1. **State the problem:** We need to solve for side $b$ in a triangle using the Law of Sines. 2. **Given:** Angle $A = 102^\circ$, side $a = 27$ opposite angle $A$, angle $B = 28^\circ$, and side $b$ opposite angle $B$ is unknown. 3. **Find the third angle $C$:** $$C = 180^\circ - A - B = 180^\circ - 102^\circ - 28^\circ = 50^\circ$$ 4. **Law of Sines formula:** $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ 5. **Use the ratio involving $a$ and $b$ to solve for $b$:** $$\frac{a}{\sin A} = \frac{b}{\sin B} \implies b = \frac{a \sin B}{\sin A}$$ 6. **Substitute known values:** $$b = \frac{27 \times \sin 28^\circ}{\sin 102^\circ}$$ 7. **Calculate sine values:** $$\sin 28^\circ \approx 0.4695, \quad \sin 102^\circ \approx 0.9781$$ 8. **Evaluate $b$:** $$b = \frac{27 \times 0.4695}{0.9781} = \frac{12.6765}{0.9781}$$ 9. **Simplify fraction:** $$b \approx 12.95$$ **Final answer:** $$b \approx 12.95$$