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### Point, Line, and Plane Postulates

- A postulate (also called an axiom) is a statement that is assumed to be true. It is accepted as fact without formal proof.
- Postulates:
- Through any two points, there is exactly one line
- Through any three points not on the same line, there is exactly one plane
- A line contains at least two points
- A plane contains at least three points not on the same line
- If two points lie in a plane, then the entire line containing those two points lies in that plane
- If two lines intersect, then they intersect in exactly one point
- If two planes intersect, then their intersection is a line

### Point, Line, and Plane Postulates

Through any two points, there is only one line.

If one point of a line lies in a plane, then the entire line is on the plane.

One line contains only one point.

2 lines intersect at 2 points.

Three points form one plane.

If three points are not on the same line, they can form more than one plane.

Two planes can intersect at more than one line.

Decide whether each statement is true or false.

Points A, B and C are coplanar.

Plane N contains line AC.

Plane N contains line AC. False.

Decide whether each statement is true or false.

Points E, D, B and F are all on plane N.

Plane M can intersect plane N on another line other than line DF.

Plane M can intersect plane N on another line other than line DF. False.

Decide whether the statement is true or not.

There is a point C on plane P that both line m and line n pass through.

*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

### Point, Line, and Plane Postulates

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
- What are Postulates? 0:09
- Definition of Postulates
- Postulates 1:22
- Postulate 1: Two Points
- Postulate 2: Three Points
- Postulate 3: Line
- Postulates, cont.. 3:08
- Postulate 4: Plane
- Postulate 5: Two Points in a Plane
- Postulates, cont.. 4:46
- Postulate 6: Two Lines Intersect
- Postulate 7: Two Plane Intersect
- Using the Postulates 6:34
- Examples: True or False
- Using the Postulates 10:18
- Examples: True or False
- Extra Example 1: Always, Sometimes, or Never 12:22
- Extra Example 2: Always, Sometimes, or Never 13:15
- Extra Example 3: Always, Sometimes, or Never 14:16
- Extra Example 4: Always, Sometimes, or Never 15:03

### Geometry Online Course

### Transcription: Point, Line, and Plane Postulates

*Welcome back to Educator.com.*0000

*In this next lesson, we are going to go over some postulates that have to do with points, lines, and planes.*0002

*First, let's talk about postulates: what is a postulate?*0011

*A postulate is a statement that is assumed to be true; this is also called an axiom.*0015

*Postulates are accepted as fact without having to be proved.*0023

*Theorems are statements that have to be proved; you have to prove that it is true.*0029

*But postulates--we can just use them without any question if it is true or not--we don't have to prove it at all; it is just true.*0035

*And some postulates in your textbook--you might see that they are titled 2-2 or Postulate 2-1 or something.*0046

*Remember: when you name a postulate, you don't name it by that number that is used in your book,*0057

*because different books use different numbers, and it is in a different order.*0062

*If it doesn't have a name--if it just has a number, like Postulate 2.2, then remember that you have to write out the whole thing.*0067

*You can't just call it by the number that your book uses.*0076

*The first postulate that we are going to go over: Through any two points, there is exactly one line.*0084

*If there are any two points--I can draw two points however I want--maybe two points like that.*0094

*Through any two points, I can only draw one line through those two points, like that.*0101

*And there is no way that I can draw any other line.*0110

*So, if I have another two points, there is only one line that can be drawn through those two points.*0113

*The next one: Through any three points not on the same line, there is exactly one plane.*0123

*Through any three points not on the same line--meaning that they are not collinear, like that, there is exactly one plane.*0130

*I can only draw one plane that covers those three points--I can't draw any other plane.*0140

*Just like this one, through any two points, I can only draw one line--I can't draw any other type of line*0148

*that is going to go through those same two points--it is the same thing here.*0154

*Through any three points, I can only draw one single plane that is going to cover those points.*0158

