Two Column Proof
The Complete Guide w/ 7 Step-by-Step Examples!

In today’s lesson, you’re going to learn all about geometry proofs, more specifically the two column proof.

You’re going to learn how to structure, write, and complete these two-column proofs with step-by-step instruction.

Let’s jump in!

What Is A Two Column Proof?

Definition

A proof is a logical argument that is presented in an organized manner.

There are many different ways to write a proof:

1. Flow Chart Proof
2. Paragraph Proof
3. Two-Column Proof

The most common form in geometry is the two column proof.

Every two-column proof has exactly two columns. One column represents our statements or conclusions and the other lists our reasons. In other words, the left-hand side represents our “if-then” statements, and the right-hand-side explains why we know what we know.

And to help keep the order and logical flow from one argument to the next we number each step.

Detailed Example

In the example below our goal we are given two statements discussing how specified angles are complementary. Additionally, we are provided with three pictures that help us to visualize the given statements. Our goal is to verify the “prove” statement using logical steps and arguments.

Remember, everything must be written down in coherent manner so that your reader will be able to follow your train of thought. While you can assume the reader has a basic understanding of geometric theorems, postulates, and properties, you must write your proof in such as way as to sequentially lead your reader to a logical and accurate conclusion.

How to write a two column proof?

So what should we keep in mind when tackling two-column proofs?

Always start with the given information and whatever you are asked to prove or show will be the last line in your proof, as highlighted in the above example for steps 1 and 5, respectively.

Remember when you are presented with a word problem it’s imperative to write down what you know, as it helps to jumpstart your brain and gives you ideas as to where you need to end up?

The same thing is true for proofs.

Start with what you know (i.e., given) and this will help to organize your statements and lead you to what you are trying to verify.

Sometimes it is easier to first write down the statements first, and then go back and fill in the reasons after the fact. Other times, you will simply write statements and reasons simultaneously. There is no one-set method for proofs, just as there is no set length or order of the statements.

As long as the statements and reasons make logical sense, and you have provided a reason for every statement, as ck-12 accurately states. As seen in the above example, for every action performed on the left-hand side there is a property provided on the right-hand side. These steps and accompanying reasons make for a successful proof.

Proofs take practice! The more your attempt them, and the more you read and work through examples the better you will become at writing them yourself.

Additionally, it’s important to know your definitions, properties, postulates, and theorems.

Consequently, I highly recommend that you keep a list of known definitions, properties, postulates, and theorems and have it with you as you work through these proofs. Again, the more you practice, the easier they will become, and the less you will need to rely upon your list of known theorems and definitions.

In the video below, we will look at seven examples, and begin our journey into the exciting world of geometry proofs.

How To Do Proofs In Geometry – Lesson & Examples (Video)

44 min

• How to Write Two-Column Proofs?
• 00:00:25 – What is a two column proof? (Example #1)
• 00:08:58 – Complete the two-column proof for congruent segments or supplementary angles (Examples #2-3)
• Exclusive Content for Member’s Only

• 00:20:07 – Complete the two column proof for congruent segments or complementary angles (Examples #4-5)
• 00:29:19 – Write a two column proof (Examples #6-7)
• 00:40:53 – List of important geometry theorems
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions