Let’s address the questions in this image. Here's the breakdown:

For problems 7-10: Determine if the three sides can form a triangle.

To determine if three sides form a triangle, use the Triangle Inequality Theorem:
The sum of the lengths of any two sides must be greater than the third side.

7. Sides: 11, 7, 21

• $11 + 7 = 18$, which is not greater than 21.
  Thus, these sides do not form a triangle.

8. Sides: 7, 6, 9

• $7 + 6 = 13$, $6 + 9 = 15$, $7 + 9 = 16$.
  Since all sums are greater than the third side, these sides form a triangle.

9. Sides: 8, 1, 7

• $8 + 1 = 9$, $1 + 7 = 8$, $8 + 7 = 15$.
  Since $1 + 7 = 8$, which is equal to the third side, these sides do not form a triangle.

10. Sides: 14, 6, 11

• $14 + 6 = 20$, $6 + 11 = 17$, $14 + 11 = 25$.
  Since all sums are greater than the third side, these sides form a triangle.

For problems 11-13: Find the range of possible measures for the third side.

To find the range of a triangle's third side:
$a - b < c < a + b$, where $a$ and $b$ are the two given sides.

11. Sides: 8, 8

• $8 - 8 < c < 8 + 8$
  $0 < c < 16$.
  Thus, the third side must be greater than 0 and less than 16.

12. Sides: 6, 7

• $7 - 6 < c < 7 + 6$
  $1 < c < 13$.
  Thus, the third side must be greater than 1 and less than 13.

13. Sides: 10, 11

• $11 - 10 < c < 11 + 10$
  $1 < c < 21$.
  Thus, the third side must be greater than 1 and less than 21.

Would you like detailed explanations for any of these solutions? Here are 5 related questions to expand your understanding:

1. What are the consequences of violating the Triangle Inequality Theorem?
2. How does the range of a third side change if the two given sides are close in length?
3. Can the third side of a triangle ever equal the sum of the other two sides? Why?
4. How do you prove the Triangle Inequality Theorem mathematically?
5. Can a triangle have all its sides with integer lengths? Explore examples.

Tip: Always compare all combinations of side sums when checking the Triangle Inequality!