Which theorem provides a fundamental understanding of the relationship between the sides of a triangle?

The Triangle Inequality Theorem is essential for understanding the relationship between the sides of a triangle because it establishes a crucial rule about the lengths of the sides. According to this theorem, the sum of the lengths of any two sides of a triangle must always be greater than the length of the third side. This principle not only helps to determine whether three lengths can form a triangle but also provides insights into the relative sizes of the sides in connection to one another.

For example, if you have three lengths, say 3, 4, and 8, you can apply the Triangle Inequality Theorem as follows: 3 + 4 is not greater than 8, which confirms that these lengths cannot form a triangle. This theorem is fundamental in geometry and has practical applications in various fields, including architecture and engineering, where the integrity of triangular structures must be ensured.

While the Pythagorean Theorem specifically pertains to right triangles and helps understand the relationship between the lengths of the sides through the equation (a^2 + b^2 = c^2), it does not apply to all triangles. The Euclidean Theorem is a broader concept related to Euclidean geometry as a whole, and the Triangle Congruence Theorem focuses