  [Abridged] We test the evolutionary model of cool close binaries on the
observed properties of near contact binaries (NCBs). Those with a more massive
component filling the Roche lobe are SD1 binaries whereas in SD2 binaries the
Roche lobe filling component is less massive. Our evolutionary model assumes
that, following the Roche lobe overflow by the more massive component (donor),
mass transfer occurs until mass ratio reversal. A binary in an initial phase of
mass transfer, before mass equalization, is identified with SD1 binary. We show
that the transferred mass forms an equatorial bulge around the less massive
component (accretor). Its presence slows down the mass transfer rate to the
value determined by the thermal time scale of the accretor, once the bulge
sticks out above the Roche lobe. It means, that in a binary with a (typical)
mass ratio of 0.5 the SD1 phase lasts at least 10 times longer than resulting
from the standard evolutionary computations neglecting this effect. This is why
we observe so many SD1 binaries. Our explanation is in contradiction to
predictions identifying the SD1 phase with a broken contact phase of the
Thermal Relaxation Oscillations model. The continued mass transfer, past mass
equalization, results in mass ratio reversed. SD2 binaries are identified with
this phase. Our model predicts that the time scales of SD1 and SD2 phases are
comparable to one another. Analysis of the observations of 22 SD1 binaries, 27
SD2 binaries and 110 contact binaries (CBs) shows that relative number of both
types of NCBs favors similar time scales of both phases of mass transfer. Total
masses, orbital angular momenta and orbital periods of SD1 and SD2 binaries are
indistinguishable from each other whereas they differ substantially from the
corresponding parameters of CBs. We conclude that the results of the analysis
fully support the model presented in this paper.

Solar and Stellar Astrophysics