Clarification of the definition of two dimensional motion of rigid body

Question

Our teacher gave the following definition:

Motion of rigid body is said to be two-dimensional if

1. Each particle of the rigid body moves in a plane parallel to some space-fixed plane.

2. The motion of particles of the rigid body which are in contact with the straight line perpendicular to the space-fixed plane are identical.

I do not understand either of the two points geometrically. Any explanation will be highly useful. Thank you.

Answer 1

For example, if the fixed plane is the $xy$-plane, then the movement of a particle with initial coordinates $(x,y,z)$ must be such that only $x$ and $y$ change, but $z$ remains constant.

I cannot really parse the sentence under 2, but suppose that the following is meant: If two particles have same $x$ and $y$ coordinates at one point in time, but differ only in their $z$ coordinate, then this relation also holds for other points in time.

Note that for a non-deformable rigid body the second condition follows from the first, which already allows only translations parallel to the plane and rotations around axes perpendicular to the plane.