1. **State the problem:** Find the perimeter of triangle $\triangle GHI$ given sides $HG=27.5$, $IG=21$, and angles $\angle H=49^\circ$, $\angle G=48^\circ$. Side $HI=x$ is unknown. 2. **Use the triangle angle sum rule:** The sum of angles in a triangle is $180^\circ$. $$\angle I = 180^\circ - 49^\circ - 48^\circ = 83^\circ$$ 3. **Use the Law of Sines to find side $HI=x$:** $$\frac{x}{\sin 48^\circ} = \frac{27.5}{\sin 83^\circ}$$ 4. **Solve for $x$:** $$x = \frac{27.5 \times \sin 48^\circ}{\sin 83^\circ}$$ Calculate the sines: $$\sin 48^\circ \approx 0.7431, \quad \sin 83^\circ \approx 0.9925$$ 5. **Substitute and simplify:** $$x = \frac{27.5 \times 0.7431}{0.9925} = \frac{20.43525}{0.9925}$$ 6. **Cancel common factors (approximate):** $$x \approx 20.59$$ 7. **Calculate the perimeter $P$:** $$P = HG + IG + HI = 27.5 + 21 + 20.59 = 69.09$$ 8. **Round to the nearest tenth:** $$P \approx 69.1$$ **Final answer:** The perimeter of $\triangle GHI$ is approximately $69.1$ units.