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Introduction to Two Column Proofs


A two column proof, usually taught during high-school Geometry in the United States, is a type of proof that visually shows steps of the proof, with each step as a row of a two-column table, the columns usually labeled "Statements" and "Reasons". As you can probably guess, the "Statements" column contains all the statements of the proof, and the "Reasons" column contains all the reasons that justifies each of the statements. A reason can be either a theorem, definition, property, or a postulate.

This note is created to tutor you on the idea of the two-column proof, promote justification of answers, and show the power of proof-writing. Therefore, this note enforces Common Core ideas. Oooooo....

Example of a two column proof

Say that we were given these facts: \(x+40=80\) and \(y=x\), and from these two facts, we are supposed to find \(y=40\).

Here is the proof in two-column fashion:

1) \(x+40=80\); \(y=x\)1) Given
2) \(40=40\)2) Reflexive Property
3) \(x=40\)3) Subtraction Property of Equality
4) \(y=40\)4) Substitution Property

Note: Because of its simple nature, the reflexive property is usually ignored and one can skip from step 1 to step 3, especially if the property is used with constants. However, it is best to justify every step possible in order to prevent mistakes while writing a proof.

Let's observe this example. First off, you can tell that this proof took only four reasons and four steps. Let's check each step one by one.

The first step is the information that was given to us, so we use the reason Given for utilizing the given facts. The second step shows that \(40=40\) because of an Algebra property: the Reflexive Property. The reason why we prove \(40\) equal is because we want to subtract \(40\) from both sides of the given equation, so we first set both sides equal in preparation of the third step. The third step uses the Subtraction Property of Equality to subtract the two integers from both expressions in both sides. Finally, by utilizing our old given fact that \(y=x\), we can prove that \(y=40\) using the Substitution Property, and we have reached our conclusion, as desired.

There are more ways to do the same problem, some resulting in a less amount of steps and some resulting in more. Below, I have noted another way of doing this example proof.

Creating A Pathway for a Proof

I didn't choose to write about a two-column proof for no reason. By using a two-column proof, it is much easier to organize and find the steps and their reasons of the proof compared to a regular paragraph proof and makes the process of creating a two column proof faster and quicker. Because we can find our steps so easily, we can also use this following idea to create the backbone of a two column proof: Come up with the statements only.

Think through how you are going to prove a statement. For example, we can use the previous example and come up with a summarized process of \(y=40\), as follows:

The problem gave me these facts: \(x+40=80\) and \(y=x\), and I am expected to prove that \(y=40\). This can go two ways: either I solve for \(x\) first and use that value as \(y\) by \(y=x\) or I can swap \(x\) for \(y\) in \(x+40=80\) and solve it that way. I'll go with the latter process because it seems more convenient. After swapping, I get \(y+40=80\). Now, I just need to 'eradicate' the 40 from both sides to yield \(y=40\)! OK, time to write the proof!

This person analyzed this problem and explored his options of attacking this problem. (Usually in Geometry textbooks in the United States, authors usually try to confine the proof to one option as to keep all answers consistent and to make answer keys line up with student responses). Then, he chose the option that he felt was most efficient and convenient (for some reason unknown to us). After that, he listed his steps one by one and even included some reasoning in his summary.

The 'Reasons' part comes naturally once you have the summary down; observe a part of his synopsis again, but this time, with italicized key terms that we will discuss after.

...After swapping, I get \(y+40=80\). Now, I just need to 'eradicate' the 40 from both sides to yield \(y=40\) ...

This clause is where the proof writer lists his steps. These italicized words are key words in creating the reasons behind the statements.

The word eradicate in this means to subtract, which means to use the Subtraction Property of Equality. The constant 40 is emphasized because we want both sides to be subtracted by this constant, so we use the Reflexive Property to prove \(40=40\) and subtract both sides by it using the Subtraction Property of Equality.

See how easy the 'why?' comes after you get the 'how?'.

Final Product

1) \(x+40=80\); \(y=x\)1) Given
2) \(y+40=80\)2) Substitution Property
3) \(40=40\)3) Reflexive Property
4) \(y=40\)4) Subtraction Property of Equality


Thank you for taking the time to read this note! I really enjoyed writing this note. So please, comment down your thoughts, suggestions, and maybe if you found some errors, please post them! Thanks again!

Note by Kevin Mo
2 years ago

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Join the fight against two-column notes. Let true Geo Proofs prosper! Daniel Liu · 1 year, 12 months ago

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@Daniel Liu @Daniel Liu I don't really like two column proofs, either, in my opinion. However, many people focus on answers, not reasons, esp. in the last few years. (Common Core is changing all of that.) Kevin Mo · 1 year, 12 months ago

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Well, in INDIA, we have to write reason beside the statement in any proof but not in a column , in a bracket ( Because I am an INDIAN) xD.. Subrata Saha · 1 year, 12 months ago

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Lol, \(y=40!\)...so you mean \(y=815915283247897734345611269596115894272000000000\) Trevor Arashiro · 1 year, 12 months ago

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@Trevor Arashiro Lolol!! Pranjal Jain · 1 year, 12 months ago

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@Trevor Arashiro XD Good one! @Trevor Arashiro Kevin Mo · 1 year, 12 months ago

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Nice post, but I don't like this style of proving. What happens if there needs to be a lemma that needs to be proven? It would be very unorganised. Or if there is a huge expression then the columns would need to be large. Joel Tan · 1 year, 12 months ago

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@Joel Tan very true, or what if induction is needed, what proof is there? Trevor Arashiro · 1 year, 12 months ago

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