Theorems and Postulates
Date: 12/02/2006 at 04:05:45 From: Alona Subject: difference between a theorem and a postulate If SSS, SAS, and AAS are theorems, why do other books still use them as postulates? And can you show me the PROOFS that were used for these theorems? :) Sorry, I know you've answered questions about the topic many times but as I was reading the answers I realized that you were trying to say that SSS, SAS, and AAS are really theorems because they were proved (theorems need proofs). But, why do books and even teachers still teach students like us SSS, SAS, and AAS "POSTULATES". Do the words "theorem" or "postulate" really matter? I am a high school student studying geometry right now and your answers to my questions would be a great help for me. Thank you very much.
Date: 12/02/2006 at 23:29:48 From: Doctor Peterson Subject: Re: difference between a theorem and a postulate Hi, Alona. These facts CAN be postulates, but they don't have to be. It's a matter of how an author chooses to present geometry to his audience. Different geometry texts choose different starting points. The best way to do geometry is to start with as few assumptions as possible, and prove everything from those. Many texts "cheat" a bit by using as postulates anything they don't want to bother proving (probably because the proofs are difficult and wouldn't really help their students understand the subject). Others use a good, small set of postulates, but state some theorems without proof, explaining that the proof is beyond the level of the text. I prefer the latter approach, but I can understand the "cheating". It is possible to take ONE of these congruence facts as a postulate and prove the others from it (so they become theorems). It is also possible to define congruence in such a way that all three can be proved from some more basic postulate about congruence. You can take them however your own text presents them; but be aware that they are all really equivalent facts, and which you take as postulates doesn't affect how you use them, which is what really matters. In other words, in answer to your question as to whether "theorem" or "postulate" really matters: it matters in presenting a specific systematic treatment of geometry, but not in USING the facts you learn, which are true one way or another regardless. See this page for more discussion: Theorem or Postulate? http://mathforum.org/library/drmath/view/62548.html See also: The Role of Postulates http://mathforum.org/library/drmath/view/62560.html If you have any further questions, feel free to write back. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 12/10/2006 at 23:07:42 From: Alona Subject: Proving theorems from theorems Dear Dr. Math, thank you very much for answering my past question about "theorems" and "postulates". I now know that one of the SSS, SAS and ASA theorems can be considered as a postulate. It just depends on the starting point of the discussion. But they are really theorems (I hope I understood it the way you want me to understand it). My question is, is it really possible to prove theorems from theorems? What I mean is, is it possible to call all the three congruency theorems "theorems" and still prove each one using each theorem? In our geometry class, it is possible to prove theorems from previous theorems. Then why do we need to assume one of the congruency theorems as "postulate" when we could really prove it using "theorems". Thank you very much for your previous reply. It answered 70% of my questions. These questions are the remaining 30%. :)
Date: 12/11/2006 at 12:49:58 From: Doctor Peterson Subject: Re: Proving theorems from theorems Hi, Alona. Certainly you can prove a theorem from a theorem; you do it all the time, I would think. You can use any known fact, whether theorem or postulate, as the basis for a proof. What you CAN'T do is prove A from B, and B from C, and C from A! Such circular reasoning is not allowed, because you have to start with something that is known to be true. So if you call ALL THREE of these "theorems", then at least one of them has to be proved on the basis of something else (such as a definition of congruency that is more powerful than what elementary texts usually use). That's why the best approach is to take one of them as a postulate, and then prove the others as theorems. It doesn't matter which one you start with, but you have to start with one without assuming another is already true. If you do merely prove each from another of them, then what you have done is to show that they are all EQUIVALENT--that is, IF one is true, then they all are. But then either they are all true, OR they are all false. You don't know which! I think the pages I referred you to answer this question, by explaining the role of postulates as starting points. You may want to reread them with this new perspective in mind. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
Date: 12/20/2006 at 01:11:37 From: Alona Subject: Thank you (Proving theorems from theorems) Thank you very very very much!! I totally understood the topic now! :)
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