By The Math Forum Drexel University, Jessica Wolk-Stanley
You, too, can comprehend geometry -- simply ask Dr. Math!
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Definition of angle congruence —Dr. Math, The Math Forum Logic and Proof 37 The Order of a Proof Dear Dr. Math, My geometry book only describes the twocolumn proof twice, and it doesn’t give too many details. I cannot figure out if the 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 is no problem.
Just make a list of facts that you can either deduce from the givens or use to get to the Logic and Proof 47 goal. This is like building a bridge by starting at both ends and working toward the middle: Given: ∠1 = ∠2 ∠3 = ∠4 Deductions (from the givens): ∠1 = ∠4 (vertical angles) ∠1 = ∠2 implies that l and m are parallel. Possibilities (for proving the goal): n is parallel to p if ∠2 and ∠3 are equal. Aha! I found a link. I have ∠2 = ∠1 and ∠1 = ∠4 and ∠4 = ∠3, so I can prove that ∠2 = ∠3. ) Now you have the idea of a proof, and you can start working out the details.
Why do you use the symbols p and q for logic statements? Yours truly, Qian Hi, Qian, Let me answer your second question first. The use of p and q in logic statements began around 1900, but it’s not clear why these letters were chosen. We might guess that it was because the beginning and end of the alphabet were already heavily used in algebra, but no one really knows. Now for the hard question: if the p in the hypothesis is false, why is the whole statement true and not undecided? Let’s take a hypothetical example: if I stay in the shower five more minutes, then I’ll miss my bus.