If `|z_1 + z_2| = |z_1| + |z_2|` is possible if :

Options

A. `z_(2) = bar(z)_(1)`
B. `z_(2) = (1)/(z_(1))`
C. arg ` (z_(1)) "= arg (z_(2))`
D. `|z_(1)|= |z_(2)|`

Answer

To solve the problem, we need to analyze the condition under which the equation \( |z_1 + z_2| = |z_1| + |z_2| \) holds true.

### Step-by-Step Solution:

1. **Understanding the Equation**:
The equation \( |z_1 + z_2| = |z_1| + |z_2| \) states that the magnitude of the sum of two complex numbers \( z_1 \) and \( z_2 \) is equal to the sum of their magnitudes. This is a property that holds true under certain conditions.

2. **Applying 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**:
The condition for equality in the triangle inequality is that \( z_1 \) and \( z_2 \) must be non-negative multiples of each other. This means:
\[ z_1 = k \cdot z_2 \quad \text{for some } k \geq 0 \]
or both \( z_1 \) and \( z_2 \) must be either both positive or both negative.

4. **Testing Cases**:
- **Case 1**: Both \( z_1 \) and \( z_2 \) are positive.
- Let \( z_1 = 3 \) and \( z_2 = 2 \):
\[ |3 + 2| = |5| = 5 \quad \text{and} \quad |3| + |2| = 3 + 2 = 5 \]
Thus, \( |z_1 + z_2| = |z_1| + |z_2| \) holds true.

- **Case 2**: Both \( z_1 \) and \( z_2 \) are negative.
- Let \( z_1 = -3 \) and \( z_2 = -2 \):
\[ |-3 - 2| = |-5| = 5 \quad \text{and} \quad |-3| + |-2| = 3 + 2 = 5 \]
Thus, \( |z_1 + z_2| = |z_1| + |z_2| \) holds true.

- **Case 3**: One is positive and the other is negative.
- Let \( z_1 = 3 \) and \( z_2 = -2 \):
\[ |3 - 2| = |1| = 1 \quad \text{and} \quad |3| + |-2| = 3 + 2 = 5 \]
Thus, \( |z_1 + z_2| \neq |z_1| + |z_2| \).

5. **Conclusion**:
The equality \( |z_1 + z_2| = |z_1| + |z_2| \) is possible if both complex numbers \( z_1 \) and \( z_2 \) are either both positive or both negative, which means they must have the same argument.