Which theorem is used to determine if three lengths can form a triangle?

The Triangle Inequality Theorem is essential for determining if three lengths can form a triangle. This theorem states that for any three lengths, let's denote them as a, b, and c, the sum of the lengths of any two sides must be greater than the length of the third side. This can be expressed as:

1. a + b > c
2. a + c > b

3. b + c > a

If all three of these inequalities hold true, then the three lengths can indeed form a triangle.

For instance, if you have three segments with lengths of 3, 4, and 5, you would check:

• 3 + 4 > 5 (True)
• 3 + 5 > 4 (True)
• 4 + 5 > 3 (True)

Since all three conditions are satisfied, these lengths can form a triangle.

Other theorems listed, such as the Pythagorean Theorem, apply specifically to right triangles and do not necessarily evaluate the general condition of forming any triangle. The Congruence Postulate addresses the relationships between triangles regarding their size and shape but does not provide a method for verifying if three lengths can create a triangle. The