1. **State the problem:** Given a triangle with angle $\angle C = 54^\circ$, side $b = 24$ km, and side $c = 23$ km, solve for side $a$ or other unknowns as needed. 2. **Formula used:** Use the Law of Sines, which states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ This law relates the sides of a triangle to the sines of their opposite angles. 3. **Find angle $B$:** Since the sum of angles in a triangle is $180^\circ$, $$B = 180^\circ - A - C$$ We need to find $A$ first or use the Law of Sines to find $A$ or $B$. 4. **Apply Law of Sines to find angle $B$:** $$\frac{b}{\sin B} = \frac{c}{\sin C} \implies \sin B = \frac{b \sin C}{c} = \frac{24 \times \sin 54^\circ}{23}$$ Calculate $\sin 54^\circ \approx 0.8090$, $$\sin B = \frac{24 \times 0.8090}{23} = \frac{19.416}{23} \approx 0.8446$$ 5. **Calculate angle $B$:** $$B = \sin^{-1}(0.8446) \approx 57.4^\circ$$ 6. **Calculate angle $A$:** $$A = 180^\circ - C - B = 180^\circ - 54^\circ - 57.4^\circ = 68.6^\circ$$ 7. **Find side $a$ using Law of Sines:** $$\frac{a}{\sin A} = \frac{c}{\sin C} \implies a = \frac{c \sin A}{\sin C} = \frac{23 \times \sin 68.6^\circ}{\sin 54^\circ}$$ Calculate $\sin 68.6^\circ \approx 0.9284$, $$a = \frac{23 \times 0.9284}{0.8090} = \frac{21.353}{0.8090} \approx 26.4$$ **Final answer:** $$a \approx 26.4 \text{ km}$$