1. **State the problem:** We have triangle EFG with angles $\angle G = 109^\circ$, $\angle F = 52^\circ$, and side $FG = 4$. We want to find side $g = EF$. 2. **Find the missing angle:** The sum of angles in a triangle is $180^\circ$. $$\angle E = 180^\circ - 109^\circ - 52^\circ = 19^\circ$$ 3. **Use the Law of Sines:** The Law of Sines states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where sides $a,b,c$ are opposite angles $A,B,C$ respectively. 4. **Assign sides and angles:** - Side $g = EF$ is opposite $\angle G = 109^\circ$ - Side $4 = FG$ is opposite $\angle E = 19^\circ$ - Side $f$ is opposite $\angle F = 52^\circ$ 5. **Set up the proportion to find $g$:** $$\frac{g}{\sin 109^\circ} = \frac{4}{\sin 19^\circ}$$ 6. **Solve for $g$:** $$g = \frac{4 \times \sin 109^\circ}{\sin 19^\circ}$$ 7. **Calculate values:** $$\sin 109^\circ \approx 0.9455$$ $$\sin 19^\circ \approx 0.3256$$ 8. **Substitute and compute:** $$g = \frac{4 \times 0.9455}{0.3256} = \frac{3.782}{0.3256} \approx 11.6$$ **Final answer:** $$g \approx 11.6$$ Thus, the length of side $g$ is approximately 11.6 units.