1. **State the problem:** We have a triangle with a base of 7 cm and two angles adjacent to the base: 40° and 20°. We need to find the length $x$ of the side opposite the base. 2. **Identify the third angle:** The sum of angles in a triangle is 180°. So, $$\text{Third angle} = 180^\circ - 40^\circ - 20^\circ = 120^\circ$$ 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 and angles:** Let the side of length 7 cm be opposite the 120° angle. Then $x$ is opposite the 40° angle, and the remaining side is opposite the 20° angle. 5. **Apply the Law of Sines to find $x$:** $$\frac{x}{\sin 40^\circ} = \frac{7}{\sin 120^\circ}$$ 6. **Calculate $x$:** $$x = \frac{7 \times \sin 40^\circ}{\sin 120^\circ}$$ Using approximate values: $$\sin 40^\circ \approx 0.6428, \quad \sin 120^\circ \approx 0.8660$$ $$x = \frac{7 \times 0.6428}{0.8660} \approx \frac{4.4996}{0.8660} \approx 5.2$$ 7. **Final answer:** The length $x$ is approximately 5.2 cm to 1 decimal place.