We present the arcsecond-size radio structure for a complete sample of 14 X-ray-selected BL Lacertae objects (XBLs) from the Einstein Extended Medium-Sensitivity Survey (EMSS). All are resolved by VLA A-configuration 20 cm observations. Most possess radio morphologies which are similar to those of nearby Fanaroff-Riley type 1 (FR 1) radio galaxies once the effects of surface brightness dimming and beam dilution as a function of redshift are taken into account. In order to test the beamed FR 1 hypothesis for BL Lac objects, the core and extended power levels are determined for these sources and compared with samples of radio-selected BL Lac objects (RBLs) and FR 1 from galaxies from the literature. Once necessary morphology-dependent corrections are made to the XBL and RBL samples, which have not been made in previous comparisons of this type, we find that statistically RBLs and XBLs possess extended (and thus unbeamed) power levels and largest angular sizes similar to those of the FR 1 galaxies both supporting the beaming hypothesis and a common parent population for XBLs and RBLs. However, a small number of RBLs have radio properties which appear incompatible with the beamed FR 1 hypothesis; these few may be either examples of beamed Fanaroff-Riley type 2 (FR 2) galaxies or gravitationally lensed sources. For a fixed extended power level, the core powers of the XBLs are intermediate between FR 1 galaxies and RBLs, supporting previous suggestions that XBLs are ''off-axis'' RBLs and that the X-ray beam is broader than the radio beam in BL Lac objects. But some RBLs have core-to-extended flux ratios comparable to those of the most extended XBLs; this is unexpected in the simplest beaming scenario, in which all RBLs would be more extreme than XBLs. In general, the XBLs appear to be a more homogeneous class than the RBLs.