1. **State the problem:** We have a triangle formed by a tent and a lamppost. Given angles are 30°, 135°, and 6°, and sides 8 m and 12 m. We need to find the distance from the lamppost to the tent. 2. **Identify known elements:** The triangle has angles 30°, 135°, and 6°. The side opposite 30° is 8 m, and the base is 12 m. We want to find the distance $x$ from the lamppost to the tent. 3. **Sum of angles in a triangle:** The sum of angles in any triangle is 180°. 4. **Check angles:** Given angles 30°, 135°, and 6° sum to 171°, which is less than 180°. This suggests the 6° angle is an angle of depression, not an interior angle of the triangle. We focus on the triangle with angles 30°, 135°, and the remaining angle $15°$ (since $180° - 30° - 135° = 15°$). 5. **Use 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. 6. **Assign sides and angles:** Let side opposite 30° be 8 m, opposite 135° be $b$, and opposite 15° be $x$ (distance from lamppost to tent). 7. **Apply Law of Sines:** $$\frac{8}{\sin 30°} = \frac{x}{\sin 135°}$$ 8. **Calculate sines:** $$\sin 30° = 0.5$$ $$\sin 135° = \sin (180° - 45°) = \sin 45° = \frac{\sqrt{2}}{2} \approx 0.7071$$ 9. **Solve for $x$:** $$x = \frac{8}{0.5} \times 0.7071 = 16 \times 0.7071 = 11.3136$$ 10. **Final answer:** The distance from the lamppost to the tent is approximately $11.31$ meters.