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What is a proof?
How do you write a two-column proof?

Students often ask about proofs: what they are, how to understand them, and, often, how to write two-column proofs. Here are some answers from our archives:

What are Proofs?
I am a home school student through American schools, and am stuck in geometry. I do not understand proofs. Can you help me out?

    The study of geometry is the first place I encountered an axiom system. You start with certain "undefined objects," in this case "point," "line," "plane," "length," "area," "between," etc. Then you are given certain statements about them which you are to accept as true. These are called Postulates or Axioms. They appear in the very first part of your book on Plane Geometry. Examples might be...

- Dr. Rob

Thinking about Proofs
How do you know what statement to write next when you're doing a proof? And what are the reasons that you use?

    Probably the most difficult part of proving something is where to start. It takes a lot of practice, and trial and error. Since you are more interested in the thought process than the solution, let me tell you what went through my mind as I solved your example...

- Doctor Pete

Learning Proofs
I'm interested in learning how to do proofs. Can you recommend some good books on the different techniques used and how they are applied?

    What a proof is depends on whether you are talking about math, science, law, politics, etc. Many good books on proof assume experience from high school or beyond. I recommend getting as much math-related experience as you can. Solving so-called word problems is best...

- Doctor Mike

Building a Geometric Proof
I'm homeschooled, and I don't understand how two-column proofs work.

    Proofs are probably something pretty new to you, and it does take time to get a feel for what makes a proof good enough and how you can find the way to prove something. It's really more like writing an essay than like the math you've done before now - more creative and less mechanical. That makes it harder, but also more rewarding and even fun.

- Doctor Peterson

Geometry Proofs
When my teacher is writing proofs I understand them, but I am having trouble writing them on my own.

    Let's take a look at each of your reasons, and see how we can improve them.

- Doctor Peterson

Geometric Proofs
I am trying to help a friend learn geometric proofs. I need a straightforward way to explain things to her. Do you have any suggestions?

    There are three preliminary steps required to construct a good proof. The first is to understand and be aware of the definitions of each of the terms associated with what you are trying to prove. Second, know and understand previous proven theorems related to what you are trying to prove. Third, know the basic rules of logic...

- Doctor Jaffee

Books about Proofs
It would be most helpful if you gave me some advice on how to understand proofs.

    You could take a look at two excellent books that helped me understand proofs...

- Tarin

Two-column Proofs
Can you explain the steps to prove geometric figures?

    A proof is meant to take the reader from a hypothetical to a conclusion, showing why we should have no doubt of the truth... The proof is built like a kind of scaffolding; once you state a premise and show why that's true, then you can confidently make another assertion which is supported by the previous premise...

- Doctor Steve

Building Two Column Proofs
I know how a certain theorem backs up each statement, but what I don't understand is, how do you know what order to write your statements?

    A proof is just an orderly way to show that something is true, by building on other things you know are true. The only way that order matters is that each thing you say must be based on something you've already said. Often it will be based on the previous statement, but sometimes you will have to use earlier statements as well. Think of it as building a tower to reach a high goal. Your "givens" are the foundation someone laid for you, and the theorems you have are the girders and rivets you have to put together to make the tower. Let's try drawing your sample proof as a building, to show how its parts are connected...

- Doctor Peterson

How to Build a Proof
Given: Triangle ABC is a right triangle, with a right angle at 3. Prove: Angle A and angle B are complementary angles.

    One thing that's important is not to sit staring at an empty two-column chart. Our goal is to make a proof, not to fill in two columns; if we think about the columns too early it can keep us from the goal. I like to think of a proof as a bridge, or maybe a path through a forest: you have to start with some facts you are given, and find a way to your destination. You have to start out by looking over the territory, getting a feel for where you are and where you have to go - what direction you have to head, what landmarks you might find on the way, how you'll know when you're getting close...
- Doctor Peterson

The Order of a Proof
Are statements and reasons completely random in their ordering (other than the "given" and the "to prove" which are always first and last), is there a particular method for the order?

