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To determine if each set of numbers can represent the sides of a triangle, we can apply the **Triangle Inequality Theorem**. This theorem states that for any three lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. 

Let's analyze each set of numbers:

1. **5, 4, and 1**:
   - 5 + 4 > 1 (True)
   - 5 + 1 > 4 (True)
   - 4 + 1 > 5 (False)
   - Since one condition fails, **No**.

2. **3, 3, and 7**:
   - 3 + 3 > 7 (False)
   - Since this fails, **No**.

3. **9, 7, and 14**:
   - 9 + 7 > 14 (True)
   - 9 + 14 > 7 (True)
   - 7 + 14 > 9 (True)
   - All conditions are satisfied, **Yes**.

4. **4, 8, and 2**:
   - 4 + 8 > 2 (True)
   - 4 + 2 > 8 (False)
   - Since one condition fails, **No**.

Thus, the answers for each set are:

- **5, 4, and 1**: No
- **3, 3, and 7**: No
- **9, 7, and 14**: Yes
- **4, 8, and 2**: No

Let me know if you'd like further explanations or have any questions.

Here are five related questions to explore further:

1. What is the Triangle Inequality Theorem, and why does it apply to triangle formation?
2. Can three sides of equal length always form a triangle?
3. What happens if two sides of a triangle are equal, but the third side is much longer?
4. How does the Triangle Inequality Theorem apply to real-world structures?
5. What are some examples of geometric shapes where the Triangle Inequality Theorem is essential?

**Tip:** To quickly check if lengths form a triangle, add each pair and compare with the remaining side. If all sums are greater, a triangle can be formed.

Decide if the lengths given in each option can be used as sides of a triangle. Select the correct column for each set of numbers.

Understand how to determine if a set of lengths can form a triangle by applying the Triangle Inequality Theorem. This problem includes examples such as (5, 4, 1), (3, 3, 7), (9, 7, 14), and (4, 8, 2) with step-by-step solutions. Ideal for students in Grades 6-8.

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