If a, b,c are the lengths of the sides of triangle , then

Options

A. `a - b gt c`
B. `a+b lt c`
C. `c = a+b`
D. `c lt a+b`

Answer

To solve the problem regarding the sides of a triangle, we will use the triangle inequality theorem. This theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

### Step-by-Step Solution:

1. **Understanding the Triangle Inequality Theorem**:
- For any triangle with sides of lengths \( a \), \( b \), and \( c \), the following inequalities must hold:
- \( a + b > c \)
- \( a + c > b \)
- \( b + c > a \)

2. **Analyzing the Options**:
- We need to evaluate the given options based on the triangle inequality theorem.

3. **Option A: \( a - b > c \)**:
- This option suggests that the difference between sides \( a \) and \( b \) is greater than side \( c \).
- This is not true according to the triangle inequality theorem. The theorem states that the sum of two sides must be greater than the third side, not the difference.

4. **Option B: \( a + b < c \)**:
- This option suggests that the sum of sides \( a \) and \( b \) is less than side \( c \).
- This contradicts the triangle inequality theorem, which states that the sum must be greater than the third side.

5. **Option C: \( b + c < a \)**:
- This option suggests that the sum of sides \( b \) and \( c \) is less than side \( a \).
- Again, this contradicts the triangle inequality theorem.

6. **Option D: \( a + b > c \)**:
- This option states that the sum of sides \( a \) and \( b \) is greater than side \( c \).
- This is true according to the triangle inequality theorem.

7. **Conclusion**:
- The only correct statement based on the triangle inequality theorem is option D: \( a + b > c \).

### Final Answer:
The correct option is **D: \( a + b > c \)**.