Final Answer:
The statement that is not used to prove that triangle LKM is similar to triangle NOM is D) The lengths of all sides in LKM are equal to the corresponding sides in NOM. Therefore corect one is option d.
Step-by-step explanation:
In order to prove that two triangles are similar, we typically rely on angle-angle (AA) or side-angle-side (SAS) criteria. The statement "All angles in LKM are congruent to the corresponding angles in NOM" (Option A) corresponds to the angle-angle criterion. Similarly, "LM is proportional to NO" (Option B) aligns with the side-angle-side criterion.
The statement LK is parallel to NM (Option C) is crucial for proving similarity using the angle-angle criterion. When two lines are parallel, corresponding angles are congruent. However, the statement "The lengths of all sides in LKM are equal to the corresponding sides in NOM" (Option D) is not a criterion for triangle similarity. While equal side lengths (congruent sides) are a property of similar triangles, having equal sides alone is not sufficient to establish similarity.
To elaborate further, similarity between two triangles implies that their corresponding angles are equal, and the ratios of their corresponding sides are in proportion. In the context of triangles, proving similarity often involves establishing relationships between angles and sides. Therefore, Option D, which focuses solely on equal side lengths without considering the angle relationships, is not a valid statement for proving the similarity of triangles LKM and NOM.