How to Do Two Column Geometry Proofs Online
When constructing two-column proofs whether in geometry or otherwise, there are several guidelines to follow. First off, make sure you understand the format of a two-column proof. It is not the only format available, though a common one. Establish your givens (starting point) and desired conclusion to be proven (destination). Write a draft of the proof with focus being on logical rigor and wording that helps get you to the end. Lastly, distinguish between theorems and axioms with help of reputable sources on the subject.
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Instructions
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Proof Construction
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Keep in mind the importance of formal reasons in steps. The whole idea of a proof in mathematics revolves around statements that are crystal clear, rigorously tested and researched, and are, based on the best mathematical knowledge available, absolutely correct. Proofs can be lengthy and convoluted, or elegant.
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List given statements with "Given" as reasons. This part is the information from which other steps and the conclusion will be deduced. State the yet-to-be-proven conclusion after leaving some white space. Do not given "reason" yet since you don't know by what logic the conclusion is true.
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Reasons should be written to help you. While strict adherence to wording from theorems is not required in the informal draft of the proof, do not ignore precision. Informality refers to wording, not mathematical laxity. Jot down these steps to the side since they should not be in the final version of a two-column proof.
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Research and revise that informality. When giving informal reasons, all may seem right and "obvious." With formal proofs, remember to differentiate between theorems and axioms. An axiom is a simple statement accepted as true a priori---the "identity property" that a value x equals itself, for example. A theorem is any statement strictly deduced from axioms.
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Turn informal logical statements into claims explicitly backed by formal geometric and algebraic theorems. Research reputable internet sources---not Wikipedia---to make sure that theorems in the proof are true and accepted. State theorems and axioms in the "Reasons" column, across from statements that are justified by them.
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Tips & Warnings
By stating the conclusion on the paper (or e-document) you're working on, you can save yourself from agonizing over potentially superfluous given information.
In case axiom-theorem distinction seems painstaking ... It is. Alfred Whitehead and Bertrand Russell's "Principia Mathematica" defined axioms that proved---after 379 pages, mind you---that 1+1 =2.
Credible sources on verifying theorems would be university or government education websites.
Beware if tempted to ponder how, or by what reason, one needs over 370 pages to justify the statement "1+1=2". Unless you have a substantial passion for set theory and/or philosophy of mathematics, this will drive you mad.
