What does the Triangle Inequality Theorem state?

The Triangle Inequality Theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining third side. This theorem is fundamental in geometry because it establishes a necessary condition for three lengths to form a triangle. For example, if you have side lengths of 3 and 4, their sum is 7, which must be greater than the length of the third side, say 5. This ensures that the three segments can indeed connect to form a closed shape, which is a triangle.

Understanding this theorem helps in various applications, such as in proofs involving triangles, determining the feasibility of constructing triangles with given side lengths, and solving problems related to triangle measurements in broader mathematical contexts.