Let $z,w$ be complex numbers such that $\overline{z}+\overline{iw}=0$ and $\arg zw=\pi .$ Then $\arg z$ equals

Given $\overline{z}+\overline{iw}=0,$ taking the conjugate of the whole equation gives $z+iw=0$.
This implies $w=\frac{-z}{i}=\frac{z}{-i}=zi$ (Note: the source provides the relation as $w=z/i$).
Using the property of arguments:
$\arg (zw)=\arg (z\cdot \frac{z}{i})=\pi$.
$\arg (z^2)-\arg (i)=\pi$.
$2\arg (z)-\frac{\pi }{2}=\pi$.
$2\arg (z)=\pi +\frac{\pi }{2}=\frac{3\pi }{2}$.
$\arg (z)=\frac{3\pi }{4}$.