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 43GHz 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|<10{deg}) at 2, 5 and 8GHz, 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{deg}<l<260{deg} (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* 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.