To prove that triangle \( \triangle ABC \) is congruent to triangle \( \triangle GHE \), we can describe a sequence of transformations that will map \( \triangle ABC \) onto \( \triangle GHE \). Here’s a step-by-step process:
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Translation: First, translate triangle \( \triangle ABC \) so that one of its vertices (say point A) coincides with the corresponding vertex (say point G) of triangle \( \triangle GHE \).
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Rotation: Next, perform a rotation about the point G (where point A was translated to) so that the line segment \( AB \) aligns with \( GH \). This will ensure that the orientation of the triangles is correct.
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Reflection (if necessary): If the triangles are mirror images of each other (depending on the orientation), apply a reflection across the appropriate line. This will ensure that the remaining vertices align (i.e., point B aligns with point H, and point C aligns with point E).
By completing these transformations (translation, rotation, and possibly reflection), we have mapped \( \triangle ABC \) onto \( \triangle GHE \), thus proving their congruence.
In summary, the sequence of transformations is:
- Translate \( \triangle ABC \) so vertex \( A \) is at vertex \( G \).
- Rotate \( \triangle ABC \) around point \( G \) to align side \( AB \) with side \( GH \).
- Reflect \( \triangle ABC \) across an appropriate line if necessary, to match points \( B \) with \( H \) and \( C \) with \( E \).
This sequence demonstrates that \( \triangle ABC \) is congruent to \( \triangle GHE \).
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