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If $z_1$ and $z_2$ are two non-zero complex numbers such that $\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right|$ then $\arg z_1-\arg z_2$ is equal to
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$\left|z_1+z_2\right|=\left|z_1\right|+\left|z_2\right| \Rightarrow z_1$ and $z_2$ are collinear and are to the same side of origin; hence $\arg z_1-\arg z_2=0$.
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