1. **State the problem:** Given that $\frac{JL}{NL} = \frac{KL}{ML}$ and $JL = ML$, prove that $\angle J \cong \angle N$. 2. **Analyze the given information:** We have two triangles, $\triangle JLN$ and $\triangle KLN$, sharing side $LN$. Also, $JL = ML$ is given. 3. **Use the given ratio:** From $\frac{JL}{NL} = \frac{KL}{ML}$ and $JL = ML$, substitute $ML$ for $JL$: $$\frac{ML}{NL} = \frac{KL}{ML}$$ 4. **Cross multiply:** $$ML \times ML = KL \times NL$$ or $$ML^2 = KL \times NL$$ 5. **Use the fact that $JL = ML$:** Since $JL = ML$, the triangles share side $LN$, and the sides satisfy the proportion, by the Side-Splitter or Side-Angle-Side similarity criteria, the triangles are similar or congruent in a way that implies corresponding angles are equal. 6. **Conclude angle equality:** Since the sides around $\angle J$ and $\angle N$ correspond and the triangles share side $LN$, it follows that $\angle J \cong \angle N$. **Final answer:** $$\boxed{\angle J \cong \angle N}$$