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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.

Answer: No

Practice problems

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

Choices: Yes · No

Show solution
  1. Warm-up: First identify exactly what the question is asking: Can 3, 4, and 5 form a triangle?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 3 + 4 is greater than 5.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Yes

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

Choices: No · Yes

Show solution
  1. Warm-up: First identify exactly what the question is asking: Can 2, 6, and 9 form a triangle?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 2 + 6 is not greater than 9.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: No

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

Choices: Yes · No

Show solution
  1. Core Practice: First identify exactly what the question is asking: Can 8, 8, and 15 form a triangle?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. 8 + 8 is greater than 15.
  4. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: Yes

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: A triangle has sides 5 and 9. The third side must be less than what number?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. The third side must be less than 5 + 9.
  4. Check the result by substituting or estimating: the response should match 14 and make sense in the original problem.

Answer: 14

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

Show solution
  1. Challenge: First identify exactly what the question is asking: A triangle has sides 6 and 11. The third side must be greater than what number?
  2. Use the relevant geometric relationship first, then set up an equation from the angle measures or side relationships.
  3. Use the difference: 11 - 6 = 5.
  4. It must be greater than 5, not equal to 5.
  5. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

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