Geometry - Parallel LinesDate: 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/ |
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