Early in our history, we answered many questions about geometric proofs, particularly the “two-column” variety. Many of these were collected into a FAQ page.
I want to briefly survey just some of what we have said about the big picture – an overall view of how to approach a proof, and how to work your way through it without getting lost. I’ll focus on three helpful analogies, and then point you to some examples where we worked through a proof with the student, helping them learn for themselves how to carry it out.
First analogy: Building a bridge
This student helpfully gave us a specific problem with which to demonstrate general principles:
Parallel Lines: Two Column Proof I was wondering if there is 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 l m / 3/ ------/---------/------n / / 1 / 2/ ----/---------/------p /4 /
My first comments were about the need, in proofs as in writing, to start with a first draft or an outline, not a final version. In particular, the rigid two-column format (statements and reasons) is not a good way to do your initial thinking:
I suggest that you first try to prove your goal without thinking about the details of the two columns. People too often get bogged down in the details ("what is the exact reason for this step?" "is this legal?") when the important thing about a proof is to learn the logic behind it.
So you should look for connections first, and explain the details later. Start with the big picture.
A proof is sort of a bridge from the "mainland" of known truth to an "island" you want to get to. In your case, you have been given a platform you are supposed to start from (the "givens"), and you have some set of definitions, postulates, and already-proven theorems that you can use. Think of those as materials you can use to build the bridge: ... Now you need to take a quick helicopter ride over the territory between the starting and ending points and see what you can recognize as useful stepping stones. It will take some practice to get used to what sorts of things you should look for. Just make a list of facts you either can deduce from the givens or can use to get to the goal. This is like building a bridge by starting at both ends and working toward the middle: ...
So you should think first about what you are given (the first steps), and about what it might take to reach the goal (possible last steps), and find ways to join these. Once you have the main flow of the proof, you can fill in the details. But never let the details distract you from the main idea. And don’t worry that the proofs you are initially assigned will often seem trivial; they’re just meant to give you practice in clear thinking:
... you should not worry about the "rivets" in your proof (as long as they pass the teacher's inspection). Think about it; you've just built a bridge to new territory. When you get further, you'll have much more significant proofs to write. None of them will be a Golden Gate, but some will be pretty impressive. Right now, you're just hopping a puddle, and it might not look like much, but it's good practice for building the big bridges.
At the end, I refer to two other discussions of proof. They share a common analogy, which is my next point:
Second analogy: Building a tower
Building Two Column Proofs My geometry teacher has just taught us how to do proving, but I don't get it. 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? Here's an example: <----|----|----|----|----> A B C D Given: AB = CD A-B-C-D Prove: AC = BD Statements Reason AB = CD given BC = BC reflexive property AB + BC = BC + CD addition property of equality A-B-C-D given AC = AB + BC definition of between AC = BC + BD transitive property of equality BC+CD = BD definition of between AC = BD transitive property of equality See, I don't understand how you know what statements to put and in what order, if there is any. Please help.
(Note: If you are not familiar with the notation, “A-B-C-D” means that the four points are in that order along a line.)
Here Crystal had a finished example that we could examine together, which in some ways is better than having to invent a proof. The best way to learn to write well is to read a lot of good examples, and that applies to proofs, too. How did the author of this proof know how to construct it?
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.
I went through the reasoning of the proof, step by step creating a diagram showing how each statement depends on those before it, resulting in a structure in which each “floor” is connected to the one below it. Here is the final result:
AC=BD
/ \
6 \
AC=BC+CD \
/ \ \
3 5 7
AB+BC=BC+CD AC=AB+BC BC+CD=BD
/ \ \ /
2 1 4
BC=BC AB=CD A-B-C-D
=========================
Do you see how each statement we put in is supported by some reason (the girder) using one or more previous statements (the rivets)? I could have built some parts in a different order (such as 7 before 5), but nothing can come before what it depends on.
At the end, we discussed different possible orders of the statements, and I pointed out that there may be several entirely different paths to the goal; as in writing a creative essay, two proofs may be quite different, yet both equally correct.
Third analogy: Finding a path
Here, Ashley asked about a specific proof she had to do, helpfully describing the ideas she had, but frustrated about finishing it. She had “prover’s block”.
Triangle Proofs in General I'm having a difficult time with proofs in general. I am able to do the really simple proofs but as they get harder and longer I get really lost, especially with the method cpctc after you prove triangles congruent. I am able to start the proof but am never able to complete it.
Here is my general advice:
Let's imagine another situation in which you have no idea what to do: you've just parachuted from a disabled plane and find yourself in unknown country, not knowing just where you are or how to get back to civilization. What would you do? I would probably first climb a tree or otherwise get the best view I can of the territory. In your case, you might start by drawing the picture and marking the known facts, as you have; then look it over and familiarize yourself with it. Think not only about what you are told, but about what you see, ... Look for triangles that MIGHT be congruent; you can't just assume that, and the picture might be deliberately drawn to hide such things (if you didn't draw your own picture, which is always a good idea!), but it helps to have an idea what you might be able to prove true. In this example, ACD and BCE are a good choice.
So you start by knowing where you are and where you are going, and making some hopeful guesses in between. Then just do something!
Next I would decide where I want to go, and choose a possible route. I might know that there is a town near the lake, and therefore decide to find a stream and follow it downstream. I wouldn't expect to know exactly where to go without a map, but I would want to have some general ideas of the direction I'm heading. In your proof, you want angles 3 and 4 to be equal; what "lake" is that fact on the "shore" of? It sounds like the same fact I mentioned before: the base angles of an isosceles triangle. So if I can prove that DCE is an isosceles triangle, I will be able to get to the goal. What I've really done so far is to think from both ends of the problem: where can I go from where I am (the givens), and how might I get to the goal? Now it's time to start trudging along. I'll never get anywhere unless I start moving; even if I don't know just where to go, I may run into something (like a stream running downhill) that I can follow. You might take a pair of triangles that look congruent and make a list of their "corresponding parts" to see if you have enough to prove congruence. ...
In the course of describing this style of thinking, I was giving various hints, without giving too much away.
Here are some examples where we worked all the way through a problem with the student, so you can see the whole process:
Geometry Proofs Two-Column Proof: Parallel Tangents
For more on bridge building, see
How to Build a Proof Building a Geometric Proof
Pingback: Non-routine Algebra Probems – The Math Doctors