Question

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

Explanation:

Step1: Recall ASA - Angle - Side - Angle

ASA criterion for congruence states that if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent. Rigid motions (rotations, reflections, translations) preserve congruence.

Step2: Analyze each option

A rotation of - 90 degrees (clock - wise) about vertex C followed by a reflection across $\overline{CA'}$ can be used to map $\triangle ABC$ onto $\triangle A''B''C''$ in a way that aligns the corresponding angles and the included side as required by ASA. A 360 - degree rotation is equivalent to no rotation at all and may not be the most effective way to show ASA - based congruence. A 90 - degree counter - clockwise rotation about C followed by reflection across $\overline{CA'}$ or a clock - wise rotation about C followed by reflection across $\overline{B'C}$ may not align the triangles in the way that directly shows ASA congruence.

Answer:

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