1. **State the problem:** We are given a triangle with angles $35^\circ$, $105^\circ$, and the side opposite the $105^\circ$ angle is 7 units long. We need to find the length of side $c$ opposite the $35^\circ$ angle using the Law of Sines. 2. **Recall the Law of Sines formula:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a$, $b$, and $c$ are sides opposite angles $A$, $B$, and $C$ respectively. 3. **Identify known values:** - Angle $B = 35^\circ$ - Angle $C = 105^\circ$ - Side opposite angle $C$ (side $AB$) = 7 - Side $c$ is opposite angle $B$ (35°) 4. **Find the missing angle $A$:** $$A = 180^\circ - B - C = 180^\circ - 35^\circ - 105^\circ = 40^\circ$$ 5. **Apply Law of Sines to find $c$:** $$\frac{c}{\sin 35^\circ} = \frac{7}{\sin 105^\circ}$$ 6. **Solve for $c$:** $$c = \frac{7 \times \sin 35^\circ}{\sin 105^\circ}$$ 7. **Calculate the sine values:** $$\sin 35^\circ \approx 0.574\quad \text{and} \quad \sin 105^\circ \approx 0.966$$ 8. **Substitute and compute:** $$c = \frac{7 \times 0.574}{0.966} = \frac{4.018}{0.966} \approx 4.16$$ **Final answer:** $$c \approx 4.16$$ units