\(z = \frac{1 + \sqrt3 i}{1 +i} \)

\(= \frac{1 + \sqrt3 i}{1 +i} \times \frac{1-i}{1-i} \)

\(= \frac{(1 + \sqrt3) + i(\sqrt 3 - 1)}{1 -i^2}\)

\(= \frac{1 + \sqrt 3}2 + i \frac{(\sqrt 3 - 1)}2\)

\(|z| = \left(\frac{\sqrt 3 + 1}2\right) ^2 + \left(\frac{\sqrt 3 - 1}2\right) ^2 \)

\(= \frac{(3 +1 + 2\sqrt3) + (3 + 1- 2\sqrt3)}{4}\)

\(= \frac 84 = 2\)

\(tan\theta =\frac ba = \cfrac{\frac{\sqrt3 - 1}2}{\frac{\sqrt3+1}2}\)

\(= \frac{\sqrt3 - 1}{\sqrt3+1}\)

\(= \frac{\sqrt3 - 1}{1 + \sqrt3 \times 1}\)

\(= \frac{tan60° - tan45°}{1 + tan60°tan45°}\) \((\because tan60° = \sqrt 3)\)

\(= tan(60° - 45° )\)

\(= tan(15° )\)

\(\therefore \theta = Arg\, z = 15°\)