The measures of two sides are given. Between what two numbers must the third side fall. 5 and 8

Solution:
1. 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 remaining side. Therefore, the following inequalities must be satisfied:
- a + b > c
- a + c > b
- b + c > a

2. Assign the given side lengths to a and b, making a = 5 and b = 8.

3. Apply the triangle inequality theorem:
- First inequality: 5 + 8 > c gives us c < 13.
- Second inequality: c + 5 > 8 gives us c > 3.
- Third inequality is redundant as c + 8 > 5 is always true for positive values of c.

4. Therefore, the third side (cmust satisfy:
- 3 < c < 13

5. Conclusion: The third side must be between 3 and 13.