Postulates & Theorems in Math: Definition & Applications

Instructor: Laura Pennington

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will define postulates and theorems in mathematics. We will look at examples of postulates and theorems and how to use them in mathematics and in real-world applications.

What is a Postulate?

Suppose you decide to participate in a group research project for a new hair growth product that a company is testing. When you go to the group meeting, the first thing you have to do is fill out a questionnaire asking for your initial hair length before using the product. As you're all doing this, a woman in the group, Angie, states that she has the longest hair in the group. You take a quick glance around, and it looks like she does have that longest hair of all of the people in the group.


Now, are you going to accept her statement as true, or are you going to whip out a tape measure and measure the length of everyone's hair to verify the truth of her statement? Most likely, you would accept her statement as true, because it is fairly obvious that she has the longest hair in the group.

In mathematics, we call a statement like this a postulate. A postulate is a statement that is accepted as true without having to formally prove it. In the same way that it was fairly obvious that Angie's hair was the longest in the group, postulates in mathematics are usually easy to accept as true using simple mathematical reasoning. This is why they don't need to be proven formally.

For example, a well-known postulate in mathematics is the segment addition postulate, which states the following:

  • Segment Addition Postulate: If a point, B, is drawn on a line segment AC, then AC is the sum of AB and BC. That is, AB + BC = AC


Looking at the image of this postulate, we see that it is obvious that if we split a line segment into two parts, then adding up those two parts will give us the original line segment. Therefore, we accept the postulate as true without having to prove it.

Now that we are familiar with what a postulate is, let's take a look at theorems!

What is a Theorem?

Now, suppose that one of the side effects of the hair product is that it can cause shortening of the user's right index finger, so another question on the questionnaire asks for the length of the participant's index finger before using the product. George, a member of the group, states that his index finger is the longest in the group. Hmmmm…George's statement isn't as obvious as Angie's statement because the lengths of everyone's index fingers look pretty similar. In this case, you go ahead and measure the right index fingers of the group, and it turns out that George's index finger is, in fact, the longest in the group.


In this case, George's statement was formally proven and then accepted as true. In mathematics, we call this a theorem. A theorem is a statement that is not always obvious but has been proven using mathematical reasoning, along with other theorems and postulates.

For instance, a well-known mathematical theorem is the midpoint theorem, and it states the following:

  • Midpoint Theorem: If M is the midpoint of a line segment AB, then AM = (1/2)AB and MB = (1/2)AB

The proof of this theorem makes use of the segment addition postulate and is shown in the image.


We see that both postulates and theorems are extremely useful in the proofs of theorems! Let's take a look at an application of postulates and theorems in a couple of real-world situations.

Real-World Applications

Another simple postulate in mathematics is the Angle Addition Postulate, which states the following:

  • Angle Addition Postulate: If we have an angle and we draw a ray splitting that angle up into two new angles, then the sum of the measures of the two new angles is equal to the measure of the original angle.


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