Understanding the Sum of Two Sides of a Triangle: The Triangle Inequality

Let's explore a fundamental rule about triangles that relates the lengths of their sides. This rule is called the Triangle Inequality Theorem.

The Triangle Inequality Theorem states that the sum of two sides of a triangle must always be greater than the length of the third side.

Think about it practically. If you have two short sticks and one very long stick, you can't connect the ends of the two short sticks to the ends of the long stick to form a triangle. The two short sticks wouldn't be long enough to meet across the length of the third side.

Mathematically, if a triangle has sides with lengths \(a\), \(b\), and \(c\), the Triangle Inequality Theorem tells us three things:

• \(a + b > c\)
• \(a + c > b\)
• \(b + c > a\)

This means that if you pick any two sides of the triangle and add their lengths together, the result will always be bigger than the length of the remaining side, which is the third side.

Let's consider an example:

Suppose we have potential side lengths 3 cm, 4 cm, and 8 cm.

Check the sum of two sides of a triangle:

• 3 + 4 = 7. Is 7 > 8? No, 7 is not greater than 8.

Since the sum of two sides (3 and 4) is not greater than the third side (8), you cannot form a triangle with these lengths. The triangle inequality is not satisfied.

Now, consider side lengths 3 cm, 4 cm, and 5 cm.

Check the sum of two sides of a triangle:

• 3 + 4 = 7. Is 7 > 5? Yes.
• 3 + 5 = 8. Is 8 > 4? Yes.
• 4 + 5 = 9. Is 9 > 3? Yes.

Since the sum of any two sides is greater than the third side in all cases, a triangle can be formed with these lengths. This demonstrates the triangle inequality in action.

The question asks about the relationship between the sum of two sides of a triangle and the third side. According to the Triangle Inequality Theorem, this relationship is always that the sum is larger than the third side.

Therefore, the sum of two sides of a triangle compared with the third side is always greater.