Question

fying rigid transformations
the two triangles created by the diagonal of the parallelogram are congruent. recall that the opposite sides of a parallelogram are congruent.
which transformation(s) could map one triangle to the other?
translation
rotation and translation
reflection
reflection and translation

Explanation:

Step1: Analyze translation

Translation moves a figure without rotation or reflection. But in a parallelogram, just translation won't map one triangle (created by diagonal) to the other as their orientations (due to the parallelogram's shape) aren't just a simple shift in position without other changes.

Step2: Analyze rotation and translation

Rotation alone (around the mid - point of the diagonal) can map one triangle to the other. And if we consider translation along with rotation (but rotation is the key as the mid - point of the diagonal is a fixed point for rotation in a parallelogram, translation is redundant here as rotation about the mid - point already maps the triangles).

Step3: Analyze reflection

Reflection over the perpendicular bisector of the diagonal (or the line that is the angle - bisector in some sense for the parallelogram's angles related to the diagonal) can map one triangle to the other.

Step4: Analyze reflection and translation

Reflection over the appropriate line (as in Step3) is sufficient. Translation is not needed as reflection itself can map the congruent triangles (since they are congruent and reflection is a rigid transformation that can flip one triangle to the position of the other within the parallelogram).

Answer:

rotation and translation, reflection