Question

How do I understand two column proofs?

Answer 1

Explanation:

When you solve an equation, you use the rules of algebra. Before you begin solving equations, you learn the rules of algebra. Specifically, you learn about rules relating to addition, multiplication, equality, identity elements, order of operations, and inverse operations.

Whenever you "show work" solving an equation, you are demonstrating at each step that you know how to apply these rules to get one step closer to a solution.

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A 2-column proof is a list of "statements" in one column, and associated "reasons" in the other column.

The first statement is generally a list of all of the things that are "given" in the particular problem. The first reason is generally, "Given".

The last statement is generally a statement of what you are trying to prove. The last reason is a description of the postulate or theorem you used to conclude the last statement is true, based on the previous statement.

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Here's a brief example:

Suppose we have line segment RT with point S on the segment. Suppose the lengths are given: RS = 3, ST = 4. We are asked to prove that RT = 7. The proof might look like this:

Statement . . . . Reason

Point S lies on RT; RS = 3; ST = 4 . . . . Given

RT = RS +ST . . . . segment addition postulate

RT = 3 + 4 . . . . substitution property of equality

RT = 7 . . . . properties of integers

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So, creating or filling in 2-column proofs requires you have a good understanding of the theorems and postulates you are allowed (or expected) to choose from, and an understanding of logical deduction. Essentially, you cannot make a statement, even if you "know it is true", unless you can cite the reason why you know it is true. Your proof needs to proceed step-by-step from what you are given to what you want to prove.

It might be useful to keep a notebook or "cheat sheet" of the names and meanings of the various properties and theorems and postulates you run across. Some that seem "obvious" still need to be justified. X = X, for example, is true because of the reflexive property of equality.

It can be helpful to read and understand proofs that you see in your curriculum materials, or that you find online--not just skim over them. This can help you see what detailed logical steps are needed, and the sorts of theorems and postulates that are cited as reasons. It is definitely helpful to pay attention when new relationships among geometrical objects are being introduced. You may have to use those later in a proof.

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Additional comment

As in the above proof, you may occasionally run across a situation where you're asked to "justify" some arithmetic fact: 3+4=7 or 2×3=6, for example. I have never been quite clear on the justification that is appropriate in such cases. In the above, I have used "properties of integers", but there may be some better, more formal reason I'm not currently aware of. This is another example of the "obvious" needing to be justified.