## 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. Hmm...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's 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, but let's quickly move through the different steps:

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.

Suppose that an architect is working on some blueprints of a project, and he has a situation where he needs to know the measurement of an angle that has been split in two by another ray. The measurements of the two angles that make up the angle are 45° and 80°.

We can use our Angle Addition Postulate to find the measurement of the angle by adding up the measurements of the two angles!

The measurement of the angle is 125°, and we see how postulates can be used in real-world applications. Let's consider one more!

A well-known theorem is the Pythagorean Theorem.

- Pythagorean Theorem: If a right triangle has side lengths
*a*, *b*, and *c*, where *c* is the longest side or the hypotenuse, then *a*2 + *b*2 = *c*2.

Suppose that on a map the roads connecting three cities form a right triangle as shown in the image appearing on your screen right now.

We show that the distance between city A and city B is 30 miles, and the distance between city B and city C is 40 miles. We can find the distance between city A and city C using the Pythagorean Theorem! We simply plug *a* = 30 and *b* = 40 into the formula and solve for *c*. As we can see here, what we get is first:

So we therefore get that the distance between city A and city C is 50 miles. Wow! These postulates and theorems really are useful in the real world!

## Lesson Summary

All right, let's take a moment or two to review what we've learned. We first learned that a **postulate** is a statement that is accepted as true without having to formally prove it. Postulates are usually easy to accept as true using a bit of simple mathematical reasoning. A **theorem**, on the other hand, is a statement that is not always obvious but has been proven using mathematical reasoning, and other theorems and postulates.

Theorems and postulates are extremely useful in mathematical applications. We can use them to prove other theorems, and we can also use them in real-world applications. Obviously, these concepts are definitely worth tucking into our mathematical toolboxes for future use!