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Math for Kids23 chapters | 325 lessons

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Lesson Transcript

Instructor:
*Allison Petrovic*

Allison has experience teaching high school and college mathematics and has a master's degree in mathematics education.

In this lesson, will learn the definition of a theorem. We will also go over the Pythagorean theorem and the triangle sum theorem. An example of each theorem in action will also be provided.

Have you ever heard of the word 'theorem'? Maybe you've heard of the Pythagorean theorem or the Triangle Sum theorem. The word 'theorem' is quite common in mathematics. As you continue to middle school, and even into high school, you'll learn many new theorems in your math classes.

So what is a theorem? Put simply, a **theorem** is a math rule that has a proof that goes along with it. In other words, it's a statement that has become a rule because it's been proven to be true. This definition will make more sense as we look over two popular theorems in mathematics. Let's get started!

The **Pythagorean theorem** might be one of the most well known theorems in mathematics. This theorem explains that if you add together the squares of the two legs of a right triangle, you'll get the square of the hypotenuse. The **hypotenuse** is the side of a right triangle that is opposite from the right angle. So if we have a right triangle, we can label one of the shorter sides as *a*, and the other shorter side as *b*. We also can label the hypotenuse or the longest side as *c*. The formula for the Pythagorean theorem is then:

The Pythagorean theorem will only work for triangles with a right angle. Let's not forget that every theorem has a proof. While there are many interesting proofs for this theorem, in this lesson we're just looking at the main idea.

Let's look at an example. In this triangle, we have one leg that's 5 inches long, another leg that's 12 inches long, and a hypotenuse that's 13 inches long. We can prove that these are the correct lengths using the Pythagorean theorem. If we square 5, we get 5 x 5 = 25. If we square 12, we get 12 x 12 = 144. If we add those squares together, we get 25 + 144 = 169. Now, if we square the hypotenuse, we get 13 x 13 = 169. We can see that adding the squares of the legs gives a number that's equal to the square of the hypotenuse.

Now let's look at the Triangle Sum theorem. The **Triangle Sum theorem** says that if you add the three interior angles (the angles on the inside of a triangle), the sum will equal 180 degrees. The word 'sum' in the name of the theorem reminds us that we're adding. This theorem can be used when working with any triangle. Like the Pythagorean theorem, the Triangle Sum theorem also has a very interesting proof.

Let's look at an example using the Triangle Sum theorem. In this triangle, we can see that the measurements of the angles inside of the triangle are 30 degrees, 70 degrees, and 80 degrees. We can prove those are correct angle measurements using the Triangle Sum theorem. If we add those measurements together we get 30 degrees + 70 degrees + 80 degrees = 180 degrees.

A **theorem** is basically a math rule that has a proof that goes along with it. The **Pythagorean theorem** and the **Triangle Sum theorem** are two theorems out of many that you will learn in mathematics.

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Math for Kids23 chapters | 325 lessons

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