Before we can do anything with theorems, we need our vocabulary to already have good, clear definitions. A good, complete definition is our jumping off point for creating and proving theorems.
One thing to note is that definitions aren’t things that are proven. I can never prove that same side interior angles are angles that are on the same side of the transversal and in the interior of the two lines. That’s just what same side interior angles are.
It is also useful to note what a definition doesn’t include. In all of our definitions of angle pairs with transversals, none of them say that the lines have to be parallel! Noting this helps us better understand theorems. A lot of the time, a theorem is just understanding how the parts of a definition work under certain conditions. A theorem puts extra constraints on a definition, and we see what the outcomes are, what the conclusions are.
For example, the Alternate Interior Angles Theorem requires that we understand the definition of Alternate Interior Angles, but we get to consider the extra condition that the lines are specifically parallel. This new condition, along with our other assumption of the corresponding angles postulate, leads us to the conclusion that Alternate Interior Angles must be congruent if the lines are parallel.
So, one difference between theorems and definitions is that theorems say something new about a definition. Theorems say how a definition works under certain circumstances.
Next, we need to think about how we use theorems and definitions.
We use definitions to simply say whether or not a term is correct or appropriate for the given situation. We use definitions to identify things. Do the angles have the same measure? Then we call them congruent. Because of the definition.
We use theorems when certain conditions arise that match the conditions of a theorem. The theorem then tells us what conclusion we can draw. This is a conclusion that has been previously proven. Theorems are usually what you use to actually solve problems and create new information.