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Triangle Inequality and Side Relationships

A free Geometry lesson from the “Triangles, Similarity, and Trigonometry” unit, with a worked example and practice problems including step-by-step solutions.

The triangle inequality says that the sum of any two side lengths of a triangle must be greater than the third side. This matters because not every set of three lengths can form a triangle. The rule reflects the idea that two shorter sides must be long enough to meet. When practicing, check all three pair sums, especially the two smallest sides compared with the largest side. A common mistake is using greater than or equal to; equality would make a flat line segment, not a triangle.

What you'll learn

Why it matters: Engineers and fabricators check triangle inequality before cutting supports; lengths that fail the rule cannot close into a rigid frame.

Worked example

Problem. Can side lengths 4, 7, and 12 form a triangle?

  1. Check the two smaller sides.
  2. 4 + 7 = 11.
  3. 11 is not greater than 12, so no triangle is possible.
  4. Connect the result back to Triangle Inequality and Side Relationships so the geometric relationship is explicit.

Answer: No

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (multiple choice and typed response). Counts include repeated prompts and variants.

Practice problems

1. Can 3, 4, and 5 form a triangle?

Choices: Yes · No

Show solution
  1. 3 + 4 is greater than 5.

Answer: Yes

2. Can 2, 6, and 9 form a triangle?

Choices: No · Yes

Show solution
  1. 2 + 6 is not greater than 9.

Answer: No

3. Can 8, 8, and 15 form a triangle?

Choices: Yes · No

Show solution
  1. 8 + 8 is greater than 15.

Answer: Yes

4. A triangle has sides 5 and 9. The third side must be less than what number?

Show solution
  1. The third side must be less than 5 + 9.

Answer: 14

5. A triangle has sides 6 and 11. The third side must be greater than what number?

Show solution
  1. Use the difference: 11 - 6 = 5.
  2. It must be greater than 5, not equal to 5.

Answer: 5

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