Question

△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 right, which is a rigid motion
△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 left, which is a rigid motion

Explanation:

Step1: Identify coordinates

Let the coordinates of $\triangle ABC$ be $A(3,4)$, $B(4,1)$, $C(2,1)$ and of $\triangle A'B'C'$ be $A'(- 3,4)$, $B'(-2,1)$, $C'(-4,1)$.

Step2: Analyze rigid - motion

A reflection across the $y$ - axis changes the sign of the $x$ - coordinate of each point. If we reflect point $(x,y)$ across the $y$ - axis, we get $(-x,y)$.
For point $A(3,4)$ reflected across the $y$ - axis, we get $A'(-3,4)$.
For point $B(4,1)$ reflected across the $y$ - axis, we get $B'(-2,1)$.
For point $C(2,1)$ reflected across the $y$ - axis, we get $C'(-4,1)$.
Since reflection is a rigid - motion and we can map $\triangle ABC$ to $\triangle A'B'C'$ using a reflection across the $y$ - axis, the two triangles are congruent.

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.