To determine which congruence theorem can be used to prove that triangles \( \triangle LON \) and \( \triangle LMN \) are congruent, we analyze the information provided:
- Triangles \( \triangle LON \) and \( \triangle LMN \) share the side \( LN \).
- Sides \( OL \) and \( ML \) are congruent (i.e., \( OL \cong ML \)).
- Sides \( ON \) and \( NM \) are congruent (i.e., \( ON \cong NM \)).
Now, we identify the sides involved:
- We have \( OL \cong ML \) (the sides from \( O \) and \( M \) to \( L \)).
- We have \( ON \cong NM \) (the sides from \( O \) and \( M \) to \( N \)).
- The shared side \( LN \) (the side connecting vertices \( L \) and \( N \)).
Given the configurations, we observe:
- The two sides from \( L \) to \( O \) and \( M \) are congruent (\( OL \cong ML \)).
- The two sides from \( N \) to \( O \) and \( M \) are congruent (\( ON \cong NM \)).
- The shared side \( LN \) is congruent to itself.
Thus, we have:
- Side \( OL \) is congruent to \( ML \).
- Side \( ON \) is congruent to \( NM \).
- Side \( LN \) is shared and therefore congruent to itself.
This fits the SAS (Side-Angle-Side) congruence criterion, which states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the two triangles are congruent.
Therefore, the correct answer is:
SAS.
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