By the <equality condition in the complex triangle inequality>,
$$
|x+y|=|x|+|y|
$$
for nonzero $x,y$ exactly when $x/y$ is a positive real number. Having the same <complex argument> is reflexive, symmetric, and transitive, so this is an equivalence relation. Its classes are the open rays from the origin.

Solution

Past exam of the mathematics course of the University of Cambridge / 2020 / ia / Paper 2 / 2D / ii / Solution

For nonzero <complex numbers> $x,y$, equality $|x+y|=|x|+|y|$ holds exactly when $x/y$ is a positive real number, equivalently when the two numbers have the same <complex argument> modulo $2\pi$.

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 Incoming links: Equality condition in the complex triangle inequality