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Parallel Lines and Three-Dimensional Space

Date: 12/18/2005 at 20:30:28
From: Tammy
Subject: Parallel lines and Space

If two lines are perpendicular to the same line or plane, are they 
parallel to each other?  Why or why not?  I do not agree with my
textbook, which states that the lines are not necessarily parallel. 
If the two lines never intersect, why can't they be parallel?  Isn't
that the definition of parallel?

Date: 12/19/2005 at 10:06:57
From: Doctor Peterson
Subject: Re: Parallel lines and Space

Hi, Tammy.

There are two questions in one here, and I think you're really talking 
only about the first:

  If two lines are perpendicular to the same line, are they parallel
  to each other?

  If two lines are perpendicular to the same plane, are they parallel
  to each other?

The answer to the first is no, and to the second is yes.

To see why the first is true, look back at the definition your book 
gave for parallel lines.  If they did their job right, they will have 
said that two lines are parallel if they are IN THE SAME PLANE and 
never intersect.  That's important: two lines in different planes that 
don't intersect are called "skew lines".

To picture the difference between parallel and skew lines, look at the 
edges of your room.  Take the vertical line up one corner, and first 
look at the top and bottom of the wall to its left.  These are both 
perpendicular to the vertical line, and they are also in the same 
plane (the wall).  They are parallel; they go in the same direction.  
Now take the BOTTOM of the wall on the left, and the TOP of the wall 
on the right.  These are NOT in the same plane, and they do NOT go in 
the same direction.  They are skew lines.

The real idea behind "parallel" is "going the same direction".  We 
don't put that in the definition for a simple reason: we'd first have 
to define what we mean by "direction", and that's not easy!  But 
having defined "parallel" as we do, you can then take that as the 
definition of "in the same direction", and that can help you see why 
we don't want to define parallel lines as ANY two lines that never 

(As for the second question, looking at your room again, all four 
vertical lines in the corners are perpendicular to the floor plane, 
and they are parallel!  Part of the reason this is true is that any 
two of them ARE in the same plane.)

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 
Associated Topics:
High School Euclidean/Plane Geometry
Middle School Two-Dimensional Geometry

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