To prove AABC ~ AA"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)
• Rotate AABC -90 degrees (clockwise) about vertex C, and then reflect AA' B'C' across CA!.
• Rotate AABC 90 degrees (counterclockwise) about vertex C, and then reflect AA' B'C" across C'A'.
• Rotate AABC 360 degrees (counterclockwise), and then reflect A A'B'C across B'C.
• Rotate A ABC -90 degrees (clockwise) about vertex C, and then reflect A A'B'C" across B'C.

Answer 1

To prove that triangle \( AABC \) is similar to triangle \( AA'B'C' \) using the ASA (Angle-Side-Angle) criterion, we need to establish that there is a sequence of rigid motions (rotations, reflections, translations) that maps triangle \( AABC \) onto triangle \( AA'B'C' \). The rigid motions must preserve angles and the lengths of sides.

The most suitable option that demonstrates the ASA criterion involves rotating and reflecting the triangles in a way that aligns the angles and sides correctly.

Among the options provided, the second option appears to be the best approach:

• Rotate \( AABC \) 90 degrees (counterclockwise) about vertex \( C \), and then reflect \( AA'B'C' \) across \( C'A' \).

This shows a combination of a rotation that maintains angle A and angle C, and a reflection that will maintain the corresponding sides as well, effectively showing that the angles and the included sides (the ones between them) align accordingly, which is essential for proving similarity via the ASA criterion.

Thus, the correct choice is:

• Rotate AABC 90 degrees (counterclockwise) about vertex C, and then reflect AA' B'C" across C'A'.