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To prove △ABC≅△A′′B′′C′′ △ 𝐴 𝐵 𝐶 ≅ △ 𝐴 ″ 𝐵 ″ 𝐶 ″ , which of the following shows how the ASA criterion follows from the definition of congruence in terms of rigid motions? (1 point) Responses Rotate △ABC △ 𝐴 𝐵 𝐶 360 degrees (counterclockwise), and then reflect △A′B′C′ △ 𝐴 ′ 𝐵 ′ 𝐶 ′ across B′C¯¯¯¯¯¯¯¯¯ 𝐵 ′ 𝐶 ¯ . Rotate triangle upper A upper B upper C 360 degrees (counterclockwise), and then reflect triangle upper A prime upper B prime upper C prime across Modifying above upper B prime upper C with bar . Rotate △ABC △ 𝐴 𝐵 𝐶 −90 − 90 degrees (clockwise) about vertex C 𝐶 , and then reflect △A′B′C′ △ 𝐴 ′ 𝐵 ′ 𝐶 ′ across CA′¯¯¯¯¯¯¯¯¯ 𝐶 𝐴 ′ ¯ . Rotate triangle upper A upper B upper C negative 90 degrees (clockwise) about vertex upper C , and then reflect triangle upper A prime upper B prime upper C prime across Modifying above upper C upper A prime with bar . Rotate △ABC △ 𝐴 𝐵 𝐶

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To prove that triangle \( \triangle ABC \) is congruent to triangle \( \triangle A''B''C'' \) using the Angle-Side-Angle (ASA) criterion and the definition of congruence in terms of rigid motions, we need to demonstrate a sequence of rigid motions that will map \( \triangle ABC \) onto \( \triangle A''B''C'' \).

Given the ASA criterion, we want to show that if two angles and the included side of one triangle are equal to the corresponding angles and side of another triangle, then the triangles are congruent.

Analysis of the options provided:

  1. Rotate \( \triangle ABC \) 360 degrees (counterclockwise): This is not effective since it would leave the triangle in the same position.

  2. Reflect \( \triangle A'B'C' \) across \( B'C' \) after a rotation of \( \triangle ABC \): This reflects triangle \( A'B'C' \) but does not imply any congruence because there is no defined relationship given the lack of positions of \( A', B', C' \).

  3. Rotate \( \triangle ABC \) -90 degrees (clockwise) about vertex \( C \) and then reflect \( \triangle A'B'C' \) across line \( CA' \): This combination may work, depending on the angle measures and sides involved.

  4. Reflect \( \triangle A'B'C' \) across line \( B'C' \) after rotating \( \triangle ABC \): Similar reasoning applies here with relationships that are not clear.

In terms of proving congruence via ASA, we want a method that demonstrates that after performing these motions, the triangles align perfectly, preserving the lengths and angles.

The best approach to illustrate the ASA criterion is:

  • First, rotate \( \triangle ABC \) or reflect it in a direction that will align \( A, B, \) and \( C \) with the corresponding points \( A'', B'', C'' \).
  • This motion should maintain the angles between the sides involved.

From the options provided, the most promising method would be the rotation followed by the reflection that aligns two angles of the triangles and the included side. However, the exact definition of the lines being reflected over should align to \( B'C'\) and \( CA'\).

Conclusion: A detailed sequence of rigid motions demonstrating that \(\angle A = \angle A''\), \(\angle B = \angle B''\), and side \( AB = A''B'' \) will show \( \triangle ABC \cong \triangle A''B''C''\) by ASA. Thus we recommend checking the most logical option based on these criteria.