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What is a Theorem?

Reference > Mathematics > Introduction to Proofs

So far we've talked mainly about algebraic properties, like "The additive property of equality," which says that if you have a true equation, you can add the same thing to both sides and it will still be a true equation. Now, we never proved that this property is true; we just accepted it as being true. In geometry, things that you just accept to be true without proving them are called "axioms" or "postulates." So when you see "axiom" in a geometry class, or "postulate," you can just think to yourself, "This is like those algebra properties that we accepted as true, without proving."

But when you take a geometry class, you won't just talk about axioms and postulates; you'll also use theorems. A theorem is anything that you've proved, that you think is useful enough that you might want to use it again sometime. Let me show you what I mean. Here's a proof for us to do:

Prove: if ax + b = 0, and a ≠ 0  then x = -

Here's how we prove it:

STATEMENT                            REASON
1. ax + b = 0                        Given
2. a ≠ 0                             Given
3. ax + b - b = 0 - b                Subtraction property of equality
4. ax = -b                           Combining like terms
5. ∴ x = -
Division property of equality (divide by a; 2, 4)

Note that I was required to specify that a ≠ 0, because the division property of equality requires that we not divide by zero. Therefore, I had to include that step as part of my reason for using the division property. Also, notice that I shortened my proof by clarifying in step 5 what I divided by, which allowed me to avoid having an extra step where I showed what I divided by, before showing the result.

At this point, all I have is just another proof, but as I consider this problem, I realize that it's very common to have to solve a linear equation in the form ax + b = 0, and it would be nice if I didn't have to do those 5 steps every single time I have to solve an equation like that. So I decide that I'm going to give this proof a name, and then I will be able to use this proof as a reason in another proof.

What should I call this proof? about "Mr. T's Linear Equation Theorem"? That's got a nice ring to it, don't you think? And just by giving the proof a name, it is no longer just a proof - it's a theorem!

So let's take a look at how we can use this in another problem:

Prove: If 2x + 6 = 0, then x = -3

We've done problems like this before, and they took us several steps to prove. Now, though, our proof is going to be very short, because I have theorem that does most of the work for us:

STATEMENT                              REASON
1. 2x + 6 = 0                          Given
2. x = -
Mr. T's Linear Equation Theorem 3. ∴ x = -3 Arithmetic

Wow! Wasn't that easy? It's a whole lot nicer than using the extra steps of subtraction property and division property. Using a theorem can make my proofs a lot shorter! In the next section we'll create another theorem, and then we'll make use of the new theorem, along with this one, to make a long proof very short!


Prove: If 3x + 9 = 0 then x = -3
Prove: If 2x + 6 = 5x - 12, then x = 6
Prove: If 5(x + 2) = 30, then x = 4
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Definitions in ProofsDefinitions in Proofs
Transitive Property in ProofsTransitive Property in Proofs

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