Let `z_(1),z_(2)` be two complex numbers such that `|z_(1)+z_(2)|=|z_(1)|+|z_(2)|`. Then,

Options

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

Explanation

To solve the problem, we need to analyze the condition given for the complex numbers \( z_1 \) and \( z_2 \):

### Step-by-Step Solution:

1. **Given Condition**: We have the condition
\[ |z_1 + z_2| = |z_1| + |z_2|. \]

2. **Understanding the Condition**: The equality \( |z_1 + z_2| = |z_1| + |z_2| \) holds true if and only if \( z_1 \) and \( z_2 \) are in the same direction in the complex plane. This means that the argument (angle) of \( z_1 \) and \( z_2 \) must be the same.

3. **Squaring Both Sides**: To explore this condition further, we can square both sides:
\[ |z_1 + z_2|^2 = (|z_1| + |z_2|)^2. \]

4. **Expanding Both Sides**:
- Left-hand side:
\[ |z_1 + z_2|^2 = |z_1|^2 + |z_2|^2 + 2 \text{Re}(z_1 \overline{z_2}). \]
- Right-hand side:
\[ (|z_1| + |z_2|)^2 = |z_1|^2 + |z_2|^2 + 2 |z_1| |z_2|. \]

5. **Setting the Two Expansions Equal**:
\[ |z_1|^2 + |z_2|^2 + 2 \text{Re}(z_1 \overline{z_2}) = |z_1|^2 + |z_2|^2 + 2 |z_1| |z_2|. \]

6. **Simplifying the Equation**: We can cancel \( |z_1|^2 + |z_2|^2 \) from both sides:
\[ 2 \text{Re}(z_1 \overline{z_2}) = 2 |z_1| |z_2|. \]

7. **Dividing by 2**:
\[ \text{Re}(z_1 \overline{z_2}) = |z_1| |z_2|. \]

8. **Interpreting the Result**: The equality \( \text{Re}(z_1 \overline{z_2}) = |z_1| |z_2| \) implies that \( z_1 \) and \( z_2 \) are in the same direction, meaning:
\[ \frac{z_1}{|z_1|} = \frac{z_2}{|z_2|} \quad \text{or} \quad z_1 = k z_2 \text{ for some } k > 0. \]
This means that \( z_1 \) and \( z_2 \) are positive scalar multiples of each other.

### Conclusion:
Thus, we conclude that \( z_1 \) and \( z_2 \) must be in the same direction in the complex plane, which can be expressed as:
\[ \text{arg}(z_1) = \text{arg}(z_2). \]