Apply the triangle inequality theorem to determine the possible whole number measures of the third side of a triangle

The triangle inequality 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.

So, if the lengths of two sides of a triangle are x and y, and the third side has length z, then according to the triangle inequality theorem:

x + y > z

y + z > x

x + z > y

From these inequalities, we can determine the possible whole number measures of the third side of the triangle by trying out different combinations of x and y:

For example, if x = 2 and y = 4:

2 + 4 > z
6 > z

4 + z > 2
z > -2

2 + z > 4
z > 2

Therefore, the possible whole number measures of the third side z are all integers greater than 2 and less than 6. So, the possible values for z are 3, 4, and 5.