Subjects geometry

Triangle Congruence 8D643D

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1. **State the problem:** Prove that triangles $\triangle LMN$ and $\triangle PON$ are congruent. 2. **Given:** - $LM \cong PO$ (Given) - $LN \cong PN$ (Given) - $N$ is the midpoint of $LO$ (Given) 3. **Recall the midpoint definition:** If $N$ is the midpoint of segment $LO$, then $LN \cong NO$. 4. **Use midpoint to find congruent segments:** Since $N$ is midpoint of $LO$, we have $$LN \cong NO$$ 5. **Identify the missing statement (#4):** Because $N$ is midpoint, segment $MN$ is congruent to segment $ON$ (or equivalently $MN \cong ON$) if $M$ and $O$ are points such that $MO \cong OM$ (which is always true by reflexive property). But here, the key is that $MN \cong ON$ because $N$ divides $LO$ into two equal parts. 6. **Complete the proof using SSS:** - $LM \cong PO$ (Given) - $LN \cong PN$ (Given) - $MN \cong ON$ (From midpoint property) Therefore, by the Side-Side-Side (SSS) congruence criterion, $\triangle LMN \cong \triangle PON$. **Final answer:** The missing #4 statement is: $$MN \cong ON$$