1. **State the problem:** Find the equation of the line passing through the point $(0, 2)$ making an angle $\frac{2\pi}{3}$ with the positive x-axis. 2. **Formula and rules:** The slope $m$ of a line making an angle $\theta$ with the positive x-axis is given by: $$m = \tan(\theta)$$ The equation of a line with slope $m$ passing through point $(x_1, y_1)$ is: $$y - y_1 = m(x - x_1)$$ 3. **Calculate slope:** $$m = \tan\left(\frac{2\pi}{3}\right) = \tan\left(120^\circ\right) = -\sqrt{3}$$ 4. **Equation of the first line:** Passing through $(0, 2)$: $$y - 2 = -\sqrt{3}(x - 0)$$ Simplify: $$y = -\sqrt{3}x + 2$$ 5. **Find equation of parallel line:** Parallel lines have the same slope, so slope $m = -\sqrt{3}$. The new line crosses the y-axis at a distance 2 units below the origin, so the y-intercept is $-2$. Equation: $$y = -\sqrt{3}x - 2$$ **Final answers:** - Equation of the first line: $$y = -\sqrt{3}x + 2$$ - Equation of the parallel line: $$y = -\sqrt{3}x - 2$$