Line dividing closed plane figure into congruent parts

Question

Today in class a student observed and asked the following:

While the diagonal of a rectangle is not a line of symmetry, it does cut the rectangle into two congruent parts. This is a noteworthy property. Is there a name for this kind of line segment (one that cuts a closed plane figure into two new congruent closed plane figures)?

This stumped me. It also brought to mind a more general question: Is there a name for a line segment that cuts a closed plane figure into two new closed plane figures with equal area?

Answer 1

A line of symmetry *including rotational symmitry as Doug M observes above ) always divides a figure into two symmitrical parts which will be called congruent figures in case the initial figure was a closed figure.

Is every line which divides thus a line of symmetry....i leave that question for now

Answer 2

A line that cuts a polygon into two congruent polygons does not necessarily correspond to a symmetry of the original polygon. Consider e.g.

A line that cuts a plane region into regions of equal area is sometimes called an area bisector of the region.