1 Corinthians 11
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. 8 For man did not come from woman, but woman from man; 9 neither 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. 11 Nevertheless, in the Lord woman is not independent of man, nor is man independent of woman. 12 For as woman came from man, so also man is born of woman. But everything comes from God.
These verses can be very confusing if not read in the proper context. First, note the context of Jewish and Gentile women worshipping together. "The matter of wearing hats or head covers, though seemingly insignificant, had become a big problem between two different cultures colliding. Jewish women always covered their heads in worship. For a woman to uncover her head in public was a sign of loose morals. On the other hand, Greek women may have been used to worshipping without head coverings." (NIV Life Application Commentary)
In terms of authority, in Jesus' culture it was the norm for a man to have authority in the house. So Paul acknowledges this but look also at what he says up above, "In the Lord woman is not independent of man, nor is man independent of woman. For as woman came from man, so also man is born of woman but everything comes from God."
This is in keeping with Paul's teaching on men and women in Ephesians where he says to men and women, "Submit to each other out of reverence for Christ." Also in Galatians, where the believers kept wanting to go back to the Law to make distinctives between men and women Paul says, "For all of you who were baptized into Christ have clothed yourselves with Christ. 28There is neither Jew nor Greek, slave nor free, male nor female, for you are all one in Christ Jesus." Galatians 3:28
So you see Paul's logic here. In one sense, because there were certain norms already existing, he wanted to avoid division over insignificant things, especially as the body of Christ gathered together. But in new sense, he is introducing the fact that in Christ, we are all the same and human divisions are gone. We are all sinners. Jesus died for all of us. And we are to serve one another both in the church and in our homes. Can you see how this could be misconstrued? We see women playing a prominent role in the story of Jesus. See Luke's gospel. Cf. Anna the prophetess and also the women being first to go and see the tomb was empty.
As human we like distinctions and power, but Jesus showed us that true power is when we give up our rights, as he emptied self as the Son of God to become human! We ought to follow his lead don't you think?