What is a proof?
How do you write a twocolumn proof?
Students often ask about proofs: what they are, how to understand them, and, often,
how to write twocolumn 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 mathrelated experience as
you can. Solving socalled word problems is best...
 Doctor Mike
Building
a Geometric Proof
I'm homeschooled, and I don't understand how twocolumn 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
Twocolumn
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 twocolumn 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 twocolumn
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...
Twocolumn 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 twocolumn 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
TwoColumn 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
TwoColumn 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 twocolumn
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 twocolumn 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
Twocolumn
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 TwoColumn Proofs
What is the point of doing twocolumn proofs? Geometry is shapes and
angles, not writing out twocolumn 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 twocolumn proofs,
search
the archives of the newsgroup geometryprecollege for the words
two column proof and browse the threads returned.
For indepth discussions of proof, see PreuveProofPrueba,
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 400yearold problem posed by the astronomer Johannes Kepler...."
