Let $Z_1$ and $Z_2$ be two complex numbers such that ${\overline{Z}}_1=i{\overline{Z}}_2$ and arg$\left(\frac{z_1}{{\overline{z}}_2}\right)$$=\pi$. Then

$$\begin{gathered} {\overline{Z}}_1={\overline{iZ}}_2 \\ {z}_1=-i{z}_2 \\ given that arg\left(\frac{Z_1}{\overline{Z_2}}\right)=\pi \\ arg\left(\frac{-i{Z}_2}{{\overline{Z}}_2}\right)=\pi , let \arg \left(z_2\right)=\theta \\ -\frac{\pi }{2}+\theta +\theta =\pi \\ 2\theta =\frac{3\pi }{2} \\ \arg \left(z_2\right)=\theta =\frac{3\pi }{4}\Rightarrow \arg z_1=\frac{\pi }{4} \end{gathered}$$