1. **Problem:** Find the measurements $m_a$, $m_b$, and $m_c$ given the relation $\frac{\sin A}{18} = \frac{\sin 54}{31}$. 2. **Step 1:** Use the Law of Sines formula: $$\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$$ 3. **Step 2:** Given $\frac{\sin A}{18} = \frac{\sin 54}{31}$, solve for $\sin A$: $$\sin A = \frac{18 \times \sin 54}{31}$$ 4. **Step 3:** Calculate $\sin 54$: $$\sin 54 \approx 0.8090$$ 5. **Step 4:** Substitute and calculate $\sin A$: $$\sin A = \frac{18 \times 0.8090}{31} = \frac{14.562}{31} \approx 0.4697$$ 6. **Step 5:** Find angle $A$ by taking inverse sine: $$A = \sin^{-1}(0.4697) \approx 28^\circ$$ 7. **Step 6:** Since $m_a = 31$, $m_b = 18$, and $m_c$ is unknown, use the Law of Sines to find $m_c$: Assuming $m_c$ corresponds to angle $C$, and knowing angles $A=28^\circ$, $B=54^\circ$, find $C$: $$C = 180^\circ - A - B = 180^\circ - 28^\circ - 54^\circ = 98^\circ$$ 8. **Step 7:** Use Law of Sines to find $m_c$: $$\frac{m_c}{\sin C} = \frac{m_a}{\sin A} \Rightarrow m_c = \frac{m_a \times \sin C}{\sin A} = \frac{31 \times \sin 98^\circ}{\sin 28^\circ}$$ 9. **Step 8:** Calculate $\sin 98^\circ \approx 0.9903$ and $\sin 28^\circ \approx 0.4695$: $$m_c = \frac{31 \times 0.9903}{0.4695} \approx \frac{30.7}{0.4695} \approx 65.4$$ **Final answers:** $$m_a = 31, \quad m_b = 18, \quad m_c \approx 65.4$$