1. **State the problem:** We need to find the perimeter of triangle $\triangle XYZ$ where two sides $XY=36$, $YZ=34.4$ and two angles $\angle Y=63^\circ$, $\angle X=57^\circ$ are given. The side $XZ=x$ is unknown. 2. **Use the triangle angle sum rule:** The sum of angles in a triangle is $180^\circ$. So, $$\angle Z = 180^\circ - \angle X - \angle Y = 180^\circ - 57^\circ - 63^\circ = 60^\circ.$$ 3. **Use the Law of Sines to find $x = XZ$:** The Law of Sines states: $$\frac{XY}{\sin \angle Z} = \frac{YZ}{\sin \angle X} = \frac{XZ}{\sin \angle Y}.$$ We want $XZ$, so: $$XZ = \frac{YZ \cdot \sin \angle Y}{\sin \angle X} = \frac{34.4 \times \sin 63^\circ}{\sin 57^\circ}.$$ 4. **Calculate the sines:** $$\sin 63^\circ \approx 0.8910, \quad \sin 57^\circ \approx 0.8387.$$ 5. **Calculate $XZ$:** $$XZ = \frac{34.4 \times 0.8910}{0.8387} = \frac{30.6384}{0.8387}.$$ 6. **Simplify the fraction:** $$XZ \approx 36.53.$$ 7. **Find the perimeter $P$:** $$P = XY + YZ + XZ = 36 + 34.4 + 36.53 = 106.93.$$ 8. **Round to the nearest tenth:** $$P \approx 106.9.$$ **Final answer:** The perimeter of $\triangle XYZ$ is approximately $106.9$ units.