Triangle Congruence 4D3A95

1. **State the problem:** We need to prove that $\triangle JKN \cong \triangle MKL$ given that $K$ is the midpoint of $JM$ and $NL$, $\angle L \cong \angle N$, and $LM \cong JN$. 2. **Given information:** - $K$ is the midpoint of $JM$ and $NL$, so $JK = KM$ and $NK = KL$. - $\angle L \cong \angle N$ (given). - $LM \cong JN$ (given). 3. **Goal:** Prove $\triangle JKN \cong \triangle MKL$. 4. **Use the Midpoint Theorem:** Since $K$ is the midpoint of $JM$, we have $$JK = KM$$ Similarly, since $K$ is the midpoint of $NL$, we have $$NK = KL$$ 5. **Identify corresponding parts:** - Side $JK$ corresponds to side $KM$. - Side $NK$ corresponds to side $KL$. - Angle $\angle N$ corresponds to angle $\angle L$ (given congruent). 6. **Check the congruence criteria:** We have two pairs of sides equal ($JK = KM$ and $NK = KL$) and the included angles $\angle N \cong \angle L$. 7. **Apply SAS (Side-Angle-Side) Congruence Postulate:** Since two sides and the included angle of $\triangle JKN$ are congruent to two sides and the included angle of $\triangle MKL$, we conclude $$\triangle JKN \cong \triangle MKL$$ **Final answer:** $\triangle JKN \cong \triangle MKL$ by SAS congruence postulate.