1. **State the problem:** Given a triangle with sides and angles, find the measure of angle $T$ when $s=42$, $t=29$, and $m \angle S=63^\circ$ using the Law of Sines. 2. **Write the Law of Sines formula:** $$\frac{\sin m \angle S}{s} = \frac{\sin m \angle T}{t}$$ 3. **Substitute known values:** $$\frac{\sin 63^\circ}{42} = \frac{\sin m \angle T}{29}$$ 4. **Solve for $\sin m \angle T$ by cross-multiplying:** $$\sin m \angle T = 29 \times \frac{\sin 63^\circ}{42}$$ 5. **Calculate $\sin 63^\circ$:** $$\sin 63^\circ \approx 0.8910$$ 6. **Evaluate the right side:** $$\sin m \angle T = 29 \times \frac{0.8910}{42} = 29 \times 0.021214 = 0.6152$$ 7. **Find $m \angle T$ by taking the inverse sine:** $$m \angle T = \sin^{-1}(0.6152) \approx 37.9^\circ$$ **Final answer:** $$m \angle T \approx 37.9^\circ$$