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To prove triangle AB C cong triangle A^ prime prime B^ prime prime C^ prime prime which of the following shows how the ASA criterion follows from the definition of congruence in terms of rigid motions?

A Rotate triangle ABC 90 degrees (counterclockwise) about vertex C, and then reflect triangle A' * B' * C' across overline C * A' . C

B Rotate triangle ABC - 90 degrees (clockwise) about vertex C, and then reflect triangle A' * B' * C' across overline B' * C .

C Rotate triangle ABC - 90 degrees (clockwise) about vertex C, and then reflect triangle A' * B' * C' across overline C * A' .

D Rotate triangle ABC 360 degrees (counterclockwise), and then reflect triangle A' * B' * C' across overline B' * C

1 answer

To prove that triangle \( ABC \) is congruent to triangle \( A''B''C'' \) using the ASA (Angle-Side-Angle) criterion through rigid motions, we need to ensure that the corresponding angles and the included side between those angles in both triangles are preserved. This means we should find a way to transform triangle \( ABC \) into triangle \( A''B''C'' \) through a series of rotations and reflections that maintains distances and angles.

Given the options:

Option A: Rotate triangle ABC 90 degrees (counterclockwise) about vertex C, and then reflect triangle \( A'B'C' \) across line \( CA' \).

Option B: Rotate triangle ABC - 90 degrees (clockwise) about vertex C, and then reflect triangle \( A'B'C' \) across line \( B'C \).

Option C: Rotate triangle ABC - 90 degrees (clockwise) about vertex C, and then reflect triangle \( A'B'C' \) across line \( CA' \).

Option D: Rotate triangle ABC 360 degrees (counterclockwise), and then reflect triangle \( A'B'C' \) across line \( B'C \).

Among these options, option A appears to correctly align with the ASA criterion because:

  1. The initial rotation around point \( C \) preserves the relative positions of points \( A \) and \( B \) with respect to \( C \).
  2. Reflecting around line \( CA' \) corresponds to matching the angles and ensuring \( A \) aligns with \( A'' \) while maintaining the side length between \( C \) and the new position of \( A \).

Thus, the proper transformation that follows the ASA criterion through rigid motions is best captured by Option A.