*A line contains at least two points--"at least" meaning infinite--it contains two and a lot more.*0166

*So, a line contains at least two points.*0178

*A plane (remember, the fourth one--the next one) contains at least three points not on the same line.*0189

*If I have a plane, then this plane is going to contain at least three points; it is actually many, many, many--*0204

*but at least three points not on the same line, because if they are collinear, then it is just going to be on the same line.*0213

*But if they are not collinear, then it is going to be on the same plane; so this plane contains at least three points not on the same line.*0223

*The next one: If two points lie in a plane, then the entire line containing those two points lies in that plane.*0233

*So again, if two points lie in a plane (let me draw a plane, and two points lying in that plane),*0244

*then the entire line containing those two points lies in that plane.*0256

*The line that I can draw through those two points is going to also be in that plane.*0262

*So, if I have two points in a plane, then the line (remember: you can only draw one line through those points)*0272

*that you can draw is also going to be in that plane.*0280

*If two lines intersect, then they intersect in exactly one point.*0288

*If I have two lines, where they intersect is right here; where they intersect is going to be one point.*0293

*There is no way that they could intersect in any more than one point, because lines, we know, go straight.*0304

*Now, if we can bend it, then maybe it can come back around and meet again.*0313

*But we know that lines can't do that; it just goes straight, so their intersection is always going to be one point.*0319

*If two planes intersect, then their intersection is a line; so if I have (now, I am a very bad draw-er, but say I have) a plane like this,*0330

*and then I have a plane like this, so this is where they are intersecting; then this, where they intersect,*0345

*right here--that place where they are touching, where they are meeting, is a line.*0361

*When two lines intersect, it is going to be a point; when two planes intersect, it is going to be a line.*0372

*We can't just say that these two planes are going to intersect at a point,*0378

*because then that is not true; it is not just a single point--it is all of this right here.*0384

*So, it is going to be a line.*0388

*OK, if you want to review over the postulates again, just go ahead and rewind, or just go back and go over them again.*0396

*We are going to use the postulates to do a few example problems.*0406

*Using the postulates, determine if each statement is true or false.*0411

*Points A, B, and E...first of all, let's actually go over this diagram.*0416

*We have a plane: this is plane N; this is point A, right here; this is point B; this is C, point D; this is plane N--*0423

*this plane is N, right there; this point is I; this is point E.*0437

*So, points A, B, and E line in plane N; points A (this is point A), B, and point E (that is point E, right there) lie in plane N.*0446

*And we know that this is false, because E does not lie in it; A lies in it; B lies in the plane; but E does not, so this is false.*0463

*The next one: Points A, B, C, and E are coplanar.*0480

*"Coplanar" means that they are on the same plane.*0488

*Well, A, B...look at this...C, and E are coplanar; now, they might not be on plane N all together,*0492

*but they actually are coplanar, because this point...2, 3...and this one right here...I can form a plane*0506

*that is going to contain these four points, so this right here is true.*0523

*They are coplanar; it is not plane N, but they can lie on some plane, a different plane.*0532

*BC does not lie in plane N: here is BC right here; BC does not lie in plane N.*0542

*Well, it does actually lie in plane N, so this one is false, because B and C both lie (this point, B, and this point, C, lie) in plane N.*0553

*Remember the postulate where it says that, if two points are in the plane, then the line containing those two points also lies in the plane.*0570

*So, since point B and point C lie in the plane, BC has to lie in the plane.*0580

*Points A, B, and D are collinear: are they collinear?*0589

*They are coplanar, because they are all on plane N; but they are not collinear,*0597

*because, for it to be collinear, they have to be on the same line; and A, B, and D are not, so this one is false.*0601

*OK, let's go over a few more: now, we are going to determine if these are true or false.*0617

*Just to go over this diagram again: this right here is plane R; this right here is plane P; these are all the points.*0628

*Points B, D, A are part of this plane, and then, it is also part of this plane.*0641

