Triangle Inequality: Theorem & Proofs

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will use definitions and proofs to learn what the triangle inequality theorem is, why it works, and how to use it to determine if three given line segments can form a triangle.

Can We Form a Triangle From Any Three Line Segments?

Most of us are familiar with the fact that triangles have three sides. However, we may not be familiar with what has to be true about three line segments in order for them to form a triangle. Is it possible to create a triangle from any three line segments? For instance, if I give you three line segments having lengths 3, 4, and 5 units, can you create a triangle from them? What about if they have lengths 3, 4, and 9 units?

Forming Triangles
triangle inequality theorem 1

Notice in the picture, when our line segments have lengths 3, 4 and 9 units, it looks like we have an incomplete triangle. It looks like our two line segments of length 3 and 4 units are too short or the line segment of length 9 is too long. This answers our question--we can't create a triangle from just any three line segments. But there's a relationship between those line segments that must be true, right? As it so happens, there is, and that relationship is explained by the triangle inequality theorem.

Triangle Inequality Theorem

The triangle inequality theorem states that the length of any of the sides of a triangle must be shorter than the lengths of the other two sides added together. This tells us that in order for three line segments to create a triangle, it must be true that none of the lengths of each of those line segments is longer than the lengths of the other two line segments combined.

For example, let's look at our initial example. We were able to create a triangle with line segments having lengths 3, 4, and 5 units. This is because those line segments satisfy the triangle inequality theorem.

  • 3 + 4 = 7 and 5 < 7
  • 4 + 5 = 9 and 3 < 9
  • 3 + 5 = 8 and 4 < 8

We see that none of the line segments are longer than the other two line segments combined.

However, if we consider the line segments with lengths 3, 4 and 9, we see that the line segment with length 9 units is longer than the other two line segments combined.

  • 3 + 4 = 7 and 9 > 7

This explains why we couldn't create a triangle with these three line segments. They don't satisfy the triangle inequality theorem.

Why Is the Triangle Inequality Theorem True?

In order to understand why the triangle inequality theorem is true, we need to recognize that the shortest distance between a point A and a line L is the length of the line segment that passes through A and is perpendicular to line L. This is easy to see in the image below.

Shortest Distance From a Point to a Line
Shortest Distance Postulate

We'll call this the shortest distance property, as we explain why the triangle inequality theorem is true.

Let's consider a general triangle RST. We'll draw a line segment starting at point R and perpendicular to the line segment ST. We let point V be where this line intersects line segment ST as shown below.

Triangle RST

Notice that the line we drew from R to ST is perpendicular to the line segment SV. Therefore, the shortest distance from S to RV is SV by our shortest distance property. Thus, it must be the case that line segment RS is longer than line segment SV, so we have SV < RS.

To unlock this lesson you must be a Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account