The Torah recounts that three angels appeared to Abraham (Genesis 18:2). Later, only two angels go down to Sodom. What happened to the third angel? And what is the significance of three versus two?
The Aish Rabbi Replies:
Although the Torah does not mention the names of the angels that went to visit Abraham, the Talmud tells us they were Raphael, Michael and Gabriel. (Bava Metzia 86b)
Throughout most of their interaction, the Torah does not refer to them on an individual basis, but rather as a group, as it is written, "[Abraham] stood over THEM beneath the tree and THEY ate. [Afterwards,] THEY said to him..." (Genesis 18:8-9)
There are, however, instances in which the angels act as individuals. For example, only the angel Michael told Sarah that she was going to have a baby (Genesis 18:10). In Sodom, only the angel Raphael told Lot to flee from the city (Genesis 19:17). Similarly, it was only the angel Gabriel that informed Lot that the city was going to be destroyed.
From these verses, we can derive that an angel can only perform one mission: Gabriel's mission was to destroy Sodom; Michael's mission was to inform Sarah that she would give birth in a year's time; Raphael's mission was to heal Abraham and save Lot. (Although Raphael's mission included two tasks, they were considered a single mission since they were both acts that saved people). (Talmud - Bava Metzia 86b, Rashi - Genesis 18:2 and 19:16)
This now helps us understand your question: Why did three angels come to Abraham, but only two go to Lot?
The answer is that since Michael's sole mission was to tell Sarah she would become pregnant, he had no business in Sodom. Therefore, upon leaving Abraham's tent he went back to his post at God's Holy Throne. But Gabriel, who was left with his task of destroying Sodom, and Raphael, who was commanded to save Lot, had not yet finished their tasks, and continued to Sodom. (Rashi - Genesis 19:1 and 9:16)