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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 (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
−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 (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 .

1 answer

To prove the congruence of triangles \( \triangle ABC \) and \( \triangle A''B''C'' \) using the ASA criterion through rigid motions, we need to establish a sequence of rotations and reflections that demonstrate that one triangle can be transformed into the other, maintaining their side lengths and angles.

The ASA (Angle-Side-Angle) criterion states that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.

When considering how a transformation using rigid motions can demonstrate ASA congruence, we look for a combination of rotation and reflection that maintains the angles and side lengths.

Analysis of Choices

  1. First Choice:

    • Rotate \( \triangle ABC \) by \(-90\) degrees about vertex \( C \), then reflect across \( B'C' \).
    • This move appears to focus on reflections that might not align the triangles correctly.

  2. Second Choice:

    • Rotate \( \triangle ABC \) by \(-90\) degrees about vertex \( C \), then reflect across \( CA' \).
    • This setup allows for potential alignment of angles and side lengths appropriately.

  3. Third Choice:

    • Rotate \( \triangle ABC \) \( 360 \) degrees (which does nothing), then reflect across \( B'C' \).
    • A \( 360 \)-degree rotation does not help establish congruence since it does not modify positions.

  4. Fourth Choice:

    • Rotate \( \triangle ABC \) \( 90 \) degrees (counterclockwise) about vertex \( C \), then reflect across \( CA' \).
    • This rotation and subsequent reflection would allow for proper alignment of the sides and angles as required by ASA.

Conclusion

Considering the need for an effective transformation that maintains the angles and side relationships in \( \triangle ABC \) and \( \triangle A''B''C'' \) based on the ASA criterion, the most effective option appears to be the Fourth Choice:

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

This choice will appropriately position the triangles to demonstrate their congruence as per the ASA criteria defined by rigid motions.