The Order of a Proof
Date: 01/29/99 at 08:14:32 From: Stan D. Subject: Geometry Dear Dr. Math, My question is about proofs. My Geometry book only describes the two- column proof twice, and it doesn't give too many details. I cannot figure out if their statements and reasons are completely random in their ordering (other than the "given" and the "to prove" which are always first and last), or if there is a particular method for the order in which they should be placed. Figuring out which theorems, postulates, and definitions to use in a proof are no problem. However, they seem to just pull statements out of thin air, not taking the time to describe how even in the study guide. Any help that you could give me would be greatly appreciated and I thank you for your time.
Date: 01/29/99 at 12:14:09 From: Doctor Rob Subject: Re: Geometry Thanks for writing to Ask Dr. Math! The only requirement for ordering the steps and reasons is that if Step A depends on Step B, then A should follow B. That is why the given is first and the conclusion is last, and the same logic applies to all the intermediate steps. For example, prove: In a triangle ABC, if side CA = side CB, then <A = <B. Proof: Steps: Reasons: 1. CA = CB. 1. Given. 2. CB = CA. 2. Step 1 and Symmetric Law of Equality. 3. <C = <C. 3. Reflexive Law of Equality. 4. triangle BCA =~ triangle ACB 4. Steps 1, 2, 3, and S.A.S. theorem. 5. <A = <B. 5. Step 4 and corresponding parts of congruent figures are congruent. Q.E.D. In this proof, Step 2 depends on Step 1, so must follow it. Step 4 depends on Steps 1, 2, and 3, so must follow them. Step 5 depends on Step 4, so must follow it. The only freedom you have is the placement of Step 3, which can go anywhere before Step 4 (and then you would renumber the steps and reasons to be in order 1 through 5). Here is a diagram of the structure of the proof: 1 ==> 2 ==> 4 1 ========> 4 ==> 5 3 ==> 4 The hard part is not figuring the order of the steps, it is figuring what steps are good ones to take! Usually the mathematician does this kind of thing backwards (yes, backwards!). He says, "If I could prove that triangles BCA and ACB are congruent, then I could conclude that <A = <B in one step. Furthermore, it seems from my diagram that this is probably true." So he tries to prove the triangle congruence. "If I could find two sides and an included angle in BCA equal to two sides and an included angle in ACB, I could use S.A.S. to prove the triangles congruent." So he hunts for equal sides and angles. This is the way it goes. When done, the proof is presented in the forward manner, not providing a hint of the backward process used to generate it. Sometimes one can work from the given forwards and from the conclusion backwards and meet somewhere in the middle. - Doctor Rob, The Math Forum http://mathforum.org/dr.math/
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