What is the difference between a postulate and a theorem?

In order to better define the geometric world around us, we've come up with postulates that simplify what we talk about in language and understanding. Theorems, on the other hand, are known to be true based on these postulates. The main difference is that postulates and axioms are assumptions while theorems are proven facts.

Example 2

Complete the following proof.

Given: A = B, C = D Prove: X(A + C) = BX + DX

First, we start with what's given. Then we can add the same number (we'll say C) to both sides according to the addition property. Using the given information and the substitution property, we can replace C with D. Then, the multiplication property allows us to multiply both sides by X and the distributive property lets us separate out the parentheses and arrive at our conclusion. Here's the formal proof.