Associated Topics || Dr. Math Home || Search Dr. Math

### Parallel Lines: Euclidean and Non-Euclidean Geometry

```
Date: 4/25/96 at 12:25:57
From: Anonymous
Subject: Parallel lines

If two lines are parallel, do they intersect?

Thanks.

Dennis
```

```
Date: 4/26/96 at 8:15:29
From: Doctor Steven
Subject: Re: Parallel lines

Well, it really depends on what kind of geometry you are looking at.

In planar geometry (geometry on a flat surface, also called Euclidean
geometry), no.  The definition of parallel lines is that they extend
infinitely long in both direction without intersection.  The fifth
postulate of Euclid says that if a straight line falling on two
straight lines makes the angles on one side of the line less than 180
degrees, then the two straight lines if extended will intersect at
some point on the side of the other straight line where there angles
added to less than 180 degrees.

The fifth postulate was a source of contention for many mathematicians
in ancient times and even in relatively modern times.  The thought was
that it could be proven from the other four postulates that Euclid
gave:

1) From any two points a straight line can be drawn.
2) Any straight line can be extended indefinitely.
3) A circle can be drawn of any radius at any point.
4) All right angles are equal to one another.

Many mathematicians spent their time trying to prove this postulate
using the others; if they could it meant that the fifth postualte
should not be a postulate but rather a theorem.  Probably Euclid would
like to have proved it, so parallel lines could be proved to intersect
a falling straight line with interior angles equal to 180 degrees.
Unfortunately for him he could neither prove or disprove the fact, so
he added it as a postulate.

Now other mathematicians also thought it should be provable from the
other four postulates and so they tried - for hundreds of years!
Finally two mathematicians decided that if the 5th postulate was
provable by the other four they should be able to "replace" the 5th
postulate with a contrary one, and they would get a contradiction
somewhere.  Well they took out the 5th postulate and added a new
contrary one and they got crazy statements like: Every triangle must
have strictly greater than 180 degrees as the sum of the interior
geometry, of course they called this Non-Euclidean geometry. This new
geometry they created perfectly described what we call spherical
geometry.

After their discovery other mathematicians decided to see what they
could get by adding different contrary postulate in the 5th postulates
place.

In a Non-Euclidean geometry such as spherical geometry, two lines can
be parallel and still intersect.  To see this think of the globe and
two lines of longitude.  Look at the lines of longitude, say 30
degrees W and 40 degrees W, note that at the equator these lines are
parallel, no look at either of the poles and you will see that they
intersect.

Such geometries as this are called non-Euclidean, and there are many.
Lobachevsky, Riemann, and Felix Klein.

-Doctor Steven,  The Math Forum
(with some help from Dr. Patrick)
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Euclidean/Plane Geometry
High School Non-Euclidean Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search