To determine the congruence transformation used on triangle \( \triangle ABC \) that resulted in triangle \( \triangle DEF \), we need to analyze the different transformation options you provided based on their implications.
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Rotation of -180° (clockwise) or 180° (counterclockwise) followed by a shift (translation) of 2 units right and 2 units up: This transformation would effectively invert the triangle and then move it. We would check the final position accordingly.
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Reflection across the y-axis followed by a shift (translation) of 2 units right and 2 units up: This reflects the triangle across the y-axis, inverting its x-coordinates, and then shifts it to a new location.
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Shift left and up followed by reflection across the y-axis: This means moving the triangle first and then reflecting it. The final position after each would have to be compared accordingly.
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Shift right and up followed by a rotation of 180° (counterclockwise): Similar to the first transformation but with the order reversed, translating first and then rotating.
To determine which of these matches the transformation from \( \triangle ABC \) to \( \triangle DEF \), you would normally look at the coordinates of the vertices of both triangles and map the transformations to see if they yield the corresponding new coordinates.
Without specific coordinates to analyze visually, it’s necessary to consider the properties of these transformations:
- A rotation will maintain distances and angles but change the orientation.
- A reflection will flip positions over a line (here the y-axis), affecting left-right positioning but maintaining vertical relationships.
- A translation shifts all points uniformly without changing their configuration.
Assuming one transformation must match \( \triangle DEF \) based on the descriptions, the answer can be gleaned from understanding how each operation affects the geometry of the triangles involved.
Typically, if \( \triangle ABC \) was flipped over the y-axis and then translated, the shape would maintain its integrity but would change placement notably in terms of coordinates.
Among the options provided, the only operations that seem to meet these criteria logically without additional spatial reasoning are the reflection followed by translation options since these preserve the necessary congruence as defined.
If forced to choose without vertices to test against, I would suggest the following option:
“△ABC was reflected across the y-axis, then shifted 2 units to the right and 2 units up.”
This operation would reflect \( \triangle ABC \) and then shift it into a position matching \( \triangle DEF \) without altering shapes or angles inconsistent with congruence.
I recommend checking the transformation outcomes against the specific coordinates of triangles if available, but based on congruence principles, this option appears most plausible.