    The only requirement for ordering the steps and reasons is that if Step A depends on Step B, then A should follow B. That is why the given is first and the conclusion is last, and the same logic applies to all the intermediate steps...

- Doctor Rob

Parallel Lines: Two Column Proof
Is there any way that you could break down the steps in doing a two-column proof? One that we had to do for homework is: Given: Angle 1 congruent angle 2, angle 3 congruent angle 4; Prove: n parallel p...

    Two-column proofs are a little foreign to most of us - even to mathematicians, who don't usually use such a rigid way of writing a proof once they have learned what it means to prove something. The idea is to force you to think very clearly and express yourself very precisely. Unfortunately, no one really thinks that way, so if you're just shown a two-column proof without an explanation of how someone produced it, it seems like either magic ("how did he do that?") or a waste of time ("why did he bother to do all that?"). I suggest that you first try to prove your goal without thinking about the details of the two columns...

- Doctor Peterson

Two-Column Proof of Congruence
Complete the following (given: a picture of two triangles put together t o make a slanty rectangle; each corner of this slanty rectangle is a letter. Top left is G, bottom left is T, bottom right is A, and top right is O.
    A proof is just an orderly explanation of why you can be sure something is true. We take one step at a time and give a reason for everything we say, so there can be no doubt. In your problem, you are given the proof (the "statements"), and just have to figure out why each step was done (filling in the "reasons"). Let's go through it together...

- Doctor Peterson

Two-Column Proof About Kites
If I have one side of the proof I can get the other side, or if I am looking at a completed proof I can see how it was done, but I don't understand how to come up with the statements.

    Having written down the givens, and without having looked at the rest of the proof yet, let's think about what we have and where we want to get to...

- Doctor Peterson

Two Column Proof of a Theorem
I homeschool, and this question really has me stuck. Q. Write a two-column proof for the following theorem. Give numbered statements with reasons. Given: AC > BC and AP = BQ; To Prove: PC > QC... how do I know what to do next and how do I figure my reason?

    The next step is "AC = AP + PC" and the reason is "The whole is the sum of its parts." The following step is...

- Doctor Rob

Proofs and Reasons
Write a two-column proof for the following theorem. Give numbered statements with reasons. Given: AC > BC and AP = BQ; To Prove: PC > QC... are steps 2 and 3 correct?

    From the diagram, I think you want me to assume that P lies between A and C and Q lies between C and B. I wouldn't want to say statement 2 is wrong. However...

- Doctor Jerry

Two-column Proof
Much as it pains me to do this :-) here's an example of a two column proof. The problem was one of our problems of the week last year...

- Annie Fetter

The Value of Two-Column Proofs
What is the point of doing two-column proofs? Geometry is shapes and angles, not writing out two-column and paragraph proofs.

    You're half right. Geometry is about shapes and angles (and some other stuff as well), but the point of geometry is to accumulate knowledge about shapes and angles. And the difference between knowing something and 'sort of' knowing it is that you can prove what you know...

- Doctor Ian

See too False Proofs, Classic Fallacies in this FAQ, and for more discussions pro and con about two-column proofs, search the archives of the newsgroup geometry-pre-college for the words   two column proof   and browse the threads returned.

For in-depth discussions of proof, see Preuve-Proof-Prueba, the international newsletter on the teaching and learning of mathematical proof.

We also recommend highly "Proofs Without Words," the "Pythagorean Theorem" (with over 20 proofs), and "Proofs in Mathematics," a discussion of the value of proofs in the classroom followed by a collection of proofs classified as "simple" or "charming," all three articles from Alexander Bogomolny's monthly interactive column Cut the Knot for MAA Online.

And don't miss "Proof is out there," an article by Keith Devlin for the science section of the Guardian Online: "What exactly is a mathematical proof? This thorny question was raised again last month when an American mathematician announced a solution to a 400-year-old problem posed by the astronomer Johannes Kepler...."

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