); it is also angle 1.1474

Is that all? That is all; I cannot say angle F, because angle F...1493

that is the vertex of so many different angles that you can't use it to name any of those angles.1499

So then, that would be all--just those three.1508

Number 2: Name a pair of adjacent angles and vertical angles.1512

Adjacent angles would be...we can say that there are a lot; as long as they share a vertex and a side...1518

I can say that angles 1 and 2 are adjacent angles; angles 2 and 3; angles 3 and 4;1530

I can also say angle 4 and angle EFA, since this one doesn't have a number; I am going to say angle...let's see, 2...and angle 3.1540

Now, remember: angle 2 and angle 4 are not adjacent, because, even though they share a vertex, they don't share a side.1556

And for vertical angles, I can say angle AFE and angle...what is vertical to AFE?1566

It has to be angle BFC, or angle 2.1591

Be careful: right here, angle 4 and angle 1 are not vertical, because it has to be straight intersecting lines that form the two vertical angles.1599

So, you can say that 4 and 3 together are vertical to angle 1, because it is formed by the same sides, the lines; but not 4 and 1, and not 4 and 2.1613

The third one: Make a statement for angle EFC, using the Angle Addition Postulate.1628

The Angle Addition Postulate: remember, that means that this angle plus the measure of this angle is going to equal the measure of the full angle.1640

An the Angle Addition Postulate is different than an angle bisector; you can still apply the Angle Addition Postulate,1653

but for an angle bisector, it has to be cut in half; the angle has to be cut into two equal parts.1665

The Angle Addition Postulate could be like this, and you can say this, plus this small part, equals the whole thing, the whole angle.1672

That is the Angle Addition Postulate; I am going to say that the measure of angle EFD,1686

plus the measure of angle DFC, equals the measure of angle EFC; and that is my statement.1699

Number 4: Name a point in the exterior of angle AFE.1717

Angle AFE is right there; a point on the exterior...this is the interior, so it is anything outside of that.1724

I can say point B; I can say point C; or I can say point D.1734

I am just going to write point C; that is on the exterior.1737

The next example: Name a pair of opposite rays.1747

Opposite rays would be two rays with a common endpoint going in opposite directions.1754

I can say rays CF and CA...now, be careful; I can't say AC.1761

I can't say AC, because the ray is going this way, so I have to label it as CA, going towards that.1777

If the measure of angle ACB is 50, and measure of angle ACE, the whole thing, is 110, find the measure of angle BCE.1787

This is what I am looking for, x; that is the angle measure that I am looking for.1808

This is using the Angle Addition Postulate; so this, the measure of this angle, plus the measure of this angle, equals the whole thing.1813

So, the measure of angle ACB, plus the measure of angle ECB, equals the measure of angle ACE.1822

This will be 50; 50 plus the measure of angle ECB equals 110.1843

Subtract the 50; so the measure of angle ECB (or BCE--the same thing) equals 60 degrees.1855

Number 3: Name two angles that form a linear pair.1870

This one was to name a pair of opposite rays, two rays that form a line; this is two angles that form a linear pair.1878

That is a pair of angles that form a line; so I can say this angle right here and this angle right here,1889

angle FCD and angle DCA: angle ACD and angle DCF, or FCD.1899

Draw angles that satisfy the following conditions: Number 1: Two angles that intersect in one point.1928

We just need two angles that intersect in exactly one point.1935

I can draw an angle like this, and I can draw an angle like this; it doesn't matter, as long as they intersect at one point.1942

If it asks for two angles that intersect in four points, then that would be like this...1952

or this actually is two points; and then, if you went the other way, then you can just do four: so 1, 2...I meant 2.1963

The next one, measure of angle DEB plus the measure of angle BEF, equals the measure of angle DEF.1974

This is the Angle Addition Postulate; I know that this is going to be the whole thing.1987

And then, E has to be the vertex; then D and F...the measure of angle DEB...DE, and then, that means that B is going to be in the interior of the angle.1996

And then, that, plus the measure of angle BEF, equals the measure of angle DEF.2011

