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?

Question

(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 360 degrees (counterclockwise), and then reflect △A′B′C′ across 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 −90 degrees (clockwise) about vertex C, and then reflect △A′B′C′ across B′C¯¯¯¯¯¯¯¯¯.

Answer 1

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

The ASA criterion states 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 triangles are congruent.

Given the choices, we need to analyze the rigid motions to see which sequence can achieve the required congruence.

Among the options, the first one states:

• Rotate \( \triangle ABC \) 90 degrees (counterclockwise) about vertex C, and then reflect \( \triangle A'B'C' \) across \( \overline{CA'} \).

This option suggests that by rotating \( \triangle ABC \) around vertex \( C \), we could properly align it with one of the angles in triangle \( \triangle A''B''C'' \), and then reflecting it across the necessary line would help achieve congruence. The reflection across the side \( \overline{CA'} \) would align the sides corresponding to those angles and make sure the included angle remains constant.

The second option also describes a similar series of motions, but rotating clockwise could potentially affect the positioning relative to the congruent angles.

The third option (360 degrees rotation) does not change the triangle's positioning significantly—this would not help in demonstrating triangle congruence effectively.

The fourth option outlines a sequence that might not properly adhere to the ASA condition with regards to the necessary alignments and angle preservation.

Lastly, the fifth option reflects across a side that would not necessarily provide the needed congruence based on how the triangles are positioned.

In conclusion, the first option effectively shows how the ASA criterion follows from the definition of congruence in terms of rigid motions, as it demonstrates a direct method for aligning angles and sides correctly.

Thus, the correct answer is:

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