Mario is trying to prove the SSS congruency theorem using the triangles pictured in image 1 below. He translated △ABC

Question

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?
which answer is correct and why
(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.
Skip to navigation

Answer 1

The correct response to Mario's situation is:

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

Explanation: In the SSS (Side-Side-Side) congruency theorem, if all three sides of one triangle are congruent to the three sides of another triangle, then the triangles are congruent. The process of translating and rotating the triangle to overlap with the second triangle indicates that the two triangles are congruent, regardless of the specific transformations used.

While it is common to complete the proof with all three types of transformations (translation, rotation, and reflection), it is not a strict requirement. In this case, Mario successfully translated and rotated the triangle to achieve congruence, which suffices to demonstrate that the two triangles are congruent. Thus, the sequence of transformations does not need to necessarily include a reflection for the triangles to be considered congruent.