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Geometry - Parallel Lines


Date: 12/09/97 at 11:48:54
From: Moty Kamenezky
Subject: Geometry - Parallel lines

Hello there, and thank you in advance for your help.

I am currently teaching 9th grade Geometry, and below is the problem 
that the class is involved with:
                                  E
                                  /\
                                 /  \
                                /____\
                              J/      \K
                              /________\ 
                             M          N

Given: EJ = EK; JK||MN
Prove: Angle M = Angle N

This is how I want to solve this problem:  I will create an auxiliary 
line extending JK to outside the triangle, and then I will have what I 
need to prove that angle M = Angle N (the alternate interior angles 
and vertical angles).  

                                  E
                                  /\
                                 /  \
                          ______/____\_______
                              J/      \K
                              /________\ 
                             M          N

Now, even though intuition says that just as JK is parallel to MN, 
so too any line I extend from JK, but what is this postulate/theorem 
called?  It's not the Ruler Postulate, and the following theorem: 
"Through a point outside a line exactly one parallel can be drawn to 
the line" doesn't prove that this line meets JK.

Thanks again!

Moty


Date: 12/24/97 at 09:36:31
From: Doctor Bruce
Subject: Re: Geometry - Parallel lines

Hello Moty,

It is a postulate of Euclidean Geometry that two given points 
determine one and only one line which passes through them.  
Mathematicians interpret this postulate as meaning that the line is 
"already there," regardless of what a diagram may show. A line segment 
joining two points is a subset of the line joining the two points.  
Thus we are always permitted to "extend the line segment" along its 
line as much as we like. The point I am making is that axiomatic 
geometry is not the same as making pencil marks on paper. The line has 
a sort of pre-existence, independent of how much of it we have
darkened in with a pencil.

Geometry texts often usually include a postulate to avoid getting 
mired in philosophical matters like these. Such a postulate might be 
phrased something like:  

   Any line segment may be extended indefinitely in either direction.

I'm not familiar with the "Ruler Postulate," at least not by that 
name. It sounds like it might be what you need, though.

By the way, you refer to "Through a point outside a line exactly one
parallel can be drawn to the line" as a theorem; but this is 
Playfair's formulation of Euclid's "Fifth Postulate," and there is a 
long history of unsuccessful attempts to derive this postulate from 
the other postulates of Geometry. You might want to read a very nice 
summary on this subject in the MacTutor Math History archives at St. 
Andrews (look for Non-Euclidean geometry):

 http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/   

Good luck,

-Doctor Bruce,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Euclidean/Plane Geometry
High School Geometry

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