Your diagrams might be a little bit different than mine; but as long as you know that this plus this...2015

as long as B is in the middle of this angle, and E is the vertex, then that is fine.2022

Ray AB and ray BC as opposite rays: opposite rays means two rays that go in opposite directions to form a line.2029

There is A, B...I know that this has to be A, because that is how it is written.2041

A has to be at the endpoint; that and BC as opposite rays...well, I can do B, and then C like that, somewhere here.2049

Or, if you want to just draw it longer...it can be like that, or this will probably be AC; and then in that case, it would just be like...2066

A is a common endpoint; AB is going that way, and then AC is going this way.2089

And then, the last one: two angles that are complementary:2096

you can draw any two angles, as long they add up to 90 degrees (complementary, remember, is 90).2099

I can draw it like that; this could be 45 and 45; they are complementary angles.2110

I can draw two angles separately, maybe like that, and then, say, if this is 60, then I have to draw a 30-degree angle.2123

As long as they add up to 90...it is any two angles that add up to 90.2138

OK, the fourth example: BA and BE are opposite rays, and BC bisects the measure of angle ABD.2145

That means that this is a line, because they are opposite rays; so they are saying that it forms a line.2165

BC bisects this angle ABD; that means, since it bisects it, they are the same measure--they are equal.2171

If the measure of angle ABC equals 4x + 1, and the measure of angle CBD equals 6x - 15, then find the measure of angle CBD.2183

We have these two angles; now, when you draw a little line like that, that just means that they congruent to each other--they are equal.2197

And then, if you have two other angles that are not the same as that, then...2211

let's say these two angles are the same; then you can just draw these two, because you did one for each of these,2219

to show that all the angles that you drew one for are congruent.2226

Then, the next pair of congruent angles--you can just draw two.2230

The measure of angle ABC plus the measure of angle CBD is going to be the measure of angle ABD.2239

But then, they are actually wanting you to find the measure of angle CBD; and this is the angle bisector.2247

I can just make them equal to each other; let me just solve it down here.2254

4x + 1 is equal to 6x - 15; I need to solve for x.2258

So, if I subtract the 6x over, I get (let's write out the answer)...-2x; subtract the 1; and I get -16, so x = 8.2267

But they want you to find the measure of angle CBD; that means that, once you find x, you have to plug it back into to the measure of angle CBD.2286

That is 6(8) - 15; this is 48 - 15; that is 33; so the measure of angle CBD equals 33.2299

Number 2: the measure of angle DBC is 12n - 8; the measure of angle ABD, the whole thing, is 22n - 11; find the measure of angle ABC.2324

They give you this whole thing right here, and they give you DBC, and they give you ABD; and they want you to find this angle right here.2346

Since we know that this and this are the same--they have the same measure--this whole thing would be two times one of these.2359

So, if this is 10, then this has to be 10; then the whole thing is 20.2370

I can do 12n - 8, plus 12n - 8 (because this is also 12n - 8--the same thing), equals 22n - 11, the whole thing.2378

Or you can just do 2 times 12n - 8, because it is just this angle, times 2, equals ABD.2390

So, I am just going to do that; number 2 is 2(12n - 8) = 22n - 11.2400

This is going to be...I will use the distributive property...this is 24n - 16 = 22n - 11.2414

If I subtract this, I get 2n; add it over; I get 5; so n = 5/2.2425

And then, n is 5/2, and then we have to find the measure of angle ABC.2438

Now, they don't give me something for ABC; but as long as I find what DBC is, then that is the same measure.2450

I just do 12...substitute in that n...minus 8; this becomes 6; 6 times 5 is 30, so 30 - 8 is 22.2461

That means that the measure of angle ABC is 22.2480

The last one: If the measure of angle EBD, this one right here, is 115, find the measure of the angle supplementary to angle EBD.2493

Supplementary: it is asking you to find the supplement of this angle.2508

Remember: supplementary is 180, so find two angles that add up to 180; it is 115 + something (which is x) is going to add up to 180.2513

You are going to subtract it: x is equal to 65 degrees.2530

This will be the supplement of angle EBD.2539

All right, that is it for this lesson; we will see you next time.2549

Thank you for watching Educator.com!2552