To apply the triangle inequality theorem, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
In this case, the first two sides measure 4 and 6.
Let's determine if the third side length options satisfy the triangle inequality theorem:
1) 2: 4 + 2 = 6 (greater than 2), 2 + 6 = 8 (greater than 4), 2 + 4 = 6 (equal to 6) - Not valid.
2) 9: 4 + 9 = 13 (greater than 6), 6 + 9 = 15 (greater than 4), 9 + 4 = 13 (greater than 6) - Valid.
3) 11: 4 + 11 = 15 (greater than 6), 6 + 11 = 17 (greater than 4), 11 + 4 = 15 (greater than 6) - Valid.
1 is not a valid option as it does not satisfy the triangle inequality theorem.
Therefore, the valid third side options are:
- 9
- 11