which statement can verify the triangle inequality theorem

Answer 1

The Triangle Inequality Theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following inequalities must hold:

1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

To verify the triangle inequality theorem, you can check that these three conditions are satisfied for any three lengths proposed as the sides of a triangle. If all three conditions hold true, then those lengths can indeed form a triangle.

For example, consider three lengths \(3\), \(4\), and \(5\):

1. \(3 + 4 > 5\) (True; \(7 > 5\))
2. \(3 + 5 > 4\) (True; \(8 > 4\))
3. \(4 + 5 > 3\) (True; \(9 > 3\))

Since all three inequalities are satisfied, the lengths 3, 4, and 5 can form a triangle, thereby verifying the triangle inequality theorem.