Which best explains whether or not \( \triangle A B C \cong \triangle L M N \) ? The figures are congruent because a \( 270^{\circ} \) rotation about the origin and then a reflection over the \( x \)-axis will map \( \triangle A B C \) onto \( \triangle L M N \). The figures are congruent because a 180 rotation about the origin and then a reflection over the \( x \)-axis will map \( \triangle A B C \) onto \( \triangle L M N \). The figures are not congruent because point \( B \) corresponds with point \( N \) and point \( C \) corresponds with point \( M \). The figures are not congruent because there is no rigid transformation or combination of rigid transformations that will map \( \triangle A B C \) onto \( \triangle L M N \).

To determine whether \( \triangle ABC \) is congruent to \( \triangle LMN \), we need to analyze the transformations mentioned in the options provided. 1. **Understanding Congruence**: Two triangles are congruent if one can be transformed into the other through rigid transformations, which include rotations, reflections, and translations. These transformations preserve the size and shape of the figures. 2. **Evaluating the Options**: - **First Option**: "The figures are congruent because a \( 270^{\circ} \) rotation about the origin and then a reflection over the \( x \)-axis will map \( \triangle ABC \) onto \( \triangle LMN \)." - This option suggests a sequence of transformations that could potentially map one triangle onto the other. If this transformation is valid, then the triangles are congruent. - **Second Option**: "The figures are congruent because a \( 180^{\circ} \) rotation about the origin and then a reflection over the \( x \)-axis will map \( \triangle ABC \) onto \( \triangle LMN \)." - Similar to the first option, this describes a sequence of transformations that could also indicate congruence if valid. - **Third Option**: "The figures are not congruent because point \( B \) corresponds with point \( N \) and point \( C \) corresponds with point \( M \)." - This option suggests a mismatch in corresponding points, which would indicate that the triangles are not congruent. - **Fourth Option**: "The figures are not congruent because there is no rigid transformation or combination of rigid transformations that will map \( \triangle ABC \) onto \( \triangle LMN \)." - This option asserts that no transformations can map one triangle to the other, indicating a lack of congruence. 3. **Conclusion**: To determine the correct explanation, we need to verify the transformations mentioned in the first two options. If either transformation can successfully map \( \triangle ABC \) onto \( \triangle LMN \), then the triangles are congruent. If not, we would lean towards the third or fourth options. Without specific coordinates or a visual representation of the triangles, we cannot definitively conclude which option is correct. However, if we assume that the transformations in the first two options are valid, then the triangles would be congruent. If they are not valid, then the third or fourth options would be correct. Thus, the best explanation would depend on the validity of the transformations described in the first two options. If they are valid, then the first or second option is correct. If not, then the third or fourth option is correct.