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 third side. This means that if you have a triangle with sides of lengths a, b, and c, then the following must hold true: a + b > c, a + c > b, and b + c > a. This theorem is essential in geometry as it helps determine whether three given lengths can form a triangle. If any of these conditions are not met, the three lengths cannot create a triangle, ensuring the validity of geometric constructions.