Question

to prove ( \triangle abc cong \triangle abc ), which of the following shows how the asa criterion follows from the definition of congruence in terms of rigid motions? (1 point) rotate ( \triangle abc ) 90 degrees (counterclockwise) about vertex ( c ), and then reflect ( \triangle abc ) across ( overline{ca} ) rotate ( \triangle abc - 90 ) degrees (clockwise) about vertex ( c ), and then reflect ( \triangle abc ) across ( overline{bc} ) rotate ( \triangle abc ) 360 degrees (counterclockwise), and then reflect ( \triangle abc ) across ( overline{bc} ) rotate ( \triangle abc - 90 ) degrees (clockwise) about vertex ( c ), and then reflect ( \triangle abc ) across ( overline{ca} )

Explanation:

Step1: Analyze rotation

A \( - 90\) - degree (clockwise) rotation about vertex \(C\) moves \(\triangle ABC\) to a new position \(\triangle A'B'C\).

Step2: Analyze reflection

Reflecting \(\triangle A'B'C\) across \(\overline{CA'}\) aligns the triangles such that the ASA (Angle - Side - Angle) criterion for congruence is satisfied. A \(90\) - degree counter - clockwise rotation would not align as needed. A \(360\) - degree rotation brings the triangle back to its original position (no net rotation). Reflecting across \(\overline{B'C}\) after a \(-90\) - degree rotation does not align for ASA.

Answer:

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