To show that triangles \( \Delta ABC \) and \( \Delta GHE \) are congruent through a sequence of transformations, you can typically use the following steps:
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Translation: Move \( \Delta ABC \) to a position over \( \Delta GHE \) by translating it in such a way that one vertex (e.g., point A) aligns with the corresponding vertex in the second triangle (e.g., point G).
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Rotation: After translation, rotate \( \Delta ABC \) around point G (or whichever vertex aligns) so that side \( AB \) aligns with side \( GH \) or one of the corresponding sides aligns.
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Reflection (if needed): If the orientation of \( \Delta ABC \) after translation and rotation does not match that of \( \Delta GHE \), apply a reflection across a line that passes through the point G and is perpendicular to the aligned side to ensure that \( \Delta ABC \) properly matches the orientation of \( \Delta GHE \).
Using these steps guarantees that the two triangles remain congruent throughout the transformations. Depending on the specific measurements and angles of the triangles, one might combine these transformations in different ways. The key point is to establish congruence by transforming one triangle into the other through rigid motions that preserve lengths and angles.