Our church also teaches this. In a recent sermon, our Pastor taught that the Angel of the Lord can be used interchangeably with Christ. As we believe in the Trinity, this means the Angel of the Lord is also God.
The following is some (but not all) of the Scriptural evidence used to support this teaching:
From Baker's Evangelical Dictionary:
The connection between the angel of the Lord and the preincarnate
appearance of the Messiah cannot be denied. Manoah meets the angel of
the Lord, and declares that he has seen God. The angel accepts worship
from Manoah and his wife as no mere angel, and refers to himself as
"Wonderful, " the same term applied to the coming deliverer in Isaiah
9:6 ( Jud 13:9-22 ). The functions of the angel of the Lord in the Old
Testament prefigure the reconciling ministry of Jesus. In the New
Testament, there is no mention of the angel of the Lord; the Messiah
himself is this person.
Scriptural instances when the Angel of the Lord speaks as if He were God:
Genesis 16:9-12 (KJV) 9 And the angel of the Lord said unto her, Return to thy mistress, and submit thyself under her hands.
10 And the angel of the Lord said unto her, I will multiply thy seed
exceedingly, that it shall not be numbered for multitude.
11 And the angel of the Lord said unto her, Behold, thou art with
child and shalt bear a son, and shalt call his name Ishmael; because
the Lord hath heard thy affliction.
12 And he will be a wild man; his hand will be against every man, and
every man's hand against him; and he shall dwell in the presence of
all his brethren.
Exodus 3:2-4 (KJV) 2 And the angel of the Lord appeared unto him in a flame of fire out of the midst of a bush: and he looked, and,
behold, the bush burned with fire, and the bush was not consumed.
3 And Moses said, I will now turn aside, and see this great sight, why
the bush is not burnt.
4 And when the Lord saw that he turned aside to see, God called unto
him out of the midst of the bush, and said, Moses, Moses. And he said,
Here am I.
Judges 2:1-2 (KJV) 1 And an angel of the Lord came up from Gilgal to Bochim, and said, I made you to go up out of Egypt, and have
brought you unto the land which I sware unto your fathers; and I said,
I will never break my covenant with you.
2 And ye shall make no league with the inhabitants of this land; ye
shall throw down their altars: but ye have not obeyed my voice: why
have ye done this?
(There are more, but I'll stop here.)
The Angel of the Lord also accepts worship:
Judges 6:19-21 (KJV) 19 And Gideon went in, and made ready a kid, and unleavened cakes of an ephah of flour: the flesh he put in a
basket, and he put the broth in a pot, and brought it out unto him
under the oak, and presented it.
20 And the angel of God said unto him, Take the flesh and the
unleavened cakes, and lay them upon this rock, and pour out the broth.
And he did so.
21 Then the angel of the Lord put forth the end of the staff that was
in his hand, and touched the flesh and the unleavened cakes; and there
rose up fire out of the rock, and consumed the flesh and the
unleavened cakes. Then the angel of the Lord departed out of his
Also, like Christ, there are times when the Angel of the Lord is treated as distinct from God.
Judges 13:9 (KJV) 9 And God hearkened to the voice of Manoah; and the angel of God came again unto the woman as she sat in the field:
but Manoah her husband was not with her.
Like Christ, the Angel has the authority to forgive sin.
Zechariah 3:3-4 (KJV) 3 Now Joshua was clothed with filthy garments, and stood before the angel.
4 And he answered and spake unto those that stood before him, saying,
Take away the filthy garments from him. And unto him he said, Behold,
I have caused thine iniquity to pass from thee, and I will clothe thee
with change of raiment.