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Using If-Then Logic

Every geometry proof is a sequence of deductions that use if-then logic. You write one of the given facts as statement 1. Then, for statement 2, you put something that follows from statement 1 and write your justification for that in the reason column. Then you proceed to statement 3, and so on, till you get to the prove statement. The way you get from statement 1 to statement 2, from statement 2 to statement 3, and so on is by using if-then logic.

A two-column geometry proof is in essence a logical argument or a chain of logical deductions, like

  1. If I study, then I’ll get good grades.

  2. If I get good grades, then I’ll get into a good college.

  3. If I get into a good college, then I’ll become a babe/guy magnet.

  4. (And so on . . .)

(Except that geometry proofs are about geometric figures, naturally.) Note that each of these steps is a sentence with an if clause and a then clause.

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Above is an example of a two-column proof from everyday life. Say you’ve got a Dalmatian named Spot, and you want to prove that he’s a mammal. Check out the proof in the figure.

On the first line of the statement column, you put down the given fact that Spot is a Dalmatian, and you write Given in the reason column. Then, in statement 2, you put down a new fact that you deduce from statement 1 — namely, Spot is a dog. In reason 2, you justify or defend that claim with the reason If something is a Dalmatian, then it’s a dog.

Here are a couple of ways of looking at how reasons work:

  • Imagine you know that Spot is a Dalmatian and then someone says to you, “Spot is a dog.” You ask them, “How do you know?” Their response to you is what you’d write in the reason column.

  • When you write a reason like If something is a Dalmatian, then it’s a dog, you can think of the word if as meaning because I already know, and you can think of the word then as meaning I can now deduce. So basically, the second reason in the above figure means that because you already know that Spot is a Dalmatian, you can deduce or conclude that Spot is a dog.

Continuing with the proof, in statement 3, you write something that you can deduce from statement 2, namely that Spot is a mammal. Finally, for reason 3, you write your justification for statement 3: If something is a dog, then it’s a mammal. Every geometry proof solution has this same, basic structure.

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