1. **Problem statement:** Given that triangles $\triangle ABC$ and $\triangle PRQ$ are similar, find the unknown values $x$ (angle $Q$) and $y$ (side $BC$). 2. **Similarity rules:** Similar triangles have corresponding angles equal and corresponding sides proportional. 3. **Identify corresponding parts:** Since $\triangle ABC \sim \triangle PRQ$, angle $A$ corresponds to angle $P$, angle $B$ to angle $R$, and angle $C$ to angle $Q$. 4. **Find angle $x$:** Given angle $A = 30^\circ$, angle $Q = x^\circ$ corresponds to angle $C$. Since the sum of angles in a triangle is $180^\circ$, find angle $C$ in $\triangle ABC$ first. 5. **Calculate angle $C$ in $\triangle ABC$:** $$\text{Sum of angles} = 180^\circ$$ $$30^\circ + \angle B + \angle C = 180^\circ$$ We don't know $\angle B$ or $\angle C$ yet, but since $\angle C$ corresponds to $\angle Q = x$, and $\angle B$ corresponds to $\angle R$, we can use the fact that angles correspond. 6. **Use side ratios to find $y$:** Corresponding sides are proportional: $$\frac{AC}{RQ} = \frac{BC}{PQ}$$ Substitute known values: $$\frac{6}{2.8} = \frac{y}{4}$$ 7. **Solve for $y$:** $$y = \frac{6}{2.8} \times 4$$ $$y = \frac{6 \times 4}{2.8}$$ $$y = \frac{24}{2.8}$$ 8. **Simplify fraction:** $$y = \frac{\cancel{24}}{\cancel{2.8}} \times \frac{1}{1} = 8.5714$$ (approx) 9. **Find angle $x$:** Since $\triangle ABC$ and $\triangle PRQ$ are similar, corresponding angles are equal. Angle $A = 30^\circ$ corresponds to angle $P$, so angle $P = 30^\circ$. Sum of angles in $\triangle PRQ$: $$30^\circ + x + \angle R = 180^\circ$$ Similarly, in $\triangle ABC$: $$30^\circ + \angle B + \angle C = 180^\circ$$ Since $\angle B$ corresponds to $\angle R$ and $\angle C$ corresponds to $x$, angles $B$ and $R$ are equal, so: $$x = \angle C$$ But we don't have $\angle C$ directly. However, since $\angle A = 30^\circ$, and $\angle B$ and $\angle C$ sum to $150^\circ$, we can only say $x = \angle C$. Without more info, $x$ remains unknown. **Final answers:** $$y \approx 8.57 \text{ cm}$$ $$x = \angle Q \text{ cannot be determined with given data}$$