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 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¯¯¯¯¯¯¯¯¯. 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′¯¯¯¯¯¯¯¯¯.

Question

GPT-4o mini · Accepted Answer

To show how the ASA (Angle-Side-Angle) criterion follows from the definition of congruence in terms of rigid motions, we need to demonstrate that a triangle can be transformed into another triangle using rotations and reflections that preserve lengths and angles.

In the case of triangles \( \triangle ABC \) and \( \triangle A'B'C' \), if we know:

Angle \( A = A' \)
Angle \( B = B' \)
Side \( AB = A'B' \)

We can use the following steps to prove that \( \triangle ABC \cong \triangle A'B'C' \):

Rotate \( \triangle ABC \) by a certain angle such that vertex \( C \) coincides with vertex \( C' \). This part of the transformation preserves the angles and positions of points \( A \) and \( B \).
After this rotation, we can reflect \( \triangle A'B'C' \) across the line that contains side \( CA' \). This ensures that angle \( C \) coincides with angle \( C' \) and that the transformation respects congruence.

Thus, the correct option that aligns with this reasoning would involve:

Rotating \( \triangle ABC \) around vertex \( C \) and then reflecting across the line containing side \( CA' \).

The option that describes this process accurately is:

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

This process demonstrates that through rigid motions (rotation and reflection), we can show that the two triangles are congruent using the ASA criterion.