1. **Stating the problem:** We are given an Argand diagram with complex numbers $z_1, z_2, z_3, z_4,$ and $z_5$ and the following conditions: (i) $\operatorname{Re}(z_1) = \operatorname{Re}(z_5)$ (ii) $z_2 = z_1 + z_3$ (iii) $z_4 = z_1 + z_5$ We need to label these points accordingly. 2. **Understanding the conditions:** - Condition (i) means $z_1$ and $z_5$ have the same real part, so they lie vertically aligned on the Argand diagram. - Condition (ii) means $z_2$ is the vector sum of $z_1$ and $z_3$. - Condition (iii) means $z_4$ is the vector sum of $z_1$ and $z_5$. 3. **Using the Argand diagram properties:** - The real part corresponds to the horizontal axis. - The imaginary part corresponds to the vertical axis. 4. **Labeling the points:** - Since $z_1$ and $z_5$ have the same real part, the two points on the left side (one higher and one lower) are likely $z_1$ and $z_5$. - $z_3$ is a point on the right side. - $z_2 = z_1 + z_3$ means $z_2$ is the vector sum of $z_1$ and $z_3$, so $z_2$ should be located by adding the coordinates of $z_1$ and $z_3$. - $z_4 = z_1 + z_5$ means $z_4$ is the vector sum of $z_1$ and $z_5$. 5. **Summary:** - $z_1$ and $z_5$ are the two points on the left side. - $z_3$ is one of the points on the right side. - $z_2$ and $z_4$ are sums as described. Since the problem does not provide explicit coordinates, this is the labeling based on the given conditions. **Final answer:** - $z_1$ and $z_5$ are the two points on the left side with the same real part. - $z_3$ is one of the points on the right side. - $z_2 = z_1 + z_3$ is the point obtained by vector addition of $z_1$ and $z_3$. - $z_4 = z_1 + z_5$ is the point obtained by vector addition of $z_1$ and $z_5$. This completes the labeling based on the given information.