Final answer:
The nucleus and mitochondria likely evolved similarly through the process described by the endosymbiotic theory, due to both having two phospholipid bilayers and their own DNA.
Step-by-step explanation:
The membrane-enclosed organelles that most likely evolved in a similar manner are the nucleus and mitochondria. This is supported by the endosymbiotic theory, which posits that these organelles were once independent prokaryotes that were engulfed by an ancestral eukaryotic cell. Both organelles are surrounded by two phospholipid bilayers and contain their own DNA, which is evidence of their prokaryotic origins. In contrast, the other listed pairs, such as the Golgi apparatus and endoplasmic reticulum, or vacuole and lysosome, are part of the endomembrane system, which is thought to have evolved through infoldings in the plasma membrane of an ancestral eukaryotic cell.