$Amp(z)+Amp(w)=\pi.\;$ Find a relation between $\;z\;$ and $\;w.$

Question

The question says:

Let $\;z\;$ and $\;w\;$ be two non-zero complex numbers such that $|z|=|w|$ and $amp(z)+amp(w)=\pi,\;$ then find a relation between $\;z\;$ and $\;w.$

In the solution they turn $ amp(z)+amp(w)$ into $amp(Z)-amp(\overline w)\;$ and equate it to $\pi$. What was the need to do so ? I think I may be missing some concept.

The final answer is $\;z+ \overline w=0.$

Answer 1

$$w=|w|e^{i\angle w}=|z|e^{i\pi-i\angle z}=e^{i\pi}|z|e^{-i\angle z}=-\overline z.$$