1. **State the problem:** We need to find which arcs have the same size central angle given their arc lengths and radii. 2. **Formula:** The central angle $\theta$ (in radians) for an arc is given by: $$\theta = \frac{s}{r}$$ where $s$ is the arc length and $r$ is the radius. 3. **Calculate each central angle:** - For arc length $15\pi$ and radius $10$: $$\theta_1 = \frac{15\pi}{10} = \frac{15}{10}\pi = 1.5\pi$$ - For arc length $30\pi$ and radius $15$: $$\theta_2 = \frac{30\pi}{15} = \frac{30}{15}\pi = 2\pi$$ - For arc length $6\pi^2$ and radius $4\pi$: $$\theta_3 = \frac{6\pi^2}{4\pi} = \frac{6}{4}\pi = 1.5\pi$$ - For arc length $10$ and radius $\frac{20}{3\pi}$: $$\theta_4 = \frac{10}{\frac{20}{3\pi}} = 10 \times \frac{3\pi}{20} = \frac{30\pi}{20} = 1.5\pi$$ - For arc length $\frac{3\pi}{2}$ and radius $2$: $$\theta_5 = \frac{\frac{3\pi}{2}}{2} = \frac{3\pi}{4} = 0.75\pi$$ 4. **Compare the central angles:** - $\theta_1 = 1.5\pi$ - $\theta_2 = 2\pi$ - $\theta_3 = 1.5\pi$ - $\theta_4 = 1.5\pi$ - $\theta_5 = 0.75\pi$ 5. **Conclusion:** The arcs with central angles $1.5\pi$ are the same size. These correspond to the arcs with: - Arc length $15\pi$, radius $10$ - Arc length $6\pi^2$, radius $4\pi$ - Arc length $10$, radius $\frac{20}{3\pi}$ Hence, these three arcs describe the same size central angle.