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### Angles

- Angle: Figure formed by two non-collinear rays with a common endpoint
- Ray: Segment with one endpoint and one end extending indefinitely
- Opposite rays: Two rays that extend in opposite directions to form a line
- An angle separates a plane into 3 parts: interior, exterior, and the angle itself
- Angles are measured in units called degrees
- The number of degrees in the angle is the measure
- Protractor Postulate: Given AB and a number
*r*between 0 and 180, there is exactly one ray with endpoint A, extending on either side of AB, such that the measure of the angle formed is*r* - Acute angle: Angles with a measure less than 90 degrees
- Right angle: Angles that measure 90 degrees
- Obtuse angle: Angles that measure more than 90 degrees
- Angle bisector: A segment, ray, or line that divides an angle into two congruent angles
- Adjacent angles: Angles with common vertex and common side
- Vertical angles: Two nonadjacent angles formed by intersecting lines
- Linear pair: Adjacent angles formed by opposite rays
- Right angles are formed by perpendicular lines
- Supplementary angles: Two angles whose measures have a sum of 180 degrees
- Complementary angles: Two angles whose measures have a sum of 90 degrees

### Angles

(A) M

^{→}N (B) N

^{→}M (C) N

^{←}M (D) ↔MN

∠BEC, ∠AEC , ∠E , ∠CED , ∠CEB .

^{o}, m∠ABD = 30

^{o}, find m∠ABC.

^{o}+ 30

^{o}= 45

^{o}.

^{o}, find m∠MOP.

^{o}= 15

^{o}.

^{o}.

^{o}, ∠2 and ∠1 are supplementary angles, ∠3 and ∠2 are complementary angles, find m∠2 and m∠3.

- m∠1 + m∠2 = 180
^{o} - m∠2 = 180
^{o}− m∠1 = 180^{o}− 125^{o}= 55^{o} - m∠2 + m∠3 = 90
^{o}

^{o}− 55

^{o}= 35

^{o}.

Name:

A pair of adjacent angles

A pair of vertical angles

linear pairs

A pair of supplementary angles.

A pair of vertical angles: ∠AOD and ∠BOC

linear pairs: ∠AOD and ∠BOD

A pair of supplementary angles: ∠BOC and ∠BOD.

^{o}, m∠DOE = 110

^{o}, find m∠AOB.

- ∠AOC and ∠DOE are vertical angles, so m∠AOC = m∠DOE = 110
^{o}

^{o}− 40

^{o}= 70

^{o}.

- m∠BOC = 2m∠COD = 2(2x + 1)
- m∠AOB + m∠BOC = 180
- 4x + 2 + 2(2x + 1) = 180
- x = 22

^{o}.

*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

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
- Angles 0:05
- Angle
- Ray
- Opposite Rays
- Angles 3:22
- Example: Naming Angle
- Angles 6:39
- Interior, Exterior, Angle
- Measure and Degrees
- Protractor Postulate 8:37
- Example: Protractor Postulate
- Angle Addition Postulate 11:41
- Example: Angle addition Postulate
- Classifying Angles 14:10
- Acute Angle
- Right Angles
- Obtuse Angle
- Angle Bisector 15:02
- Example: Angle Bisector
- Angle Relationships 16:43
- Adjacent Angles
- Vertical Angles
- Linear Pair
- Angle Relationships 20:31
- Right Angles
- Supplementary Angles
- Complementary Angles
- Extra Example 1: Angles 24:08
- Extra Example 2: Angles 29:06
- Extra Example 3: Angles 32:05
- Extra Example 4 Angles 35:44

### Geometry Online Course

### Transcription: 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 AB*0552

*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 ray*0575

*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 PQR*0748

*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 other*1070

*("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 write*1467

*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

1 answer

Last reply by: Tamara Orujzade

Wed Jul 15, 2015 7:54 AM

Post by Cristal Barrios on June 15, 2015

I am having problems with the videos they load but they get stuck...and our internet is working fine and I am paying for this program

0 answers

Post by Aravinda Fernando on November 18, 2013

An obtuse angle is:

More than 90Â° and less than 180Â°.

Great Lecture, Thanks

1 answer

Last reply by: Professor Pyo

Fri Aug 2, 2013 1:44 AM

Post by Shahram Ahmadi N. Emran on July 14, 2013

The pair of vertical angles you drew at 18:38 are sharing both a vertex and a side. The only difference seems to be that they are on opposite sides of the intersecting line. Could you clarify this a bit?

1 answer

Last reply by: Professor Pyo

Fri Aug 2, 2013 1:42 AM

Post by Manfred Berger on May 27, 2013

The pair of vertical angles you drew at 18:38 are sharing both a vertex and a side. The only difference seems to be that they are on opposite sides of the intersecting line. Could you clarify this a bit?

1 answer

Last reply by: Ruby Zhang

Wed Jun 14, 2017 11:10 PM

Post by Abdihakim Ibrahim on November 20, 2011

oh no the popo

0 answers

Post by Abdihakim Ibrahim on November 20, 2011

c.c

1 answer

Last reply by: Ruby Zhang

Wed Jun 14, 2017 11:10 PM

Post by Abdihakim Ibrahim on November 20, 2011

>:D hahahah im a kid!

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Last reply by: Mary Pyo

Fri Feb 3, 2012 11:32 PM

Post by Abdihakim Ibrahim on November 19, 2011

?

1 answer

Last reply by: Mary Pyo

Sat Oct 29, 2011 11:20 PM

Post by Teresa Thumbi on October 19, 2011

Amazing , but i still dont get how she got 2n-5 ?

1 answer

Last reply by: Carol Corrigan

Fri Jul 8, 2011 9:34 AM

Post by Ahmed Shiran on June 5, 2011

Good Lesson,