1. **State the problem:** We need to find the length of side AC (denoted as $x$) in triangle ABC where \(\angle A = 80^\circ\), \(\angle B = 55^\circ\), and side BC = 3.8 cm. 2. **Find the missing angle:** The sum of angles in a triangle is $180^\circ$. So, $$\angle C = 180^\circ - 80^\circ - 55^\circ = 45^\circ.$$ 3. **Use the Law of Sines:** The Law of Sines states: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a$, $b$, and $c$ are sides opposite angles $A$, $B$, and $C$ respectively. 4. **Assign sides:** Side BC is opposite angle $A$, so $a = BC = 3.8$ cm. Side AC is opposite angle $B$, so $b = AC = x$ cm. 5. **Set up the ratio:** $$\frac{3.8}{\sin 80^\circ} = \frac{x}{\sin 55^\circ}$$ 6. **Solve for $x$:** $$x = \frac{3.8 \times \sin 55^\circ}{\sin 80^\circ}$$ 7. **Calculate values:** $$\sin 80^\circ \approx 0.9848, \quad \sin 55^\circ \approx 0.8192$$ 8. **Compute $x$:** $$x = \frac{3.8 \times 0.8192}{0.9848} \approx \frac{3.11296}{0.9848} \approx 3.16$$ **Final answer:** The length of side AC is approximately $3.16$ cm.