3 But I want you to realize that the head of every man is Christ, and the head of the woman is man, and the head of Christ is God. 4Every man who prays or prophesies with his head covered dishonors his head. 5But every woman who prays or prophesies with her head uncovered dishonors her head—it is the same as having her head shaved. 6For if a woman does not cover her head, she might as well have her hair cut off; but if it is a disgrace for a woman to have her hair cut off or her head shaved, then she should cover her head.
7 A man ought not to cover his head, since he is the image and glory of God; but woman is the glory of man. 8For man did not come from woman, but woman from man; 9neither was man created for woman, but woman for man. 10 It is for this reason that a woman ought to have authority over her own head, because of the angels.
Does this mean that angels observe our church meetings?
In the same way, I tell you, there is rejoicing in the presence of the angels of God over one sinner who repents.
The NT describes the angels as "ministering spirits" in a significant way:
Heb 1:14 - Are not the angels ministering spirits sent to serve those who will inherit salvation?
We see evidence many times of this in the Bible -
Ps 91:11 - For he will command his angels concerning you to guard you in all your ways
Dan 6:22 - My God sent his angel, and he shut the mouths of the lions.
Acts 12:11 - Then Peter came to himself and said, “Now I know without a doubt that the Lord has sent his angel and rescued me from Herod’s clutches and from everything the Jewish people were hoping would happen.”
Acts 12:15 - “You’re out of your mind,” they told her. When she kept insisting that it was so, they said, “It must be his angel.”
Gen 24:40 - “He replied, ‘The LORD, before whom I have walked faithfully, will send his angel with you and make your journey a success, so that you can get a wife for my son from my own clan and from my father’s family.
Dan 3:28 - Then Nebuchadnezzar said, “Praise be to the God of Shadrach, Meshach and Abednego, who has sent his angel and rescued his servants! They trusted in him and defied the king’s command and were willing to give up their lives rather than serve or worship any god except their own God.
Luke 15:10 - In the same way, I tell you, there is rejoicing in the presence of the angels of God over one sinner who repents.”
Eph 3:10 - His intent was that now, through the church, the manifold wisdom of God should be made known to the rulers and authorities in the heavenly realms
1 Cor 4:9 - For it seems to me that God has put us apostles on display at the end of the procession, like those condemned to die in the arena. We have been made a spectacle to the whole universe, to angels as well as to human beings.
Ps 34:7 - The angel of the LORD encamps around those who fear him, and he delivers them.
Thus, it appears that the angels of God take a keen interest in the affairs of the world in various ways and thus see all the events on earth, including those in the church.
Do angels observe our church meetings according to 1 Corinthians 11:10?
Yes, they are very interested in what happens here on earth, especially with the Christian congregation.
This was so that now, through the congregation, there might be made known to the governments and the authorities in the heavenly places the greatly diversified wisdom of God.-Eph 3:10 (bold mine)
It was revealed to them that they were ministering, not to themselves, but to you, regarding what has now been announced to you by those who declared the good news to you with holy spirit sent from heaven. Into these very things, angels are desiring to peer.-1 Peter 1:12 (bold mine)
For it seems to me that God has put us the apostles last on exhibition as men condemned to death, because we have become a theatrical spectacle to the world, and to angels and to men.-1 Cor 4:9 (bold mine)
In the web article Who or What Are Angels? the subheading "Do angels help people?" gives additional scriptural points:
Yes, God uses his faithful angels to help people today.