Why do Jehovah's Witnesses make their women cover their heads when leading a prayer/preaching to a man?

they have brought in this teaching that it all depends on whether a male is geographically present when she does this. 1 Cor 11:10 says "Therefore the woman has to have authority at her head for/because of the angels" - the words "sign of authority" have been added in here by a lot of bibles, but this is not in the original Koine provided in the Kingdom Interlinear Translation of the Greek Scriptures (of Jehovah's Witnesses) - they have admitted to me that they did in fact consult other bible translations to come to this conclusion and did not compare internal linguistic evidence from within the Greek scriptures to come to this conclusion. I think this teaching is in breach of Mt 15:2, 8,9 and is treating the man as like a lord or an idol.