We investigate the characteristics of two newly discovered short-period, double-lined, massive binary systems, VFTS 450 (O9.7$;$II--Ib$,$+$,$O7::) and VFTS 652 (B1$;$Ib$,+,$O9:$;$III:). We perform model-atmosphere analyses to characterise the photospheric properties of both members of each binary (denoting the `primary as the spectroscopically more conspicuous component). Radial velocities and optical photometry are used to estimate the binary-system parameters. We estimate $T_{rm eff}=27$ kK, $log{(g)}=2.9$ (cgs) for the VFTS 450 primary spectrum (34kK, 3.6: for the secondary spectrum); and $T_{rm eff} = 22$kK, $log{(g)}=2.8$ for the VFTS 652 primary spectrum (35kK, 3.7: for the secondary spectrum). Both primaries show surface nitrogen enrichments (of more than 1 dex for VFTS 652), and probable moderate oxygen depletions relative to reference LMC abundances. We determine orbital periods of 6.89d and 8.59d for VFTS 450 and VFTS 652, respectively, and argue that the primaries must be close to filling their Roche lobes. Supposing this to be the case, we estimate component masses in the range $sim$20--50M$_odot$. The secondary spectra are associated with the more massive components, suggesting that both systems are high-mass analogues of classical Algol systems, undergoing case-A mass transfer. Difficulties in reconciling the spectroscopic analyses with the light-curves and with evolutionary considerations suggest that the secondary spectra are contaminated by (or arise in) accretion disks.
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