Consider Z₁ and Z₂ are two complex numbers Such that $|{\mathrm{Z}}_1+\mathrm{Z}{}_2 |=\left|{\mathrm{Z}}_1\right|+\left|\mathrm{Z}{}_2 \right|$
Statement 1:  $\mathrm{amp}\left({\mathrm{Z}}_1\right)-\mathrm{amp}\left({\mathrm{Z}}_2\right)=0$
 Statement 2: The complex numbers Z₁ and Z₂ are collinear with origin

$\left|Z_1+Z_2\right|=\left|Z_1\right|+\left|Z_2\right|$

If $\mathrm{amp}\left({\mathrm{Z}}_1\right)={\mathrm{\theta }}_1$ and $\mathrm{amp}\left({\mathrm{Z}}_2\right)={\mathrm{\theta }}_2$, then 

$\begin{array}{l}{\left|z_1+z_2\right|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2+2\left|z_1\right|\left|z_2\right|\cos \left({\theta }_1-{\theta }_2\right)\\ \Rightarrow {\left(\left|z_1\right|+\left|z_2\right|\right)}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2+2\left|z_1\right|\left|z_2\right|\cos \left({\theta }_1-{\theta }_2\right)\\ \Rightarrow \cos \left({\theta }_1-{\theta }_2\right)=1\Rightarrow \left({\theta }_1-{\theta }_2\right)=0\\ \Rightarrow \mathrm{amp}\left(Z_1\right)-\mathrm{amp}\left(Z_2\right)=0\therefore \text{ S-I istrue }\end{array}$

S-II: Since $\mathrm{amp}\left(\frac{{\mathrm{Z}}_1}{{\mathrm{Z}}_2}\right)=0$ and 0(origin) are collinear then

$\mathrm{Comp}\left(\frac{0-{\mathrm{Z}}_1}{0-{\mathrm{Z}}_2}\right)=0 \Rightarrow \mathrm{amp}\left(\frac{{\mathrm{Z}}_1}{{\mathrm{Z}}_2}\right)=0 \mathrm{amp}\left({\mathrm{Z}}_1\right)-\mathrm{amp}\left({\mathrm{Z}}_2\right)=0$

S-II is True, which is not a correct explanation for S-I