Triangle Inequality

Can any three lengths form a triangle? The answer is no, and the Triangle Inequality Theorem tells you exactly when three side lengths can and cannot form a triangle. This theorem appears frequently on the GED and is easy to apply once you know the rule.

What Is the Triangle Inequality Theorem?

The Triangle Inequality Theorem states: the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. For a triangle with sides a, b, and c:

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• \(\color{blue}{a + b > c}\)
• \(\color{blue}{a + c > b}\)
• \(\color{blue}{b + c > a}\)

All three conditions must hold. If any one fails, the three lengths cannot form a triangle.

How to Apply the Triangle Inequality

Quick check using the two smaller sides

In practice, the only test that can fail is the one involving the two smallest sides vs. The largest side. If the two smallest sides sum to more than the largest, all three conditions are automatically satisfied.

• Sides 3, 4, 5: smallest pair \(\color{blue}{3 + 4 = 7}\) > 5 ✓ → valid triangle
• Sides 2, 3, 6: smallest pair \(\color{blue}{2 + 3 = 5}\) < 6 → NOT a valid triangle
• Sides 5, 8, 12: \(\color{blue}{5 + 8 = 13}\) > 12 ✓ → valid triangle
• Sides 4, 4, 7: \(\color{blue}{4 + 4 = 8}\) > 7 ✓ → valid triangle
• Sides 1, 2, 10: \(\color{blue}{1 + 2 = 3}\) < 10 → NOT a valid triangle

Finding the range for an unknown third side

If two sides are known, say a and b, the third side c must satisfy:

|\(\color{blue}{a – b}\)| < c < \(\color{blue}{a + b}\)

For sides 4 and 7: \(\color{blue}{|4 – 7| = 3 < c < 11}\), so the third side must be between 3 and 11 (not including 3 or 11).

Step-by-Step Summary

1. Order the three side lengths from smallest to largest.
2. Add the two smaller sides.
3. Compare that sum to the largest side.
4. If the sum is greater than the largest side, the triangle is valid.
5. If the sum equals or is less than the largest side, no triangle can be formed.

Watch: Triangle Inequality Theorem (Video Lesson)

Khan Academy demonstrates the theorem with multiple examples and explains the geometric reasoning:

Worked Examples

Example 1: Can sides of length 3, 4, and 5 form a triangle?

\(\color{blue}{3 + 4 = 7}\) > 5 ✓; \(\color{blue}{3 + 5 = 8}\) > 4 ✓; \(\color{blue}{4 + 5 = 9}\) > 3 ✓. Yes, they form a valid triangle (and it is a right triangle).

Example 2: Can sides 2, 3, and 6 form a triangle?

\(\color{blue}{2 + 3 = 5}\) which is NOT greater than 6. No, these lengths cannot form a triangle.

Example 3: Two sides of a triangle are 4 and 7. What are the possible whole-number values for the third side?

\(\color{blue}{|4 – 7| < c < 4 + 7 \rightarrow 3 < c < 11}\). Whole-number values: 4, 5, 6, 7, 8, 9, 10.

Example 4: Sides 5, 8, and 12, valid triangle?

\(\color{blue}{5 + 8 = 13}\) > 12 ✓. Yes, valid triangle.

More Practice: Triangle Inequality Theorem (Video)

Math with Sohn explains how the theorem works and demonstrates classifying triangle sides:

Exercises

1. Can sides 6, 8, and 14 form a triangle?
2. Can sides 7, 10, and 15 form a triangle?
3. Two sides of a triangle are 5 and 9. What is the range of possible values for the third side?
4. Two sides of a triangle are 6 and 6. What are the possible whole-number values for the third side?
5. Can sides 3, 3, and 3 form a triangle?
6. A triangle has sides 4, 4, and 9. Is this valid? Explain.

Answers

1. \(\color{blue}{6 + 8 = 14}\) which is NOT greater than 14. No (equality is not allowed).
2. \(\color{blue}{7 + 10 = 17}\) > 15. Yes.
3. |\(\color{blue}{5 – 9}\)| = 4 < c < 14. Third side: between 4 and 14 (exclusive).
4. 0 < c < 12; whole-number values: 1 through 11.
5. \(\color{blue}{3 + 3 = 6}\) > 3. Yes, equilateral triangle.
6. \(\color{blue}{4 + 4 = 8}\) < 9. No, not a valid triangle.

Frequently Asked Questions

What is the triangle inequality theorem?

The Triangle Inequality Theorem says the sum of any two sides of a triangle must be greater than the third side. For sides a, b and c, all three of these must be true: a + b > c, a + c > b and b + c > a. If even one fails, the lengths cannot close up into a triangle, which is why 1, 2 and 10 never form one.

How do you use the triangle inequality theorem?

Put the three lengths in order and add the two smallest. If their sum is greater than the largest side, the lengths make a triangle; if not, they do not. For 6, 9 and 14, the check is 6 + 9 = 15, and 15 > 14, so a triangle is possible. For 4, 5 and 11, 4 + 5 = 9 is less than 11, so no triangle can be built.

How do you solve triangle inequality theorem problems with a missing side?

When two sides are known, the third side must be greater than their difference and less than their sum. With sides of 7 and 12, the third side x satisfies 12 − 7 < x < 12 + 7, so 5 < x < 19. If the side must be a whole number, it can be anything from 6 to 18. GED questions often ask you to pick the one answer choice that fits this range.

Does the triangle inequality work with subtraction?

Yes. A related rule, sometimes called the reverse triangle inequality, says any side must be longer than the difference of the other two. That is where the lower limit for a missing side comes from. If two sides are 10 and 3, the third side must be more than 10 − 3 = 7 and less than 10 + 3 = 13.

When is the triangle inequality equal?

The two sides add up to exactly the third side only when the three points lie on one straight line. Lengths of 2, 3 and 5 give 2 + 3 = 5, so the ‘triangle’ collapses flat and has zero area. That is why the theorem uses a strict greater-than sign for a real triangle: equal is not enough.

Is the triangle inequality theorem always true?

Yes, it holds for every triangle in ordinary flat geometry, whether it is acute, right, obtuse, isosceles or scalene. It reflects the simple idea that a straight line is the shortest path between two points, so going around through a third corner is always longer. A 3-4-5 right triangle passes all three checks: 3 + 4 > 5, 3 + 5 > 4 and 4 + 5 > 3.

Related Topics

• Triangles
• Relationship Between Sides and Angles in a Triangle
• Area of a Triangle
• Polygons
• Area of a Parallelogram