Rigid motion follows these criteria because the motion is rigid meaning that everything that is moving stays the same except for the location of the entire figure.  There are three common types of rigid motion; translation, reflection, and rotation.

This is the correct definition in terms of rigid motions.  Some of the other options are correct definitions for congruence but do not mention the criteria of there being rigid motion between the two figures.  An example of this is that  and  are congruent because they are a reflection of one another.  Their vertices that map to each other are

First, we need to establish a vector that maps at least one pair of vertices.  We will use  to establish a translation between the two figures.  This also maps  to .

Now they share a vertex and we are able to rotate them together mapping  to  and  to .

Now we can reflect across  mapping  to ,  to , and  to .

So the order of the series of rigid motions is translation, rotation, reflection.

Consider the triangles  and , where . We are able to rotate them together mapping  to  and  to .

Now we can reflect across  mapping  ro , and  to .

Now we are left with the two congruent triangles lying on top of one another, proving that the rigid motions that map these two triangles to one another are rotation and reflection.

Consider the following triangles,  and . They share the side .  If we reflect triangle  across , we match up all congruent sides, mapping them to one another and mapping  to ,  to , and  to , proving these two triangles congruent.

Let’s first begin by showing that these two triangles are congruent through a series of rigid motions.  Let’s use our point of reference be  and since we know that these angles are congruent through the information given in the picture.  We are able to map  to  by reflecting  along the line .  So these two triangles are congruent.

Now we will show that these two triangles are congruent through another theorem.  We see that there are two pairs of corresponding congruent angles,  , and the angles’ included sides are congruent as well, .  The ASA Theorem states that if two triangles share two pairs of corresponding congruent angles and their included sides are congruent, then these two triangles are congruent.  So by ASA, these triangles are congruent.

We can see that .  We know that  by reflexive property.  So by the SSS Theorem, these two triangles are congruent.  We can also reflect triangle  across line  to map the remaining angles to one another;  to .  So these triangles are proven congruent through reflection as well.

This becomes more clear with the orange line between the two triangles.  If flipped over this orange line, the two figures would match up their corresponding congruent parts creating the same triangle.

The SAS Theorem states that if two triangles share two pairs of corresponding congruent sides are congruent and their included angle is also congruent, then these two triangles are congruent.  We are given that   and so these two triangles are congruent by the SAS theorem. We are able to map  to  by rotating  180 degrees clockwise. So these two triangles are congruent by rigid motion.

First, the two triangles  and  share a vertex, so we know that  maps to  by the reflective property. Knowing this, we are able to rotate  to match the congruent sides  and .  This maps  to .  We can also note that  maps to .

Now we can reflect the triangle  across  to map  to ,  to  and  to .

So the order of rigid motions is rotation, reflection.