What does the triangle inequality theorem state?

The triangle inequality theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side. This is essential for maintaining the shape of a triangle, as it ensures that the third side is long enough to connect the endpoints of the two sides considered.

In practical terms, if you have a triangle with sides of lengths ( a ), ( b ), and ( c ), the triangle inequality theorem asserts the following three relationships:

1. ( a + b > c )

2. ( a + c > b )

3. ( b + c > a )

If any of these conditions is not met, the lengths cannot form a triangle. This theorem is foundational in geometry and helps in understanding the properties of triangles and their side lengths.

The other statements do not capture the essence of the theorem. For instance, saying that the sum of any two sides must be less than the third or equal to the third violates the necessary conditions for triangle formation. Therefore, the assertion that the sum of any two sides must be greater than the third side is the correct interpretation of the triangle inequality theorem.