Question

△abc is congruent to △abc because you can map △abc to △abc using a reflection across the y - axis, which is a rigid motion
△abc is congruent to △abc because you can map △abc to △abc using translation 6 units to the right, which is a rigid motion
△abc is not congruent to △abc because there is no sequence of rigid motion that maps △abc to △abc
△abc is congruent to △abc because you can map △abc to △abc using translation 6 units to the left, which is a rigid motion

Explanation:

Step1: Identify rigid - motion rules

Rigid motions include translations, rotations, and reflections. Translations move a figure without changing its shape, size, or orientation. Reflections flip a figure over a line.

Step2: Analyze the transformation

The coordinates of points in $\triangle ABC$ and $\triangle A'B'C'$: For point $A(3,4)$ and $A'(- 3,4)$, $B(4,1)$ and $B'(-4,1)$, $C(2,1)$ and $C'(-2,1)$. The transformation from $\triangle ABC$ to $\triangle A'B'C'$ is a reflection across the $y$ - axis. The rule for reflecting a point $(x,y)$ across the $y$ - axis is $(x,y)\to(-x,y)$. Reflection is a rigid motion, and rigid motions preserve congruence.

Answer:

$\triangle ABC$ is congruent to $\triangle A'B'C'$ because you can map $\triangle ABC$ to $\triangle A'B'C'$ using a reflection across the $y$-axis, which is a rigid motion.