Share ThisMastering the Formal Geometry Proof
Suppose you need to solve a crime mystery. You survey the crime scene, gather the facts, and write them down in your memo pad. To solve the crime, you take the known facts and, step by step, show who committed the crime. You conscientiously provide supporting evidence for each statement you make.
Amazingly, this is the same process you use to solve a proof. The following five steps will take you through the whole shebang.
1. Get or create the statement of the theorem.
The statement is what needs to be proved in the proof itself. Sometimes this statement may not be on the page. That's normal, so don't fret if it's not included. If it's missing in action, you can create it by changing the geometric shorthand of the information provided into a statement that represents the situation.
2. State the given.
The given is the hypothesis and contains all the facts that are provided. The given is the what. What info have you been provided with to solve this proof? The given is generally written in geometric shorthand in an area above the proof.
3. Get or create a drawing that represents the given.
They say a picture is worth a thousand words. You don't exactly need a thousand words, but you do need a good picture. When you come across a geometric proof, if the artwork isn't provided, you're going to have to provide your own. Look at all the information that's provided and draw a figure. Make it large enough that it's easy on the eyes and that it allows you to put in all the detailed information. Be sure to label all the points with the appropriate letters. If lines are parallel, or if angles are congruent, include those markings, too.
4. State what you're going to prove.
The last line in the statements column of each proof matches the prove statement. The prove is where you state what you're trying to demonstrate as being true. Like the given, the prove statement is also written in geometric shorthand in an area above the proof. It references parts in your figure, so be sure to include the info from the prove statement in your figure.
5. Provide the proof itself.
The proof is a series of logically deduced statements — a step-by-step list that takes you from the given; through definitions, postulates, and previously proven theorems; to the prove statement.
Remember the following:
- The given is not necessarily the first information you put into a proof. The given info goes wherever it makes the most sense. That is, it may also make sense to put it into the proof in an order other than the first successive steps of the proof.
- The proof itself looks like a big letter T. Think T for theorem because that's what you're about to prove. The T makes two columns. You put a Statements label over the left column and a Reasons label over the right column.
- Think of proofs like a game. The object of the proof game is to have all the statements in your chain linked so that one fact leads to another until you reach the prove statement. However, before you start playing the proof game, you should survey the playing field (your figure), look over the given and the prove parts, and develop a plan on how to win the game. Once you lay down your strategy, you can proceed statement by statement, carefully documenting your every move in successively numbered steps. Statements made on the left are numbered and correspond to similarly numbered reasons on the right. All statements you make must refer back to your figure and finally end with the prove statement. The last line under the Statements column should be exactly what you wanted to prove.
Comments (20)
The goal of geometric proofs isn't for you to be able to graduate from school with a firm grasp of how to create a proof, it's to teach you how to deconstruct complex mathematical concepts logically so you can better understand how and why they work.
It's a lot like driving a car. You can learn to drive a car without any understanding of how an engine works, in the same way that you can learn, accept, and use theorems and axioms without understanding how or why they work. But when your engine starts acting funny, you'll have to pay through the nose to get it fixed. In the same way, practicing geometric proofs will give you the ability to examine a geometric statement and evaluate it (or fix it) yourself.
On the whole, though, the proofs you'll do in a high school math class amount to learning how to change the spark plugs or the oil, not rebuild the engine.
@Zack Glafanakis: If the information here seems obvious to you, great! You're ready to move on to more difficult topics, perhaps somethnig like this . But don't assume that what is obvious to you is obvious to everyone.
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