The sum of length of two sides of a triangle is always

Options

A. less than third
B. greater than third
C. equal to third
D. `(2)/(3)` to third

Answer

To solve the question regarding the sum of the lengths of two sides of a triangle, we can follow these steps:

### Step 1: Understand the Triangle Inequality Theorem
The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

### Step 2: Define the Sides of the Triangle
Let's denote the sides of the triangle as \( A \), \( B \), and \( C \). According to the theorem, we can express the following inequalities:
1. \( A + B > C \)
2. \( A + C > B \)
3. \( B + C > A \)

### Step 3: Analyze the Inequalities
From the inequalities derived from the Triangle Inequality Theorem:
- The first inequality \( A + B > C \) indicates that the sum of sides \( A \) and \( B \) is greater than side \( C \).
- The second inequality \( A + C > B \) indicates that the sum of sides \( A \) and \( C \) is greater than side \( B \).
- The third inequality \( B + C > A \) indicates that the sum of sides \( B \) and \( C \) is greater than side \( A \).

### Step 4: Conclusion
From these inequalities, we can conclude that the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. Therefore, the correct answer to the question is:
**The sum of the length of two sides of a triangle is always greater than the length of the third side.**