The Triangle Inequality Theorem states that the lengths of any two sides of a triangle sum to a length greater than the third leg. This gives us the ability to predict how long a third side of a triangle could be, given the lengths of the other two sides.

Example: Two sides of a triangle have measures 9 and 11. Find the possible range for the third side.

Solution: Let’s call the third side \(x\). Then, by the triangle inequality theorem, we have the following three true statements.

\(x + 9 > 11\)
\(x + 11 > 9\)
\(9 + 11 > x\)

These three statements come from the fact that the sum of any two sides of a triangle must sum to a length greater than the length of the third leg. We solve for \(x\) in all three of the inequalities.

\(x > 2\)
\(x > - 2\)
\(20 > x\)

ALL THREE of these must be true at the same time. That is \(x > 2\) AND \(x > - 2\) AND \(x < 2\). Two of these statements can be compressed into one, though. For, if \(x > 2\) AND \(x > - 2\), we can just say that \(x > 2\), because that accounts for both of those inequalities.

Then we are left with only two statements. \(x > 2\) AND \(x < 20\), which can be written as

\(2 < x < 20\)

That is, the length of the third leg of this triangle must be between 2 and 20.

Example: Two sides of a triangle have measures 10 and 12. Find the range of possible measures for the third side.

Solution: Again, we need to formulate three inequalities. By the triangle inequality theorem, we have

\(x + 10 > 12\)
\(x + 12 > 10\)
\(12 + 10 > x\)

Solving for \(x\), we have

\(x > 2\)
\(x > - 2\)
\(22 > x\)

Then, by combining these three statements, we obtain \(2 < x < 22\).

Below you can download some free math worksheets and practice.