Considering the teaching of 1 Corintians 11, which begins with the command, "You are to imitate me, just as I imitate Christ." and the following exhortation to, "understand that the head of every man is Christ, and the head of the woman is man, and the head of Christ is God."
We can go to John 8 for an idea of the example of that which Paul imitates in Christ honoring God as His head, as He explains in verse 49 saying, "I do not have a demon," Jesus replied, "but I honor my Father, and you dishonor Me." and in verse 50 saying, "I do not seek My own glory, there is One who seeks it, and He is the Judge." 51"Truly, truly, I tell you, if anyone keeps My word, he will never see death."
And then after the Jews doth protested too much, Jesus continues in verse 54 answering, "If I glorify Myself, My glory means nothing. The One who glorifies Me is My Father, of whom you say, 'He is our God.'..."
Now, in keeping the order of authority and relative to understanding that the head of the every man being Christ, we see Jesus transferring His authority saying, "Behold, I give you authority to trample serpents and scorpions, and over all the power of the enemy..." and can examine how this might look relative to the head of the woman being man (note to self, research whether usage of 'man' is correctly understood as singular or plural relative to the use of the singular in reference to 'the woman').
And arrive at 1 Corinthians 11:10 which discusses the roles of worship which says, "For this reason a woman ought to have a sign of authority on her head, because of the angels.", and wherein the KJV says, "For this cause ought the woman to have power on [her] head because of the angels."
This 'reason' or 'cause' is explained as her handicap, methinks, that she, indeed, is considered the weaker vessel, not unlike that man is the weaker in comparison to Christ, I would suppose, and so greatly benefits by that [sign of] authority bestowed to her by the man. What then, of her, and the state of the church in general, if this is systematically withheld from her?