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To prove △ABC≅△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 90 degrees (counterclockwise) about vertex C, and then reflect △A′B′C′ across CA′¯¯¯¯¯¯¯¯¯. Rotate triangle upper A upper B upper C 90 degrees (counterclockwise) 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 −90 degrees (clockwise) about vertex C, and then reflect △A′B′C′ across CA′¯¯¯¯¯¯¯¯¯. 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 −90 degrees (clockwise) about vertex C, and then reflect △A′B′C′ across B′C¯¯¯¯¯¯¯¯¯. 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 B prime upper C with bar . Rotate △ABC 360 degrees (counterclockwise), and then reflect △A′B′C′ across B′C¯¯¯¯¯¯¯¯¯.

1 answer

To prove that triangles \( \triangle ABC \) and \( \triangle A''B''C'' \) are congruent using the Angle-Side-Angle (ASA) criterion through rigid motions, we need to show that we can map \( \triangle ABC \) onto \( \triangle A''B''C'' \) using a series of rotations and reflections.

The ASA criterion tells us that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent. The definition of triangle congruence in terms of rigid motions (rotations, reflections, and translations) states that one triangle can be transformed into another without altering its shape or size.

To illustrate the ASA criterion:

  1. Rotate \( \triangle ABC \) around point \( C \) to align one of the angles of \( \triangle ABC \) with the corresponding angle of \( \triangle A''B''C'' \) (this will align angle \( A \) with angle \( A'' \)).

  2. Reflect the triangle across the line that contains edge \( CA' \) to match with \( C'A' \).

This sequence of operations will ensure that both triangles properly align through a series of rigid motions, hence demonstrating congruence.

Therefore, the procedure that aligns with this reasoning would be:

Rotate \( \triangle ABC \) 90 degrees (counterclockwise) about vertex C, and then reflect \( \triangle A'B'C' \) across \( CA' \) (which is the line containing vertex \( C \) and vertex \( A' \)).

This guarantees that we utilize both the angle and side alignment as per the ASA criterion for congruence.

So the correct response from your options involves the rotation about point C and the reflection across the line CA'.