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Question

Which answer is correct and why
Question

Mario is trying to prove the SSS congruency theorem using the triangles pictured in image 1 below. He translated △ABC
to create △A′B′C′
as shown in image 2 and then performed a rotation to create △A"B"C"
as shown in image 3. Surprisingly, he found that △A"B"C"
overlapped △LMN
without him having to do a reflection. What happened?

(1 point)
Responses

Mario made a mistake. He should have translated △ABC
so that point B moved to point N, then done a reflection, then a rotation
Mario made a mistake. He should have translated △ABC
so that point B moved to point N, then done a reflection, then a rotation

Mario has shown that the two triangles are not congruent because if the sequence of transformations is not a translation, a rotation and then a reflection, then there is no way that the triangles can be congruent.
Mario has shown that the two triangles are not congruent because if the sequence of transformations is not a translation, a rotation and then a reflection, then there is no way that the triangles can be congruent.

Mario made a mistake. He should have completed a reflection before the rotation so that there were three transformations in the sequence, 1 of all three types.
Mario made a mistake. He should have completed a reflection before the rotation so that there were three transformations in the sequence, 1 of all three types.

Mario has shown that the trwo triangles are congruent. Three transformations are not always necessary when moving 1 triangle on top of the other.
Mario has shown that the trwo triangles are congruent. Three transformations are not always necessary when moving 1 triangle on top of the other.
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Answer 1

The correct answer is:

Mario has shown that the two triangles are congruent. Three transformations are not always necessary when moving 1 triangle on top of the other.

Explanation:

The SSS (Side-Side-Side) congruency theorem states that if three sides of one triangle are congruent to the three sides of another triangle, then the two triangles are congruent. The fact that Mario was able to translate triangle \( \triangle ABC \) to \( \triangle A'B'C' \) and then rotate it to overlap \( \triangle LMN \) suggests that the corresponding sides are indeed equal, confirming the congruence of the two triangles without needing a reflection.

It's important to note that while the specific sequence of transformations (translation, rotation, reflection) can provide insight into congruence, having just any combination of translation and rotation can also lead to congruence, especially if the triangles coincide exactly after those transformations. Thus, it's not necessary to perform all three transformations to verify the congruency of the triangles.