Gen 18:20; And the LORD said, "The outcry of Sodom and Gomorrah is indeed great, and their sin is exceedingly grave.
Gen 18:21; "I will go down now, and see if they have done entirely according to its outcry, which has come to Me; and if not, I will know."
Gen 18:22; Then the men turned away from there and went toward Sodom, while Abraham was still standing before the LORD.
We see the LORD is not an angel, because the "other men turned away" but the Lord was still standing there.
Gen 19:1; Now the two angels came to Sodom in the evening as Lot was sitting in the gate of Sodom. When Lot saw them, he rose to meet them and bowed down with his face to the ground.
Gen 19:2; And he said, "Now behold, my lords, please turn aside into your servant's house, and spend the night, and wash your feet; then you may rise early and go on your way." They said however, "No, but we shall spend the night in the square."
We also see that angels can physically touch us.
Gen 19:10; But the men reached out their hands and brought Lot into the house with them, and shut the door.
Gen 19:11; They struck the men who were at the doorway of the house with blindness, both small and great, so that they wearied themselves trying to find the doorway.
Gen 19:13; for we are about to destroy this place, because their outcry has become so great before the LORD that the LORD has sent us to destroy it.
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The "burning bush" that Moses saw is also called an Angel.
Exod 3:2; The angel of the LORD appeared to him in a blazing fire from the midst of a bush; and he looked, and behold, the bush was burning with fire, yet the bush was not consumed.
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God can hide angels from us, and He can reveal them to us also. (not to mention make donkey's talk).
Num 22:21; So Balaam arose in the morning, and saddled his donkey and went with the leaders of Moab.
Num 22:22; But God was angry because he was going, and the angel of the LORD took his stand in the way as an adversary against him. Now he was riding on his donkey and his two servants were with him.
Num 22:23; When the donkey saw the angel of the LORD standing in the way with his drawn sword in his hand, the donkey turned off from the way and went into the field; but Balaam struck the donkey to turn her back into the way.
Num 22:24; Then the angel of the LORD stood in a narrow path of the vineyards, with a wall on this side and a wall on that side.
Num 22:25; When the donkey saw the angel of the LORD, she pressed herself to the wall and pressed Balaam's foot against the wall, so he struck her again.
Num 22:26; The angel of the LORD went further, and stood in a narrow place where there was no way to turn to the right hand or the left.
Num 22:27; When the donkey saw the angel of the LORD, she lay down under Balaam; so Balaam was angry and struck the donkey with his stick.
Num 22:28; And the LORD opened the mouth of the donkey, and she said to Balaam, "What have I done to you, that you have struck me these three times?"
Num 22:29; Then Balaam said to the donkey, "Because you have made a mockery of me! If there had been a sword in my hand, I would have killed you by now."
Num 22:30; The donkey said to Balaam, "Am I not your donkey on which you have ridden all your life to this day? Have I ever been accustomed to do so to you?" And he said, "No."
Num 22:31; Then the LORD opened the eyes of Balaam, and he saw the angel of the LORD standing in the way with his drawn sword in his hand; and he bowed all the way to the ground.
Num 22:32; The angel of the LORD said to him, "Why have you struck your donkey these three times? Behold, I have come out as an adversary, because your way was contrary to me.
Num 22:33; "But the donkey saw me and turned aside from me these three times. If she had not turned aside from me, I would surely have killed you just now, and let her live."