*E is on plane P; H and I are not on either of them.*0647

*Points A, B, and D lie in plane R: is that true?*0656

*Here is plane R; B lies in it, D, and A; yes, it is true.*0665

*Points B, D, E, and F are coplanar; B, D, E, and F...well, B, D, and E are coplanar,*0674

*and B, D, and F are coplanar; but all four of them together--they are not coplanar.*0694

*So, there is no way that we can draw those four points on the same plane, so this one is false.*0701

*BA lies in plane P; BA, this segment right here, lies in P.*0711

*Well, I know that point B lies in plane P; point A lies in plane P; so the line containing those two points also has to be on that plane.*0723

*So, this one is true.*0736

*OK, we are going to go over a few more examples.*0741

*Use "always," "sometimes," or "never" to make each statement a true statement.*0747

*Intersecting lines are [always, sometimes, or never] coplanar.*0752

*If we have intersecting lines, no matter how we draw them (we can have them like this, or maybe like this),*0760

*intersecting lines are actually always going to be coplanar.*0777

*Can you draw a plane that contains those two points? Yes, so this one is always--always coplanar.*0780

*They are always going to be on the same plane.*0790

*Two planes [always/sometimes/never] intersect in exactly one point.*0798

*So, again, let me try to draw this out; I have a plane, and I have another plane...something like that.*0804

*Do they intersect? They intersect right here; when they intersect, are they intersecting at one point?*0824

*No, they intersect at a line; so this one is never: two planes never intersect in exactly one point.*0834

*It is always going to be a line.*0848

*Three points are [always, sometimes, or never] coplanar.*0858

*Well, if I have three points, are they going to be coplanar?*0865

*Yes, they are always going to be coplanar, because no matter how I draw these three points, I can always draw a plane around them.*0878

*Whether it is like that, or whether it is like they are collinear--they are going to be coplanar, the three points.*0891

*The next one: A plane containing two points of a line [always, sometimes, or never] contains the entire line.*0904

*A plane containing two points of a line contains the entire line--this is always.*0917

*As long as the two points are in that plane, the line has to also be in that plane.*0934

*Four points are [always, sometimes, or never] coplanar.*0943

*Well, this is actually going to be...let's see...if I have a plane like this, say I draw a line through that plane;*0949

*I can have point, point...if I have points A, B, C, and then right here, D, are all four points coplanar?*0966

*Now, what if I have this point right here, E? E is on this plane.*0983

*So, in this case, A, B, C, and E are coplanar; but A, B, C, and D are not coplanar; so this would be sometimes.*0988

*Two lines [always/sometimes/never] meet in more than one point.*1006

*Two lines, when they intersect...do they always meet at one point? Sometimes? Or never?*1013

*This is always at one point; can they meet in more than one point? No, so this one is never.*1022

*They can never meet in more than one point; they always have to meet in one point.*1032

*That is it for this lesson; thank you for watching Educator.com!*1040

1 answer

Last reply by: Jennifer Sinclair

Sat Jul 29, 2017 3:35 PM

Post by Julian Xiao on July 10, 2016

At 17:10 in the lecture, what if the two lines happen to be the same line? That way they would intersect in an infinite amount of points!

2 answers

Last reply by: John Stedge

Fri Jul 13, 2018 3:37 PM

Post by sahro AbdiOmar on November 10, 2015

Is this 6 grade math

1 answer

Last reply by: Julian Xiao

Sun Jul 10, 2016 11:41 AM

Post by Austin Cunningham on June 11, 2013

Around 11:49, you said that there was no way you could draw a plane containing points B,D,E,and F, but why couldn't you? I thought (based off of what you did at 8:38) that you could draw a plane around any four points.

2 answers

Last reply by: julius mogyorossy

Mon Jul 22, 2013 6:01 PM

Post by Larry Riley on September 5, 2012

Can't a line continue outside a plane? (A plane containing two points of a line always contains the entire line)