Question

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?

Answer 1

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