Writing Two-Column Geometric Proofs

As we begin our study of geometry,
it will be necessary to first learn about two-column proofs and how they will us
aid in the display of the mathematical arguments we make. All areas of math become
quite complex or confusing in one way or another. However, writing solutions in
the form of a two-column proof will not only allow us to organize our thoughts in
an efficient way, but it will also show that we have reasons for every claim we
make. It’s kind of like when children ask “why” over and over again. In those situations,
their questioning can be annoying and may seem to go without end. Using two-column
proofs in geometry, however, will allow us to answer all the “why’s” and our problems
will have a conclusion!

Two-column geometric proofs are essentially just tables with a “Statements” column
on the left and a column for “Reasons” on the right. The statements we make are
going to be the steps we take toward solving our problem. With each statement, we
must give a reason for why the statement is true. Reasons can consist of information
given within the problem itself, definition, postulates, or theorems.

Writing geometric proofs does require work and some planning, but with some practice,
you’ll see that it is a very effective way to write mathematical arguments. Below
is a list of steps to consider to help you begin writing two-column proofs.

Step-by-Step Instructions for Writing Two-Column Proofs

1. Read the problem over carefully. Write down the information that is given
to you because it will help you begin the problem. Also, make note of the conclusion
to be proved because that is the final step of your proof. This step helps reinforce
what the problem is asking you to do and gives you the first and last steps of your
proof.

2. Draw an illustration of the problem to help you visualize what is given
and what you want to prove. Oftentimes, a diagram has already been drawn for you,
but if not, make sure you draw an accurate illustration of the problem. Include
marks that will help you see congruent angles, congruent segments, parallel
lines, or other important details if necessary.

3. Use the information given to help you deduce the preliminary steps of
your proof. Every step must be shown, regardless of how trivial it appears to be.
Beginning your proof with a good first step is essential to arriving at a correct conclusion.

4. Use the conclusion, or argument to be proven, to help guide the statements
you make. Remember to support your statements with reasons, which can include definitions,
postulates, or theorems.

5. Once you have arrived at your solution, you may choose to read through
the two-column proof you’ve written to be assured that each step has a reason. This
helps emphasize the clarity and effectiveness of your argument.

The steps above will help guide you through the rest of the geometry sections you
encounter. While they may seem painful and frustrating at times, two-column proofs
are extremely helpful because they break things down that seem trivial or intuitive
into steps that answer the question “why.”