From Flappers to Rappers: American Youth Slang by Dr. Thomas Dalzell cites the 1930s expression "cheek it" as meaning to bluff. I don't quite understand why and I'm hoping someone on here may help me to better understand this.
I tried to ngrams this guy but nothing much came up to be honest -- the only remotely close thing I could find was this
I may find myself momentarily beset by an irrational impulse; but it does not reach my muscles because I cheek it, or my ingrained habits cheek it for me. And so again if you wish to train me in automatic writing and ask me to let my hand rest ...
The Harvey Lectures, Volume 3 Academic Press, 1909
doesn't really seem to fit the criteria though....
In current UK and Irish slang, 'cheek' is to be naughty or mischievous. It can also be trying to get away with something. It is used in relation to children and adults but sort of in a childish way. A child can be 'cheeky' when trying to get away with something. 'They've some cheek', would mean being bold about being disingenuous. That would be used in regard to politicians when they are probably telling lies about past knowledge of said event, for instance.
I would say that this reference stems from this use of the work 'cheek'.