  We have measured the angular sizes of radio cores of active galactic nuclei
(AGN) and analyzed their sky distributions and frequency dependencies to study
synchrotron opacity in AGN jets and the strength of angular broadening in the
interstellar medium. We have used archival very long baseline interferometry
(VLBI) data of more than 3000 compact extragalactic radio sources observed at
frequencies, $\nu$, from 2 to 43 GHz to measure the observed angular size of
VLBI cores. We have found a significant increase in the angular sizes of the
extragalactic sources seen through the Galactic plane ($|b|\lesssim10^\circ$)
at 2, 5 and 8 GHz, about 1/3 of which show significant scattering. These
sources are mainly detected in directions to the Galactic bar, the Cygnus
region, and a region with galactic longitudes $220^\circ\lesssim
l\lesssim260^\circ$ (the Fitzgerald window). The strength of interstellar
scattering of the AGNs is found to correlate with the Galactic H$\alpha$
intensity, free-electron density, and Galactic rotation measure. The dependence
of scattering strengths on source redshift is insignificant, suggesting that
the dominant scattering screens are located in our Galaxy. The observed angular
size of Sgr A$^\ast$ is found to be the largest among thousands of AGN observed
over the sky; we discuss possible reasons of this strange result. Excluding
extragalactic radio sources with significant scattering, we find that angular
size of opaque cores in AGN scales typically as $\nu^{-1}$ confirming
predictions of a conical synchrotron jet model with equipartition.
