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The Structure of a Proof

Geometric proofs can be written in one of two ways: two columns, or a paragraph. A paragraph proof is only a two-column proof written in sentences. However, since it is easier to leave steps out when writing a paragraph proof, we'll learn the two-column method.

A two-column geometric proof consists of a list of statements, and the reasons that we know those statements are true. The statements are listed in a column on the left, and the reasons for which the statements can be made are listed in the right column. Every step of the proof (that is, every conclusion that is made) is a row in the two-column proof.

Writing a proof consists of a few different steps.

  1. Draw the figure that illustrates what is to be proved. The figure may already be drawn for you, or you may have to draw it yourself.
  2. List the given statements, and then list the conclusion to be proved. Now you have a beginning and an end to the proof.
  3. Mark the figure according to what you can deduce about it from the information given. This is the step of the proof in which you actually find out how the proof is to be made, and whether or not you are able to prove what is asked. Congruent sides, angles, etc. should all be marked so that you can see for yourself what must be written in the proof to convince the reader that you are right in your conclusion.
  4. Write the steps down carefully, without skipping even the simplest one. Some of the first steps are often the given statements (but not always), and the last step is the conclusion that you set out to prove. A sample proof looks like this:


Segment AD bisects segment BC.
Segment BC bisects segment AD.
Triangles ABM and DCM are congruent.

Notice that when the SAS postulate was used, the numbers in parentheses correspond to the numbers of the statements in which each side and angle was shown to be congruent. Anytime it is helpful to refer to certain parts of a proof, you can include the numbers of the appropriate statements in parentheses after the reason.

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