If Luke was worried about his story sounding believable, he might have left the angel out! Even people in the first century would have been sceptical about this event. But Luke had done his research, and he knew that he must include the angel, whether or not it seemed believable. God sent a messenger to inform people of what was going to happen.
Two pregnancies were announced; firstly, Zechariah’s wife Elizabeth was going to have a son, despite husband and wife being elderly. As an adult, this son would prepare people for the coming of the Lord (verse 17).
Secondly, six months later, Elizabeth’s relative Mary was told that she was also going to have a son. But this was bewildering to Mary, because she was a virgin. In Mary’s day, remaining a virgin until marriage was expected. Failure to maintain her virginity would have resulted in disgrace or worse. Mary would have been extremely guarded in her interactions with men.
But Mary was told that this child would not have a biological father, but would be conceived by God's power, and would be called the Son of God! Also, as a descendent of King David, Jesus would be the King who reigns forever.*
The angel also told Mary that her elderly relative Elizabeth was 6 months pregnant, since “nothing will be impossible with God" (sometimes translated, "No word from God will ever fail", verse 37.) Those words are so important, because these were not typical events. Normally, they are impossible! But if the everlasting Creator God intervened in human history by sending His own Son, what would that look like? Surely the people living at the time would have some pretty incredible stories to tell!