# I have a hard time understanding this simple theorem: “If two lines intersect, then exactly one plane contains the lines.”

I'm sorry if this is an extremely simple question, but I'm honestly having a hard time understanding a theorem in my geometry book. Here is the theorem:

"If two lines intersect, then exactly one plane contains the lines."

Now, each line contains two points, and according to another theorem in my book:

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

and three noncollinear points define a plane.

Now, a line endlessly continues in two opposite directions, if two lines were to intersect, shouldn’t that create $5$ points? And I'm also wondering if that would create two different planes (with both planes sharing one point at the intersection.)

• As written, the statement in the title is false. Write instead "If two distinct lines..." – John B Feb 24 '16 at 21:12
• What do you mean by "create 5 points?" – Mark Viola Feb 24 '16 at 21:12
• Let distinct lines $a,b$ be given with point of intersection $C$ and point $A$ on $a$ but not on $b$ and $B$ on $b$ but not on $a$, then $A,B,C$ are not collinear. – abiessu Feb 24 '16 at 21:17
• Two distinct lines in space that intersect form an infinitely extended $\not|$ shape. Imagine gluing a sheet of paper to all four arms of that $\not|\,$: if the $\not|$ is fixed in space, there’s only only one possible orientation and location for the plane. – Brian M. Scott Feb 24 '16 at 21:19

I think I can clear up some misunderstanding. A line contains more than just two points. A line is made up of infinitely many points. It is however true that a line is determined by 2 points, namely just extend the line segment connecting those two points.

Similarly a plane is determined by 3 non-co-linear points. In this case the three points are a point from each line and the point of intersection. We are not creating a new point when the lines intersect, the point was already there.

This is not the same thing as saying that there are 5 points because there are two from each line and the point from their intersection.

• Thank you, this will definitely help. Also, couldnt there be two planes? A plane on one side of the intersection and a plane on the opposite side. Since a line is made up of infinite points, its would require three noncollinear points to create a plane(the point of intersection, and two other points on each side of the intersection). – HTMLNoob Feb 24 '16 at 21:41
• That is still a part of the original plane, just like a line extends forever a plane does too. – Michael Menke Feb 24 '16 at 21:44
• I'm sorry if my way of explaining my problem is not clearly defined. – HTMLNoob Feb 24 '16 at 21:47
• Ok, thank you Michael, this will truly help me. :) – HTMLNoob Feb 24 '16 at 21:48

Two distinct lines intersecting at one point are contained in some plane: simply take the intersection point and one other in each line; the three noncollinear points define a plane and the plane contains the lines.

In order to see that there is no other plane containing the two lines, notice that any such plane necessarily contains the three former points and since three noncollinear points define a plane, it must be the plane in the former paragraph.

First, a line contains infinitely many points. The idea here is that if you have two distinct lines which intersect, there is only one (unique) plane that contains both lines and all of their points.

Try visualizing a plane that contains two intersecting lines:

Notice that if you then try to "twist" that plane in some way that it will no longer contain both lines. In other words, there is no other plane that could contain both lines, there is only one.

Think of a chair's 4 legs. To check that the 4 legs have the same length. Pull two strings connecting pairs of opposite legs, each string is attached at the bottom of the legs. If the strings touch each other in the middle then the chair is stable (the one plane), otherwise it is wobbly (no plane).