# 3.5: Triangle Inequality Theorem

**At Grade**Created by: CK-12

## Learning Objectives

- Apply the Triangle Inequality Theorem to solve problems.

## Triangle Inequality Theorem

For any triangle, for ANY sides you choose: when you add one side to another side, your answer is more than the third side. Try this for any combination of sides using the triangle below:

\begin{align*}& \mathbf{One \ side} \ \ + \ \ \mathbf{Another \ side}\ \ > \ \ \mathbf{The third \ side}\\ & \qquad \qquad \qquad \ 7 + 8 \ > 10\\ & \qquad \qquad \qquad \ \ \ \quad 15 > 10 \ \qquad \quad \text{yes, that’s true!}\\ & \qquad \qquad \qquad \ 8 + 10 > 7\\ & \qquad \qquad \qquad \qquad 18 > 7 \qquad \quad \ \ \ \text{yes, that’s true!}\\ & \text{and} \ \qquad \qquad \ 7 + 10 > 8\\ & \qquad \qquad \qquad \quad \quad 17 > 8 \qquad \quad \ \ \ \text{yes, that’s true too!}\end{align*}

This relationship is called the *Triangle Inequality Theorem*

**Triangle Inequality Theorem**

The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

- The sum of two sides of a triangle must be ______________________ than the third side.

**Example 1**

*Can you have a triangle with sides having lengths 4, 5, and 10?*

Without a drawing we can still answer this question — the answer is NO. It is an impossible situation; we cannot have such a triangle. By the Triangle Inequality Theorem, the sum of the lengths of any two sides of the triangle must be greater than the length of the third side. However, when you add 4 and 5 together, your result is NOT more than 10!

\begin{align*}4 + 5 = 9\end{align*}, but \begin{align*}9 > 10\end{align*} is a FALSE statement

**Example 2**

*Can you have a triangle with sides having lengths 6, 6, and 11?*

Again, we can answer this question without drawing a picture using the **Triangle Inequality Theorem.** Though this is an *isosceles* triangle, the theorem still works (and it actually makes it easier for us because we only have to test 2 pairs of sides because one repeats itself.) Let’s add up each pair of sides and compare the sum with the third side:

\begin{align*}6 + 6 & > 11\\ 12 & > 11 \ \qquad \text{yes, that’s true!}\\ \text{and} \qquad 6 + 11 & > 6\\ 17 & > 6 \ \qquad \ \ \text{yes, that’s true too!}\end{align*}

We do not need to test \begin{align*}6 + 11\end{align*} twice, since it is the same for the third pair of sides. Based on the Triangle Inequality Theorem, this triangle DOES exist!

**Reading Check:**

1. *Create a triangle that has side lengths that SATISFY the Triangle Inequality Theorem. Draw the triangle, label the sides, and show your work to prove that your triangle exists.*

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2. *Create a triangle that has side lengths that DO NOT SATISFY the Triangle Inequality Theorem. Draw the triangle, label the sides, and show your work to prove that your triangle DOES NOT exist.*

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3. *Can you have a triangle with side lengths of 13, 8, and 5? Show your work to support your answer.*

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