This worksheet involves applying the Triangle Inequality Theorem, which states that for any three sides to form a triangle, the sum of any two sides must be greater than the third side.

Let's go through the questions.

Part 1: Can the numbers be the sides of a triangle?

For each set of three numbers, check if they satisfy the Triangle Inequality Theorem:

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

For example:

1. $10, 6, 19$: $10 + 6 = 16 \quad \text{(less than 19, so this is not possible)}$ Answer: No

2. $12, 8, 25$: $12 + 8 = 20 \quad \text{(less than 25, so this is not possible)}$ Answer: No

We can proceed similarly for the rest of the questions.

Part 2: Range of the third side

For each set of two sides, the length of the third side must satisfy: $b - a < x < b + a$ where $a$ and $b$ are the given lengths.

For example: 11. $6, 13$: $13 - 6 = 7 \quad \text{and} \quad 13 + 6 = 19$ So, the third side must be between 7 and 19. Answer: 7 to 19

Do you want to go through all the steps for each problem? Feel free to ask questions!

Here are 5 follow-up questions to think about:

1. Can you explain the logic behind the Triangle Inequality Theorem?
2. How would the range change if the two given sides are equal?
3. What happens if one of the sides is 0? Can you form a triangle?
4. How would you apply the Triangle Inequality in 3D geometry?
5. Can a triangle have sides with non-integer lengths?

Tip: When checking for triangle side lengths, always confirm that the sum of the two smallest sides is greater than the largest side!