Triangles A B C and D E F are shown. Triangle A B C is rotated to the left about point A and then is shifted up and to the right to form triangle D E F.

To determine the rigid transformations that will map triangle \( \triangle ABC \) to triangle \( \triangle DEF \), we need to analyze the transformations described in the options provided.

1. Translate vertex A to vertex D, and then reflect \( \triangle ABC \) across the line containing AC.

   • This transformation first translates \( A \) to \( D \) and then reflects, which might not align \( \triangle ABC \) with \( \triangle DEF \)'s orientation.

2. Translate vertex B to vertex D, and then rotate \( \triangle ABC \) around point B to align the sides and angles.

   • Translating \( B \) to \( D \) is incorrect since we need to move \( A \) to \( D \) to maintain the triangle structure.

3. Translate vertex B to vertex D, and then reflect \( \triangle ABC \) across the line containing AC.

   • Similar to option 2, translating \( B \) to \( D \) isn't a fit for our required transformation.

4. Translate vertex A to vertex D, and then rotate \( \triangle ABC \) around point A to align the sides and angles.

   • This transformation starts with translating \( A \) to \( D \), which is appropriate, and rotating around \( A \) can help align triangle \( ABC \) with triangle \( DEF \) correctly.

Considering all options, the most appropriate transformation that correctly matches the description of moving triangle \( \triangle ABC \) to \( \triangle DEF \) is:

Translate vertex A to vertex D, and then rotate \( \triangle ABC \) around point A to align the sides and angles.