1. **State the problem:** We are given two triangles VWU and YZX with some side lengths and angles, and we need to find the length of side YZ. 2. **Given data:** - Triangle VWU: $VW=85$, $WU=90$, angles $\angle W=53^\circ$, $\angle V=67^\circ$ - Triangle YZX: $ZX=54$, angles $\angle Y=67^\circ$, $\angle X=60^\circ$ 3. **Find the missing angle in each triangle:** - For triangle VWU, sum of angles is $180^\circ$: $$\angle U = 180^\circ - 53^\circ - 67^\circ = 60^\circ$$ - For triangle YZX: $$\angle Z = 180^\circ - 67^\circ - 60^\circ = 53^\circ$$ 4. **Check for similarity:** Triangles VWU and YZX have angles $67^\circ$, $60^\circ$, and $53^\circ$ each, so they are similar by AAA similarity. 5. **Set up ratio of corresponding sides:** Corresponding sides opposite equal angles are proportional. - $VW$ corresponds to $YZ$ (opposite $U$ and $Z$ which are both $60^\circ$) - $WU$ corresponds to $ZX$ (opposite $V$ and $Y$ which are both $67^\circ$) So, $$\frac{VW}{YZ} = \frac{WU}{ZX}$$ 6. **Plug in known values:** $$\frac{85}{YZ} = \frac{90}{54}$$ 7. **Solve for $YZ$:** Multiply both sides by $YZ$: $$85 = \frac{90}{54} \times YZ$$ Divide both sides by $\frac{90}{54}$: $$YZ = \frac{85}{\frac{90}{54}} = 85 \times \frac{54}{90}$$ Simplify: $$YZ = 85 \times \frac{54}{90} = 85 \times \frac{3}{5}$$ Calculate: $$YZ = 85 \times 0.6 = 51$$ 8. **Final answer:** The length of side $YZ$ is $51$ (no rounding needed).