1. **State the problem:** We are given two triangles, $\triangle TUV$ and $\triangle WXY$, with some angles known. We need to determine if these triangles are similar. 2. **List the given angles:** - In $\triangle TUV$: $\angle T = 52^\circ$, $\angle V = 43^\circ$, and $\angle U$ is unknown. - In $\triangle WXY$: $\angle W = 52^\circ$, $\angle X = 85^\circ$, and $\angle Y$ is unknown. 3. **Find the missing angles using the triangle sum rule:** - For $\triangle TUV$, sum of angles is $180^\circ$: $$\angle U = 180^\circ - 52^\circ - 43^\circ = 85^\circ$$ - For $\triangle WXY$, sum of angles is $180^\circ$: $$\angle Y = 180^\circ - 52^\circ - 85^\circ = 43^\circ$$ 4. **Compare corresponding angles:** - $\angle T = 52^\circ$ and $\angle W = 52^\circ$ - $\angle V = 43^\circ$ and $\angle Y = 43^\circ$ - $\angle U = 85^\circ$ and $\angle X = 85^\circ$ 5. **Conclusion:** Since all corresponding angles are equal, by the Angle-Angle (AA) similarity criterion, $\triangle TUV$ is similar to $\triangle WXY$. **Final answer:** $\triangle TUV$ is similar to $\triangle WXY$.