The sum of any two sides of a triangle is greater than the third side.

The statement "The sum of any two sides of a triangle is greater than the third side" is known as the triangle inequality theorem. This theorem is a fundamental property of triangles, and it is used to determine whether a set of three side lengths can form a valid triangle.

Let's break down the theorem in detail:

Consider a triangle with sides a, b, and c. The triangle inequality theorem states the following three conditions must be true:

The sum of side a and side b must be greater than side c: a + b > c

The sum of side a and side c must be greater than side b: a + c > b

The sum of side b and side c must be greater than side a: b + c > a

These three inequalities must hold true for a set of three lengths to form a valid triangle. If any of these conditions are not satisfied, then the three lengths cannot form a triangle.

Example:
Let's check if the following lengths can form a triangle:

a = 5, b = 7, c = 10
Now, let's verify the triangle inequality:

a + b = 5 + 7 = 12 > c (10) ✔
a + c = 5 + 10 = 15 > b (7) ✔
b + c = 7 + 10 = 17 > a (5) ✔
Since all three conditions are satisfied, the sides 5, 7, and 10 can form a valid triangle.

Conclusion:
The triangle inequality theorem ensures that the sum of any two sides of a triangle is always greater than the third side. This is essential for the existence of a valid triangle.