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

Possible Answers:
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 . Rotate △ABC −90 degrees (clockwise) about vertex C, and then reflect △A′B′C′ across B′C¯¯¯¯¯¯¯¯¯.

Answer 1

To show that triangles \( \triangle ABC \) and \( \triangle A''B''C'' \) are congruent using the ASA (Angle-Side-Angle) criterion, we can use rigid motions that preserve angles and distances.

1. Identify the angles and sides involved: In a scenario where you want to prove that \( \triangle ABC \cong \triangle A''B''C'' \), the ASA criterion involves showing that we have two angles and the included side of one triangle equal to the corresponding two angles and included side of the other triangle.

2. Use of rigid motions: The rigid motions involve rotation and reflection, which do not change the shape or size of the triangles.

From the options provided, we need to carefully evaluate the series of rigid movements that align one triangle with another while preserving the angle relationships and side lengths.

The logical choice will involve:

• A rotation to align a vertex with its corresponding vertex.
• A reflection to flip the triangle over an appropriately chosen line, while preserving the angle and side measures.

Correct option:

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

This movement effectively positions \( C \) in the correct spot (after a -90 degree rotation, which aligns \( A \) with \( A'' \) and maintains the original angle at \( C \)), and then reflects across line \( CA' \) ensures that the angles and sides are preserved and matches up with \( A''B''C'' \).

Thus, the answer is:

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