1. The problem states that triangles $\triangle CJW \cong \triangle AGS$, with $m\angle A = 50^\circ$, $m\angle J = 45^\circ$, and $m\angle S = 16x + 5$. We need to find $x$. 2. Since the triangles are congruent, corresponding angles are equal. Therefore, $m\angle J = m\angle A = 50^\circ$ and $m\angle S = m\angle W$. 3. However, the problem gives $m\angle J = 45^\circ$ and $m\angle A = 50^\circ$, so these are corresponding angles but not equal. This suggests the matching angles are $\angle A$ with $\angle J$ and $\angle S$ with $\angle W$. 4. The sum of interior angles in a triangle is $180^\circ$. For $\triangle AGS$, the angles are $50^\circ$, $16x + 5$, and the third angle $m\angle G$. 5. For $\triangle CJW$, the angles are $45^\circ$, $m\angle W$, and the third angle $m\angle C$. 6. Since the triangles are congruent, $m\angle S = m\angle W$. So, $m\angle S = 16x + 5$ and $m\angle W = 16x + 5$. 7. Using the triangle angle sum for $\triangle CJW$: $$45 + (16x + 5) + m\angle C = 180$$ 8. Using the triangle angle sum for $\triangle AGS$: $$50 + (16x + 5) + m\angle G = 180$$ 9. Since the triangles are congruent, $m\angle C = m\angle G$. Subtracting the two equations: $$[45 + (16x + 5) + m\angle C] - [50 + (16x + 5) + m\angle G] = 0$$ 10. Simplify: $$45 + 16x + 5 + m\angle C - 50 - 16x - 5 - m\angle G = 0$$ $$ (45 + 5 - 50 - 5) + (16x - 16x) + (m\angle C - m\angle G) = 0$$ $$0 + 0 + 0 = 0$$ 11. This confirms the angles correspond correctly. Now, use the fact that the sum of angles in $\triangle AGS$ is 180: $$50 + (16x + 5) + m\angle G = 180$$ 12. Simplify: $$55 + 16x + m\angle G = 180$$ 13. So, $$m\angle G = 180 - 55 - 16x = 125 - 16x$$ 14. Similarly, for $\triangle CJW$: $$45 + (16x + 5) + m\angle C = 180$$ 15. Simplify: $$50 + 16x + m\angle C = 180$$ 16. So, $$m\angle C = 180 - 50 - 16x = 130 - 16x$$ 17. Since $m\angle C = m\angle G$, set equal: $$130 - 16x = 125 - 16x$$ 18. Subtract $125 - 16x$ from both sides: $$130 - 16x - (125 - 16x) = 0$$ $$130 - 16x - 125 + 16x = 0$$ $$5 = 0$$ 19. This is a contradiction, so the assumption about angle correspondence must be reconsidered. 20. Instead, since $\triangle CJW \cong \triangle AGS$, angles correspond as $C \leftrightarrow A$, $J \leftrightarrow G$, $W \leftrightarrow S$. 21. Given $m\angle A = 50^\circ$, $m\angle J = 45^\circ$, and $m\angle S = 16x + 5$, then $m\angle C = 50^\circ$, $m\angle G = 45^\circ$, and $m\angle W = 16x + 5$. 22. Use the triangle sum for $\triangle CJW$: $$m\angle C + m\angle J + m\angle W = 180$$ $$50 + 45 + (16x + 5) = 180$$ 23. Simplify: $$95 + 16x + 5 = 180$$ $$100 + 16x = 180$$ 24. Subtract 100 from both sides: $$16x = 80$$ 25. Divide both sides by 16: $$x = \frac{\cancel{16}x}{\cancel{16}} = \frac{80}{16} = 5$$ 26. The value of $x$ is 5. **Final answer:** $x = 5$ (Option D)