The word angel is not an English word, but a Greek word, ἄγγελος (angelos). It means messenger. Throughout the story of Christ’s birth in the Gospels of Matthew and Luke, you’ll find God sending His angels with His messages to explain to people exactly what He is doing. Notice also the messages given before the birth of Christ begin by the angel saying don’t be afraid.
Luke records the appearances of angels to Zacharias, Mary, and the shepherds. Matthew records the appearances of an angel of the Lord to Joseph in three different dreams. Matthew also states the Magi were warned in a dream not to return to Herod, but doesn’t specifically say the warning was given by an angel.
The angel Gabriel is sent to Zacharias (Luke 1:11–20) about the coming birth of John, the forerunner of Messiah; six months later Gabriel is sent from God to Mary (Luke 1:26–38) to tell her that she will be the mother of the Son of God; and then an angel of the Lord appears to Joseph (Matthew 1:18–21) in a dream after he hears Mary is pregnant and tells him not to be afraid to marry her, explaining her child is conceived of the Holy Spirit, and “He will save His people from their sins.”
When Jesus is born an angel of the Lord appear to shepherds in the fields (Luke 2:8–14) near Bethlehem,

Then all at once:
And on earth peace among men with whom He is pleased.”
Matthew tells us of the final appearances of angels in the Christmas story. After the Magi leave, having been warned in a dream not to return to Herod (Matthew 2:12), an angel of the Lord appears to Joseph (Matthew 2:13) in a dream and warns him to take Jesus and Mary and flee to Egypt and stay there until the angel tells him because Herod is going to search for the young Jesus to kill Him. The angel appears to Joseph (Matthew 2:19) a last time after Herod is dead to tell him to return to Israel.
So when we speak of the birth of the Lord Jesus, the angels are always recalled, and when we sing Christmas songs, in many of them we sing about angels, because at that first Christmas God sent those angels with messages to encourage, to guide, to guard, and He sent them with the message of good news of great joy—for Zacharias, for Mary, for Joseph, for the shepherds, and for all of us—that a Savior was born.
Archangel Gabriel struck dumb Zachariah: Alexandr Ivanov. Public Domain.
The Annunciation: Henry Ossawa Tanner. Public Domain.
Dream of Saint Joseph: Antonio Ciseri. Public Domain.
The Shepherds and the Angel: Carl Bloch. Public Domain.
From YouTube: “This Flashmob was performed by members and friends of the “Berliner Stadtmission” in a department store in Berlin back in 2014. It is a traditional Christmas Carol with new German lyrics written by Martin Buchholz who also directed the Video. You find the translation of the lyrics in the subtitles. Merry Christmas everybody! And Greetings from Germany!”
Via the German Wikipedia: “the Berliner Stadtmission is an independent association under the umbrella of the Evangelical Church Berlin-Brandenburg-Silesian Upper Lusatia.”
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