  We study the parity of rank of $2$-${\rm Selmer}$ groups associated to
$\pi/3$ and $2\pi/3$-congruent numbers. Our second result gives some positive
densities about $\pi/3$ and $2\pi/3$ non-congruent numbers which can support
the even part of Goldfeld's conjecture. We give some necessary conditions such
that $n$ is non $\pi/3$-congruent number for elliptic curves $E_n$ whose
Shafarevich-Tate group is non-trivial. In the last section, we show that for
$n=pq\equiv 5(resp. \ 11)\pmod{24}$, the density of non $\pi/3$($resp.$
$2\pi/3$)-congruent numbers is at least 75\%, where $p,q$ are primes.
