We present a comparison of the observed, spatially integrated stellar and ionized gas velocity dispersions of $sim1000$ massive ($log M_{star}/M_{odot}gtrsim,10.3$) galaxies in the Large Early Galaxy Astrophysics Census (LEGA-C) survey at $0.6lesssim,zlesssim1.0$. The high $S/Nsim20{rmAA^{-1}}$ afforded by 20 hour VLT/VIMOS spectra allows for joint modeling of the stellar continuum and emission lines in all galaxies, spanning the full range of galaxy colors and morphologies. These observed integrated velocity dispersions (denoted as $sigma_{g, int}$ and $sigma_{star, int}$) are related to the intrinsic velocity dispersions of ionized gas or stars, but also include rotational motions through beam smearing and spectral extraction. We find good average agreement between observed velocity dispersions, with $langlelog(sigma_{g, int}/sigma_{star, int})rangle=-0.003$. This result does not depend strongly on stellar population, structural properties, or alignment with respect to the slit. However, in all regimes we find significant scatter between $sigma_{g, int}$ and $sigma_{star, int}$, with an overall scatter of 0.13 dex of which 0.05 dex is due to observational uncertainties. For an individual galaxy, the scatter between $sigma_{g, int}$ and $sigma_{star, int}$ translates to an additional uncertainty of $sim0.24rm{dex}$ on dynamical mass derived from $sigma_{g, int}$, on top of measurement errors and uncertainties from Virial constant or size estimates. We measure the $zsim0.8$ stellar mass Faber-Jackson relation and demonstrate that emission line widths can be used to measure scaling relations. However, these relations will exhibit increased scatter and slopes that are artificially steepened by selecting on subsets of galaxies with progressively brighter emission lines.
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