Final answer:
The range of possible measures for the third side of a triangle can be found using the triangle inequality theorem. For the given sides 4, 12, the range is x < 16. For the given sides 5, 8, the range is x < 13.
Step-by-step explanation:
When given the lengths of two sides of a triangle, we can find the range of possible measures for the third side using 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. Let's consider the given sides:
a. 4, 12
The range of possible measures for the third side, let's call it x, can be found by setting up the inequality: 4 + 12 > x. Simplifying this inequality, we get 16 > x. Therefore, the range of possible measures for the third side is x < 16.
b. 5, 8
Similarly, for these given sides, we set up the inequality: 5 + 8 > x. Simplifying this inequality, we get 13 > x. Therefore, the range of possible measures for the third side is x < 13.
In summary, the range of possible measures for the third side are x < 16 for the given sides 4, 12 and x < 13 for the given sides 5, 8.