Use the Polygonal Shape Creation (S) tool 
to split shapes. When the first and last vertex are snapped to an edge or vertex of the same shape, and the whole line is contained in this face, it is automatically split into two parts.
Automatic closing
When the first vertex snaps to a shape, and the last vertex snaps to an edge that is indirectly connected to the first vertex, the polygon is automatically closed along the indirect connection, as shown here:
