Does the Triangle Inequality Theorem work if the sum of two sides equals the third?

No. When the sum exactly equals the third side, such as 3 + 5 = 8, the three points lie on a straight line and form a degenerate segment, not a triangle. The inequality must be strict: the sum must be strictly greater than the third side.

Triangle Inequality Theorem — Definition, Formula & Examples

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the remaining third side. If this condition fails for even one combination of sides, the three lengths cannot form a triangle.

For any triangle with side lengths $a$ , $b$ , and $c$ , all three of the following strict inequalities must hold simultaneously: $a + b > c$ , $a + c > b$ , and $b + c > a$ . Equivalently, if $c$ is the longest side, the single condition $a + b > c$ is sufficient to guarantee all three inequalities.

Key Formula

a + b > c \qquad a + c > b \qquad b + c > a

Where:

$a$ = Length of the first side of the triangle
$b$ = Length of the second side of the triangle
$c$ = Length of the third side of the triangle

How It Works

To check whether three given lengths can form a triangle, add every pair and compare the sum to the third length. In practice, you only need to test the shortest two sides against the longest: if their sum exceeds the longest side, all three inequalities are satisfied automatically. When the sum equals the third side exactly, the three points are collinear and no triangle is formed — the inequality is strict. You can also use the theorem in reverse to find the range of possible lengths for an unknown side when two sides are known.

Worked Example

Problem: Can a triangle be formed with side lengths 5, 8, and 14?

Step 1: Identify the longest side. Here the longest side is 14.

Step 2: Add the two shorter sides and compare to the longest side.

5 + 8 = 13

Step 3: Check the inequality: 13 is not greater than 14, so the condition fails.

13 < 14 \quad \Rightarrow \quad 5 + 8 \not> 14

Answer: No, these three lengths cannot form a triangle because 5 + 8 = 13, which is less than 14.

Another Example

Problem: Two sides of a triangle measure 7 and 10. Find the range of possible lengths for the third side.

Step 1: Let the unknown side be

x

. Apply the theorem to get three inequalities.

7 + 10 > x \qquad 7 + x > 10 \qquad 10 + x > 7

Step 2: Simplify each inequality.

x < 17 \qquad x > 3 \qquad x > -3

Step 3: Since side lengths are positive, the binding constraints are

x > 3

and

x < 17

. The third inequality (

x > -3

) is automatically satisfied.

3 < x < 17

Answer: The third side must be greater than 3 and less than 17.

Why It Matters

The Triangle Inequality Theorem appears throughout high-school geometry, from proofs involving congruence to coordinate-geometry distance problems. In engineering and computer science, it underpins algorithms for shortest-path problems and is used to prune distance calculations in GPS routing and database searches. Mastering it also lays the groundwork for understanding metric spaces in more advanced mathematics.

Common Mistakes

Mistake: Only checking one pair of sides instead of confirming the critical pair (the two shortest sides vs. the longest).

Correction: Always identify the longest side first and check whether the sum of the other two exceeds it. If you pick a different pair, you may get a true inequality that masks the failing one.

Mistake: Allowing the sum to equal the third side and concluding a triangle exists.

Correction: The inequality is strict. If

a + b = c

, the result is a straight line, not a triangle. You need

a + b > c

, not

a + b \geq c