I agree with you that context is essential, and I agree that the context here is headship and authority, but why would women need to do something in obedience to angels? For obedience to her husband, and in turn to God, absolutely, yet it is a puzzling phrase, it sounds like a warning.
Those first three words of verse 11 are indeed good reason to examine context. Like Kent Hovind once said "whenever you read the word 'therefor', you'd better find out what its there for!"
1Co 11:7 For a man indeed ought not to cover his head, forasmuch as he is the image and glory of God: but the woman is the glory of the man.
1Co 11:8 For the man is not of the woman; but the woman of the man.
1Co 11:9 Neither was the man created for the woman; but the woman for the man.
1Co 11:10 For this cause ought the woman to have power on her head because of the angels.
'For this cause', to me anyways, links it to verse 9, which references creation. The woman was created for the man, not the other way around.
This leads me to ponder Christ as the last Adam.
Bear with me for a moment;
All of creation was made to glorify God, right? To point towards His glory.
Adam was created perfect, then God created Eve
for Adam, to make the relationship complete, to realize Adam's full potential through love.
Here's the mindbender: Christ wasn't sent only because of mankind's sin, but God allowed mankind to sin in order for Christ to be glorified in his act of redemption.
Christ wasn't prepared for his bride, rather his bride was prepared for him: to God be the glory!
So, since Eve was deceived by an (fallen) angel in the garden, it seems like the admonition for the woman is to guard herself and remain under the authority of her husband. After all, the commandment from the LORD came directly to Adam not to eat of the tree of the knowledge of good and evil before Eve was even created. Its presumable then, that Adam repeated this commandment to Eve once she came on the scene.
So then, had Eve been obedient unto Adam, she would in turn have been obedient unto God.
Gen 2:15 And the LORD God took the man, and put him into the garden of Eden to dress it and to keep it.
Gen 2:16 And the LORD God commanded the man, saying, Of every tree of the garden thou mayest freely eat:
Gen 2:17 But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.
Gen 2:18 And the LORD God said, It is not good that the man should be alone; I will make him an help meet for him.
Here, the LORD God said that Adam (type of Christ) should not be alone, so he prepared an help meet (bride) for him.
So, all this to wonder why a woman "ought the woman to have power on her head because of the angels"?
Does power on her head mean that she herself exerts power over her own head (self-control, obedience, wear a hat in church?), or does it mean that the power being referred to is actually the headship of her husband?
As for the angels, it could be for fallen angels seeking to deceive, or it could also be for God's good angels. Either way, both the fallen and unfallen angels seem to desire to look into the things of mankind. (1 Peter 1:12)
If a woman therefor abides under headship, she trounces the fallen ones, and if a woman allows herself to be deceived out from under the headship, displeases the unfallen ones.
I hope I havn't angered anyone with these thoughts. I just thought I'd share my thoughts, and in no way am I saying its a doctrine. I think Winman brought up a very interesting passage that I had been curious about for a while as well. Before posting, I read each chapter slowly implicated here so far, and then this idea solidified. I'm interested to see what others think of it.
Does this mean then that all women should wear hats in church? I know some Baptists do this: I honestly don't know if we should or not. I'm sure God is more concerned with our preserving of the headship ordinance he laid out, but it does beg one to consider it.
And just in case I'm not going too far off topic
yet, here's a side-note:
I had very long hair for a few years, and it was by coming across verse 14 of this chapter that made me go and get it cut short and proper, I also had my gold earnings taken out at the same time. (about 2.5 years ago).