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ZingPath: Fundamental Axioms and Theorems

Forms of Proofs

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Fundamental Axioms and Theorems

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Forms of Proofs

Geometry

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You will get to learn the fundamental ideas you need to understand in order to make sound geometric conjectures and prove them.

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Now You Know

After completing this tutorial, you will be able to complete the following:

  • Explain a proof in two-column format.
  • Explain a proof in flow chart format.
  • Explain a proof in paragraph format.

Everything You'll Have Covered

Summarize the two-column proof format.

~ The two-column proof format uses two columns to organize the argument. The first column has statements and the second column has the reasoning behind each statement. The proof is written step-by-step with each step numbered. Each statement in the first column is a proposition, and each proposition follows from earlier propositions and is based on results that are already known. The propositions all lead up to the proposition that is trying to be proved. The reasons in the second column state why each proposition follows from the ones before it; these can include theorems, axioms, properties, or definitions.

Summarize the flow chart proof format.

~ In the flow chart proof format, the proof is written by writing the main part of the proof in boxes, and by using arrows to show how the different parts of the proof imply each other. Each box includes a step of the proof and a reason for that step. The reason is often written below the step and could be a theorem, axiom, property, or definition.

Summarize the paragraph proof format.

~ In the paragraph proof format, the proof is written with complete sentences that explain in ordinary language why the proposition is true. A paragraph proof still follows a step-by-step process with reasons for each step, but the content of the proof is integrated into one or more paragraphs.

Which of the three proof methods discussed in this Animation do you feel most comfortable using? Which one do you feel least comfortable using?

~ Answers will vary.

Is it possible that a proposition can be proved using the two-column proof format, but not the paragraph proof format?

~ No, it is not possible that a proposition can be proved using the two-column proof format, but not by using the paragraph proof format. If we can prove something in one of the proof formats discussed, then we can prove it in any of proof formats discussed. This is because there is no difference in the content of the proof, only in its presentation and formatting.

Tutorial Details

Approximate Time 2 Minutes
Pre-requisite Concepts Students should be able to define a proof format, flow chart proof format, and paragraph proof format.
Course Geometry
Type of Tutorial Animation
Key Vocabulary proof format, flow chart proof format, paragraph proof format