A geometry problem by Yahia El Haw

Triangle $ABC$ is isosceles as indicated. This implies that $\angle{ABC}\cong\angle{ACB}=\theta$ .

$\angle{ABC}+2\diamond=180$ and $\angle{ACB}+2x=180$

$\theta+2\diamond=180$ and $\theta+2x=180$

$\theta +2\diamond = \theta+2x$

$2\diamond=2x$

$\diamond=x$

$\angle{DBC}\cong\angle{DCB}$ thus making triangle $BCD$ isosceles