1. **State the problem:** We need to find the length of side CD in triangle BCD, given angles $\angle B = 15^\circ$, $\angle C = 31^\circ$, $\angle D = 95^\circ$, and side BD = 13, side CD = 8.3 (given but we will verify). 2. **Identify the knowns and unknowns:** - Angles: $\angle B = 15^\circ$, $\angle C = 31^\circ$, $\angle D = 95^\circ$ - Side BD = 13 (opposite $\angle C$) - Side CD = ? (opposite $\angle B$) 3. **Use the Law of Sines:** The Law of Sines states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a,b,c$ are sides opposite angles $A,B,C$ respectively. 4. **Assign sides to angles:** - Side BD is opposite $\angle C = 31^\circ$, so $b = 13$ - Side CD is opposite $\angle B = 15^\circ$, so $a = ?$ 5. **Apply Law of Sines to find CD:** $$\frac{a}{\sin 15^\circ} = \frac{13}{\sin 31^\circ}$$ 6. **Solve for $a$:** $$a = \frac{13 \times \sin 15^\circ}{\sin 31^\circ}$$ 7. **Calculate values:** - $\sin 15^\circ \approx 0.2588$ - $\sin 31^\circ \approx 0.5150$ 8. **Substitute:** $$a = \frac{13 \times 0.2588}{0.5150} = \frac{3.3644}{0.5150}$$ 9. **Simplify:** $$a \approx 6.53$$ 10. **Round to nearest tenth:** $$a \approx 6.5$$ **Final answer:** The length of side CD is approximately $6.5$ units.