Triangle inequality theorem
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 third side. It determines which side lengths can and cannot form a triangle.
The triangle inequality theorem says that in any triangle, the sum of any two side lengths must be greater than the third side. For sides a, b, and c, all three conditions must hold: a + b > c, a + c > b, and b + c > a. If even one fails, the three lengths cannot form a triangle — the two shorter sides simply can't reach each other.
The intuition is physical: a triangle's third side is a straight path between two vertices, and the other two sides form a detour through the third vertex. A detour can never be shorter than the straight path, and if the "detour" exactly equals the straight path, the three points collapse onto a single line (a degenerate triangle). That's why the inequality is strict.
The theorem's most tested application is finding the range of the third side. Given two sides a and b, the third side c must satisfy |a − b| < c < a + b. For sides of 5 and 8, the third side must be greater than 3 and less than 13. Test questions exploit both ends: asking for the possible integer values of c, or for the smallest or largest integer length that works.
The GRE and ACT both test the triangle inequality regularly — sometimes directly ("which of these could be the sides of a triangle?") and sometimes buried inside a quantitative comparison or multi-step geometry problem. Achievable's GRE and ACT courses cover the theorem alongside the rest of triangle geometry.
Key takeaways
- In any triangle, the sum of any two sides must be greater than the third side.
- Given two sides a and b, the third side c satisfies |a − b| < c < a + b.
- If the sum of two sides equals the third, the points are collinear and no triangle exists.
- Exam questions typically ask which side lengths can form a triangle or what integer values the third side can take.