To determine the correct congruence transformation, let's analyze the transformations mentioned in your options:
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Reflection across the y-axis: This would change the x-coordinates of the points of triangle ABC to their negatives while leaving the y-coordinates unchanged.
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Rotation of 180°: This would move each point \( (x, y) \) to \( (-x, -y) \), essentially flipping both coordinates.
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Translation (shift): Shifting 2 units to the right adds 2 to the x-coordinates, and shifting 2 units up adds 2 to the y-coordinates.
Now, we need to carefully evaluate the transformations in the provided options:
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If △ABC is reflected across the y-axis, its new coordinates (let's say for points A, B, C) would be \( (-x_1, y_1), (-x_2, y_2), (-x_3, y_3) \). After shifting 2 units right and 2 units up, we'd get new coordinates that do not match those of △DEF unless the original ABC's vertices were tailored to match after these transformations.
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Rotating 180° then shifting would effectively move points to different positions. If done correctly, this might match with △DEF, especially if those coordinates fall into a range that matches after applying these transformations.
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A translation followed by rotation could lead to a different set of coordinates as well, depending on the direction and degree of rotation.
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A shift left followed by a reflection across the y-axis has a very specific outcome and would likely not generalize well to match many transformations.
Without the specific coordinates from △ABC and △DEF, it’s hard to definitively conclude, so it’s essential to visualize how each of those transformations would impact the triangle's position.
Given all this, based on the transformations detailed in your options, the most systematic approach would be △ABC was reflected across the y-axis, then shifted 2 units to the right and 2 units up, assuming that your visualization shows that as the most plausible set of transformations influencing the final shape and orientation of △DEF after analysis.