By the Triangle Inequality Theorem, if two sides of a triangle have lengths of 6 and 13, what are the possible lengths of the third side?

The Triangle Inequality Theorem establishes that the length of the triangle is shorter than the sum of the two lenghts of the others two sides. Then, you have:

 a=6   (the lenght of a side of the triangle).
 b=13   (the lenght of a side of the triangle).
 c=x    (the length of the third side).

 Therefore:

 c<a+b
 c<6+13
 c<19

 The lenght of the third side is:

 (13-6)<c<19
 7<c<9