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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.

- 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.
- List the given statements, and then list the conclusion to be proved. Now
you have a beginning and an end to the proof.
- 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.
- 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:

Given:

Segment AD bisects segment BC.

Segment BC bisects segment AD.

Prove:

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.