We are given two complex numbers in polar form:

• $z_1 = r_1(\cos \theta_1 + i \, \sin \theta_1)$
• $z_2 = r_2(\cos \theta_2 + i \, \sin \theta_2)$

with constraints $r_1, r_2$ being real numbers, $0 \leq \theta_1 \leq \frac{\pi}{2}$, and $0 \leq \theta_2 \leq \frac{\pi}{2}$. We need to determine the condition that is necessarily true if $|z_1 + z_2| = |z_1| + |z_2|$.

Geometric Interpretation of Complex Number Sum

The condition $|z_1 + z_2| = |z_1| + |z_2|$ represents the equality case of the triangle inequality for complex numbers. Geometrically, this equality holds if and only if the vectors representing $z_1$ and $z_2$ in the complex plane point in the same direction. This implies their arguments must be equal, or at least one of the complex numbers must be zero.

Condition for Angle Equality

Let's consider the case where both $z_1$ and $z_2$ are non-zero, meaning $r_1 > 0$ and $r_2 > 0$.

• The argument of $z_1$ is $\arg(z_1) = \theta_1$.
• The argument of $z_2$ is $\arg(z_2) = \theta_2$.

For the vectors $z_1$ and $z_2$ to point in the same direction, their arguments must be equal. Thus, $\arg(z_1) = \arg(z_2)$, which implies $\theta_1 = \theta_2$.

The given constraints are $0 \leq \theta_1 \leq \frac{\pi}{2}$ and $0 \leq \theta_2 \leq \frac{\pi}{2}$. If $\theta_1 = \theta_2$, this condition is satisfied within the given range.

If either $r_1=0$ or $r_2=0$, the condition $|z_1 + z_2| = |z_1| + |z_2|$ holds trivially. However, the argument of a zero complex number is undefined, making the equality $\theta_1 = \theta_2$ not strictly necessary in those specific edge cases. In the context of problems involving geometric interpretations of the triangle inequality, the condition usually implies alignment of non-zero vectors.

Analysis of Options

We examine the given options based on the derived condition $\theta_1 = \theta_2$ for non-zero complex numbers:

• Option 1 & 2 ($\theta_1 = 0, \theta_2 = \frac{\pi}{2}$ or vice-versa): These represent vectors pointing in different directions (along the real axis and the imaginary axis respectively). The equality $|z_1 + z_2| = |z_1| + |z_2|$ generally does not hold in this case unless $r_1=0$ or $r_2=0$. For example, if $z_1 = 1$ and $z_2 = i$, then $|z_1+z_2| = \sqrt{2}$ but $|z_1|+|z_2|=2$.
• Option 3 ($r_1 = r_2$): This condition is not necessarily implied. Consider $z_1 = 1(\cos 0 + i \sin 0)$ ($r_1=1, \theta_1=0$) and $z_2 = 2(\cos 0 + i \sin 0)$ ($r_2=2, \theta_2=0$). Here, $\theta_1=\theta_2$, satisfying $|z_1+z_2| = |z_1|+|z_2|$ ($3=1+2$), but $r_1 \neq r_2$.
• Option 4 ($\theta_1 = \theta_2$): This condition means $z_1$ and $z_2$ have the same argument. If $r_1>0$ and $r_2>0$, this directly leads to the equality $|z_1 + z_2| = |z_1| + |z_2|$. This is the necessary condition derived from the geometric interpretation for non-zero complex numbers.

Based on the analysis, the condition $\theta_1 = \theta_2$ is the one that must necessarily hold true when $|z_1 + z_2| = |z_1| + |z_2|$, assuming $z_1$ and $z_2$ are non-zero complex numbers, which aligns with the constraints provided for the angles.