  Strong lensing time-delay systems constrain cosmological parameters via the
so-called time-delay distance and the angular diameter distance to the lens. In
previous studies, only the former information was used. In this paper, we show
that the cosmological constraints improve significantly when the latter
information is also included. Specifically, the angular diameter distance plays
a crucial role in breaking the degeneracy between the curvature of the Universe
and the time-varying equation of state of dark energy. Using a mock sample of
55 bright quadruple lens systems based on expectations for ongoing/future
imaging surveys, we find that adding the angular diameter distance information
to the time-delay distance information and the cosmic microwave background data
of Planck improves the constraint on the constant equation of state by 30%, on
the time variation in the equation of state by a factor of two, and on the
Hubble constant in the flat $\Lambda$CDM model by a factor of two. Therefore,
previous forecasts for the statistical power of time-delay systems were
significantly underestimated, i.e., time-delay systems are more powerful than
previously appreciated.
