WEBVTT mathematics/geometry/pyo
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Welcome back to Educator.com.
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In this next lesson, we are going to go over some postulates that have to do with points, lines, and planes.
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First, let's talk about postulates: what is a postulate?
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A postulate is a statement that is assumed to be true; this is also called an axiom.
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Postulates are accepted as fact without having to be proved.
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Theorems are statements that have to be proved; you have to prove that it is true.
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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.
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And some postulates in your textbook--you might see that they are titled 2-2 or Postulate 2-1 or something.
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Remember: when you name a postulate, you don't name it by that number that is used in your book,
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because different books use different numbers, and it is in a different order.
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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.
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You can't just call it by the number that your book uses.
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The first postulate that we are going to go over: Through any two points, there is exactly one line.
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If there are any two points--I can draw two points however I want--maybe two points like that.
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Through any two points, I can only draw one line through those two points, like that.
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And there is no way that I can draw any other line.
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So, if I have another two points, there is only one line that can be drawn through those two points.
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The next one: Through any three points not on the same line, there is exactly one plane.
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Through any three points not on the same line--meaning that they are not collinear, like that, there is exactly one plane.
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I can only draw one plane that covers those three points--I can't draw any other plane.
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Just like this one, through any two points, I can only draw one line--I can't draw any other type of line
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that is going to go through those same two points--it is the same thing here.
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Through any three points, I can only draw one single plane that is going to cover those points.
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A line contains at least two points--"at least" meaning infinite--it contains two and a lot more.
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So, a line contains at least two points.
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A plane (remember, the fourth one--the next one) contains at least three points not on the same line.
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If I have a plane, then this plane is going to contain at least three points; it is actually many, many, many--
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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.
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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.
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The next one: If two points lie in a plane, then the entire line containing those two points lies in that plane.
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So again, if two points lie in a plane (let me draw a plane, and two points lying in that plane),
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then the entire line containing those two points lies in that plane.
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The line that I can draw through those two points is going to also be in that plane.
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So, if I have two points in a plane, then the line (remember: you can only draw one line through those points)
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that you can draw is also going to be in that plane.
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If two lines intersect, then they intersect in exactly one point.
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If I have two lines, where they intersect is right here; where they intersect is going to be one point.
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There is no way that they could intersect in any more than one point, because lines, we know, go straight.
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Now, if we can bend it, then maybe it can come back around and meet again.
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But we know that lines can't do that; it just goes straight, so their intersection is always going to be one point.
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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,
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and then I have a plane like this, so this is where they are intersecting; then this, where they intersect,
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right here--that place where they are touching, where they are meeting, is a line.
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When two lines intersect, it is going to be a point; when two planes intersect, it is going to be a line.
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We can't just say that these two planes are going to intersect at a point,
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because then that is not true; it is not just a single point--it is all of this right here.
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So, it is going to be a line.
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OK, if you want to review over the postulates again, just go ahead and rewind, or just go back and go over them again.
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We are going to use the postulates to do a few example problems.
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Using the postulates, determine if each statement is true or false.
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Points A, B, and E...first of all, let's actually go over this diagram.
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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*--
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this plane is *N*, right there; this point is I; this is point E.
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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*.
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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.
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The next one: Points A, B, C, and E are coplanar.
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"Coplanar" means that they are on the same plane.
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Well, A, B...look at this...C, and E are coplanar; now, they might not be on plane *N* all together,
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but they actually are coplanar, because this point...2, 3...and this one right here...I can form a plane
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that is going to contain these four points, so this right here is true.
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They are coplanar; it is not plane *N*, but they can lie on some plane, a different plane.
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BC does not lie in plane *N*: here is BC right here; BC does not lie in plane *N*.
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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*.
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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.
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So, since point B and point C lie in the plane, BC has to lie in the plane.
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Points A, B, and D are collinear: are they collinear?
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They are coplanar, because they are all on plane *N*; but they are not collinear,
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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.
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OK, let's go over a few more: now, we are going to determine if these are true or false.
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Just to go over this diagram again: this right here is plane *R*; this right here is plane *P*; these are all the points.
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Points B, D, A are part of this plane, and then, it is also part of this plane.
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E is on plane *P*; H and I are not on either of them.
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Points A, B, and D lie in plane *R*: is that true?
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Here is plane *R*; B lies in it, D, and A; yes, it is true.
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Points B, D, E, and F are coplanar; B, D, E, and F...well, B, D, and E are coplanar,
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and B, D, and F are coplanar; but all four of them together--they are not coplanar.
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So, there is no way that we can draw those four points on the same plane, so this one is false.
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BA lies in plane *P*; BA, this segment right here, lies in *P*.
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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.
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So, this one is true.
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OK, we are going to go over a few more examples.
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Use "always," "sometimes," or "never" to make each statement a true statement.
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Intersecting lines are [always, sometimes, or never] coplanar.
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If we have intersecting lines, no matter how we draw them (we can have them like this, or maybe like this),
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intersecting lines are actually always going to be coplanar.
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Can you draw a plane that contains those two points? Yes, so this one is always--always coplanar.
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They are always going to be on the same plane.
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Two planes [always/sometimes/never] intersect in exactly one point.
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So, again, let me try to draw this out; I have a plane, and I have another plane...something like that.
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Do they intersect? They intersect right here; when they intersect, are they intersecting at one point?
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No, they intersect at a line; so this one is never: two planes never intersect in exactly one point.
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It is always going to be a line.
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Three points are [always, sometimes, or never] coplanar.
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Well, if I have three points, are they going to be coplanar?
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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.
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Whether it is like that, or whether it is like they are collinear--they are going to be coplanar, the three points.
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The next one: A plane containing two points of a line [always, sometimes, or never] contains the entire line.
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A plane containing two points of a line contains the entire line--this is always.
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As long as the two points are in that plane, the line has to also be in that plane.
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Four points are [always, sometimes, or never] coplanar.
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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;
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I can have point, point...if I have points A, B, C, and then right here, D, are all four points coplanar?
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Now, what if I have this point right here, E? E is on this plane.
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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.
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Two lines [always/sometimes/never] meet in more than one point.
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Two lines, when they intersect...do they always meet at one point? Sometimes? Or never?
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This is always at one point; can they meet in more than one point? No, so this one is never.
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They can never meet in more than one point; they always have to meet in one point.
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That is it for this lesson; thank you for watching Educator.com!