What Are Proofs?Date: 08/12/97 at 02:39:14 From: Femail Subject: Geometry proof I am a home school student through American schools, and am stuck in geometry. I do not understand proofs. Can you help me out? Thank you. S. Schmit Date: 08/18/97 at 09:09:44 From: Doctor Rob Subject: Re: Geometry proof The study of geometry is the first place I encountered an axiom system. You start with certain "undefined objects," in this case "point," "line," "plane," "length," "area," "between," etc. Then you are given certain statements about them which you are to accept as true. These are called Postulates or Axioms. They appear in the very first part of your book on Plane Geometry. Examples might be: Any two distinct points lie on one and only one line. Any two lines contain at most one common point. Given a line and a point not on it, there is one and only one line not intersecting the given line and containing the given point. The idea is now to develop as many true statements (called theorems) as you can about points, lines, etc., from these few postulates by using logical reasoning. A proof is a detailed description of the logical reasoning used to deduce a theorem from either the postulates or other previously proven theorems. Along the way, more objects will be defined in terms of the original undefined objects, such as an angle, a line segment, a triangle, a circle, and so on. Theorems about these new objects are also deduced in the same way. The first theorems you prove from the postulates are very simple statements. Using them, you prove more and more complicated theorems, including the one which says that the sum of the measures of the interior angles in a triangle is 180 degrees, and the Pythagorean Theorem (the sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse). One confusing thing about geometry proofs is that there is no single correct answer. Any logical sequence of true statements which ends with the desired theorem is a valid proof. Always there are many correct proofs of any theorem. Some are shorter and simpler than others, and those are usually preferred by teachers, students, and mathematicians. Some are clever, some are tedious, and some are even considered beautiful or elegant. (I'll bet you didn't think that esthetics could enter into mathematics, but it definitely does!) Often proofs are constructed by working backwards. Starting with the desired conclusion T, you could say, "If I could prove statement A, then using previously proved theorem (or postulate) B, I could conclude that T is true." This reduces your proof to proving statement A, then saying at the end of that proof, "Using Theorem B, T is true." Often there are many possibilities for A (and B). The trick is to pick one you can prove! Often several different plausible choices for A (and B) are tried to find one which works (for you). Some useful hints: Review carefully the hypothesis and conclusion of the theorem you want to prove. Keep in mind the postulates and previously proved theorems which might apply to the situation at hand. Consider working backwards, as described above. Draw a figure (or several) to illustrate the situation. Consider constructing helping lines, points, circles, etc., which might be useful. I hope that this is of some use to you. If you have specific questions, write back and we'll try to help you find solutions. -Doctor Rob, The Math Forum Check out our web site! http://mathforum.org/dr.math/ Date: 08/19/97 at 00:13:44 From: femail Subject: Re: Geometry proof Thank you! I will give it a shot and get back to you if I have further problems. |
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