Prove It – how (and why) to write proofs

For some reason, when geometry students get to the proof writing segments of their classes, they freak out. “Why do we have to learn how to write proofs?” “This is useless.” Etc, etc.

Well, first off, let me tell you that writing proofs is not useless, and you do it every day (whether effectively or ineffectively) without even realizing it.

Remember how you wanted that new video game that just came out and your parents said “No” and you tried to convince them to let you buy the video game? Did you just say “I need the video game.”? Probably not. That’s not very convincing. What you probably did was talk about all the reasons you “needed” the video game (or car, movie ticket, date, you name it – whatever it is that you had to convince your parents you “needed”)

Well, little did you know – but you were (verbally) writing a proof! Who would’ve thought! You already knew how to write proofs!

The tricky part about writing proofs in geometry is that there are so many rules, and most of them don’t usually make sense (if you have a bad attitude about it, like so many kids do). The thing is, you know more math than you think, and most of the “reasons” for geometric things being the way they are come pretty naturally to us as human beings.

The way I like to think about all those rules we use in Geometry proofs are as tools that I keep in my Proof Toolbox. There are a lot of rules. Such as, when two parallel lines are cut by a transversal, the alternate exterior angles are congruent. (WHAT!?)

The fascinating thing about using words in Math is that we usually use words that make sense. Don’t think about all these crazy terms like “alternate exterior angle” and freak out – just calm down and look at each word in the term as an individual word, and check out what it means.

First let’s take the word “alternate” – what does the word “alternate” mean? Well, in a selection process, such as when someone gets chosen for a play or for a spot on a team, an “alternate” would be someone that would take that person’s place if for some reason they weren’t able to perform. But that’s not really what we’re looking for. What about when you skateboard through some cones in the road “alternating” sides of the cones? Now that’s more like it. So “alternate” just means on opposite sides of the transversal (the line that cuts the parallel lines).

Now let’s look at “exterior” – well, that one is pretty easy. It just means “on the outside”. If we think of our parallel lines like a tunnel, then the “exterior” angles are on the outside of the tunnel.

We’ll talk more about how to figure out what all these crazy math words mean in another post.

Now let’s talk about how to write a proof. Since we have been talking about parallel lines, let’s start there.

Usually a proof problem has two parts – the “Givens” (i.e., the things you know for sure) and the “Prove” (what you are supposed to show is true based on the things you know for sure)

Given: and are congruent

Prove: lines l and m are parallel.

Now there are a lot of theorems and properties (i.e. tools) that will help us write this proof, but a tool won’t help you unless you actually have it. In our case, that means that a theorem or a property isn’t going to help us unless we know it. So if you have been studying your tools, you will know right away that angles 1 and 2 are “alternate exterior angles.” We know that If two parallel lines are cut by a transversal, then their alternate exterior angles are congruent. The converse (which just means switching the parts of the if-then statement) is also true: If the alternate exterior angles of two lines cut by a transversal are congruent, then the two lines are parallel. Which means that this proof is easy – we basically just state the converse of the alternate exterior angles property.

I will be starting a series on proof writing soon that will have useful information for writing proofs all the way from beginning geometry into college analysis. If you have specific questions about writing proofs, let me know in a comment, via email or Facebook or twitter, and I will try to incorporate your answer into the posts for the series.

If you have questions, you can ask them in the comments, email me, ask on Facebook, or on Twitter. I am on Facebook and Twitter live from 3:00-3:30 pm Mon-Thurs MDT.