1. **Stating the problem:** We have two triangles $\triangle PQR$ and $\triangle VUT$ that are similar ($\triangle PQR \sim \triangle VUT$). We know the angles and some side lengths of $\triangle PQR$ and some side lengths of $\triangle VUT$. We want to find the measure of angle $\angle V$ in $\triangle VUT$ and the unknown side lengths $x$ in $\triangle PQR$ and $y$ in $\triangle VUT$. 2. **Using similarity properties:** Since the triangles are similar, corresponding angles are equal and corresponding sides are proportional. This means: $$\angle P = \angle V, \quad \angle Q = \angle U, \quad \angle R = \angle T$$ and $$\frac{x}{y} = \frac{9}{8} = \frac{12}{16}$$ 3. **Find $\angle V$:** The angles in $\triangle PQR$ are given as 42° and 35°, so the third angle is: $$\angle R = 180^\circ - 42^\circ - 35^\circ = 103^\circ$$ Since $\triangle PQR \sim \triangle VUT$, corresponding angles are equal, so: $$\angle V = \angle P = 42^\circ$$ 4. **Find $x$ and $y$:** Using the side ratios: $$\frac{x}{y} = \frac{9}{8}$$ and $$\frac{12}{16} = \frac{3}{4}$$ Since $\frac{9}{8} \neq \frac{3}{4}$, check which sides correspond. Assuming sides 9 and 8 correspond, and sides 12 and 16 correspond, the scale factor from $\triangle PQR$ to $\triangle VUT$ is: $$k = \frac{8}{9}$$ Then: $$y = k \times x = \frac{8}{9} x$$ Using the other pair: $$16 = k \times 12 = \frac{8}{9} \times 12 = \frac{96}{9} = 10.67$$ This is inconsistent, so the sides must correspond differently. Let's match 12 with 16 and 9 with 8: Scale factor: $$k = \frac{16}{12} = \frac{4}{3}$$ Then: $$y = k \times x = \frac{4}{3} x$$ Using the other pair: $$8 = k \times 9 = \frac{4}{3} \times 9 = 12$$ This is inconsistent too. So the unknown side $x$ corresponds to side 8, and side 9 corresponds to side $y$. Then: $$\frac{x}{8} = \frac{9}{y} = \frac{12}{16} = \frac{3}{4}$$ From $\frac{x}{8} = \frac{3}{4}$, we get: $$x = 8 \times \frac{3}{4} = 6$$ From $\frac{9}{y} = \frac{3}{4}$, cross-multiplied: $$9 \times 4 = 3 \times y$$ $$36 = 3y$$ $$y = \frac{36}{3} = 12$$ 5. **Final answers:** $$\angle V = 42^\circ, \quad x = 6, \quad y = 12$$