If $$z_1$$ and $$z_2$$ are two non-zero complex numbers such that $$|z_1 + z_2| = |z_1| + |z_2|$$ then $$\arg z_1 - \arg z_2$$ is equal to

The triangle inequality for any two complex numbers $$z_1$$ and $$z_2$$ states
$$|z_1 + z_2| \le |z_1| + |z_2|.$$

Equality in the triangle inequality occurs only when the two complex numbers point in exactly the same direction on the Argand diagram, i.e. when their ratio is a positive real number:
$$|z_1 + z_2| = |z_1| + |z_2| \;\Longleftrightarrow\; \frac{z_1}{z_2} \text{ is a positive real}.$$

Because the ratio $$\dfrac{z_1}{z_2}$$ is positive real, its argument is $$0$$ (modulo $$2\pi$$). Therefore
$$\arg\!\left(\frac{z_1}{z_2}\right)=\arg z_1-\arg z_2=0.$$

Hence, $$\arg z_1-\arg z_2 = 0.$$

Option C which is: $$0$$