Question 379050
I hope you assume the side lengths are integers, otherwise there would be infinitely many triangles.

If the side lengths are integers, the smallest such "base" is 2, resulting in the other sides being 31 and 31. The base must be even, as the other two lengths are equal and are integers. Thus, we have the following sets as side lengths:

(2, 31, 31)
(4, 30, 30)
(6, 29, 29)
.
.
.
(30, 17, 17)

If the base were any larger (e.g. 32, 16, 16) the triangle would either be degenerate or non-existent. This gives us 15 different triangles.