`|z_(1)+z_(2)|=|z_(1)|+|z_(2)|` is possible, if

Options

A. `z_(2) = barz_(1)`
B. `z_(2)=(1)/(z_(1))`
C. arg`(z_(1))=arg(z_(2))`
D. `|z_(1)|+|z_(2)|`

Explanation

To solve the problem \( |z_1 + z_2| = |z_1| + |z_2| \), we need to analyze the condition under which this equality holds true for complex numbers \( z_1 \) and \( z_2 \).

### Step-by-Step Solution:

1. **Understanding the Modulus of Complex Numbers**:
The modulus of a complex number \( z = x + iy \) is given by \( |z| = \sqrt{x^2 + y^2} \). For two complex numbers \( z_1 \) and \( z_2 \), we can express them in polar form:
\[ z_1 = r_1 (\cos \theta_1 + i \sin \theta_1) \]
\[ z_2 = r_2 (\cos \theta_2 + i \sin \theta_2) \]

2. **Using the Triangle Inequality**:
The triangle inequality states that for any two complex numbers \( z_1 \) and \( z_2 \):
\[ |z_1 + z_2| \leq |z_1| + |z_2| \]
The equality \( |z_1 + z_2| = |z_1| + |z_2| \) holds if and only if \( z_1 \) and \( z_2 \) point in the same direction in the complex plane.

3. **Condition for Equality**:
For the equality \( |z_1 + z_2| = |z_1| + |z_2| \) to hold, the angles \( \theta_1 \) and \( \theta_2 \) must be equal. This means:
\[ \text{arg}(z_1) = \text{arg}(z_2) \]
or equivalently,
\[ z_1 \text{ and } z_2 \text{ must be in the same direction.} \]

4. **Conclusion**:
Therefore, the condition under which \( |z_1 + z_2| = |z_1| + |z_2| \) is possible is:
\[ \text{arg}(z_1) = \text{arg}(z_2) \]

### Final Answer:
The condition is that \( z_1 \) and \( z_2 \) must have the same argument, i.e., \( \text{arg}(z_1) = \text{arg}(z_2) \).

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