In a triangle $\bigtriangleup ABC$ is $\widehat A=30^{\circ}$, $|AB|=10$ and $|BC|\in\{3,5,7,9,11\}$.
How many non-congruent trangles $\bigtriangleup ABC$ exist?
The possible answers are $3,4,5,6$ and $7$.
Is there a quick solution that doesn't require much writing?