1. **State the problem:** Prove that triangles $\triangle UVX$ and $\triangle SYW$ are congruent given that $\angle UXV \cong \angle SWY$, $\angle U \cong \angle S$, and $XY \cong VW$. 2. **Given information:** - $\angle UXV \cong \angle SWY$ (Statement 1) - $\angle U \cong \angle S$ (Statement 2) - $XY \cong VW$ (Statement 3) 3. **Use segment addition to express $WY$ and $VX$:** - $WY = XY + WX$ (Statement 4) - $VX = VW + WX$ (Statement 5) 4. **Substitute $XY \cong VW$ into the expressions:** Since $XY \cong VW$, replace $XY$ with $VW$ in $WY$: $$WY = \cancel{XY} + WX = \cancel{VW} + WX$$ (Statement 6) 5. **Compare $VX$ and $WY$:** From Statements 5 and 6: $$VX = VW + WX$$ $$WY = VW + WX$$ Therefore, $$VX = WY$$ (Statement 7) 6. **Identify the congruent parts for triangle congruence:** - $\angle UXV \cong \angle SWY$ (given) - $\angle U \cong \angle S$ (given) - $VX \cong WY$ (from step 5) 7. **Apply the ASA (Angle-Side-Angle) congruence postulate:** Two angles and the included side of $\triangle UVX$ are congruent to two angles and the included side of $\triangle SYW$. 8. **Conclusion:** $$\triangle UVX \cong \triangle SYW$$ (Statement 8)