What theorem states that the sum of the measures of any two sides of a triangle is always greater than the measure of the third side?

The Triangle Inequality Theorem is a fundamental principle in geometry that describes the relationship between the lengths of the sides of a triangle. It states that the sum of the lengths of any two sides must be greater than the length of the third side. This theorem is essential for understanding the properties of triangles and is applicable in various geometric proofs and constructions.

For instance, if you have a triangle with sides of lengths a, b, and c, the theorem guarantees that a + b > c, a + c > b, and b + c > a must all hold true. This means that it is geometrically impossible to form a triangle if this inequality is not satisfied, thereby ensuring the structural integrity of the triangle.

In contrast, the other choices relate to different concepts within geometry. The Pythagorean Theorem addresses right triangles specifically, establishing a relationship between the squares of the lengths of the legs and the hypotenuse. The Triangle Congruence Theorem deals with conditions for triangles to be congruent, and the Angle Sum Theorem states that the sum of the angles in a triangle is always 180 degrees.