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Intersecting Lines Coplanar?

Date: 08/23/2002 at 10:19:53
From: Alice Matthews
Subject: Geometry - Intersecting lines

Do two intersecting lines have to be coplanar? 

The definition in our Geometry book for intersecting lines states that 
two intersecting lines must be coplanar, but we can't understand why 
they couldn't lie in more than one plane and still intersect one 
another.

Mrs. Matthews' 8th grade Geometry class


Date: 08/23/2002 at 16:18:19
From: Doctor Rick
Subject: Re: Geometry - Intersecting lines

Hi, Alice.

Each line is in more than one plane. For instance, a vertical line is 
in lots of different vertical planes. Thus you could say that one 
line is in plane A and the other line is in a different plane B, and 
the two can still intersect. For instance, you could have two 
horizontal lines that intersect, and say that one line is in one 
vertical plane and the other line is in a different vertical plane.

That's not the point. The point is that if the lines intersect, there 
must be ONE plane that BOTH lines are in. The two horizontal lines 
are in the SAME horizontal plane, even though they are in different 
vertical planes.

Has the class learned that three points determine a unique plane? 
Pick one point at the intersection of the lines, and pick one other 
point on each line. These three points determine a plane. Since two 
points on each line are in that plane, both entire lines must lie in 
that plane.

Two lines that are not coplanar can be pictured as the center line of 
a road and the center line of another road that passes above the 
first on a bridge. They never intersect. Each is in a horizontal 
plane, but the planes are parallel, one above the other. The closest 
that the lines ever get to one another is the distance between the 
planes (the height of the bridge). In order to get the lines to 
intersect, you have to reduce the height of the bridge to zero - 
and then the planes aren't parallel any more; they are the same plane!

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/ 
Associated Topics:
High School Higher-Dimensional Geometry
Middle School Higher-Dimensional Geometry

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