1. **Problem statement:** Given triangle ABC with angles $\angle C = 58^\circ$, $\angle A = 93^\circ$, and side $AC = 16$, find the length of side $BC$. 2. **Step 1: Find the missing angle $\angle B$.** Since the sum of angles in a triangle is $180^\circ$, $$\angle B = 180^\circ - \angle A - \angle C = 180^\circ - 93^\circ - 58^\circ = 29^\circ.$$ 3. **Step 2: Use the Law of Sines to find $BC$.** The Law of Sines states: $$\frac{BC}{\sin \angle A} = \frac{AC}{\sin \angle B}.$$ Rearranged to solve for $BC$: $$BC = AC \times \frac{\sin \angle A}{\sin \angle B}.$$ 4. **Step 3: Substitute known values and calculate.** $$BC = 16 \times \frac{\sin 93^\circ}{\sin 29^\circ}.$$ Calculate the sine values: $$\sin 93^\circ \approx 0.9986, \quad \sin 29^\circ \approx 0.4848.$$ 5. **Step 4: Simplify the fraction and multiply:** $$BC = 16 \times \frac{0.9986}{0.4848} = 16 \times 2.059.$$ 6. **Step 5: Final calculation:** $$BC \approx 32.94.$$ **Answer:** The length of side $BC$ is approximately $32.94$ units.