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neither entirely open nor entirely shut; partly open:
The door was ajar.
in contradiction to; at variance with:
a story ajar with the facts.
Contemporary Examples

The former girlfriend of ajar, Ames was also friends with Roy Lopez, the other alleged “fence,” and buddies with Lee.
The Real Bling Ring: Where Are They Now? Tricia Romano May 20, 2013

Postscript: According to his Instagram and Facebook feed, ajar is still living large.
The Real Bling Ring: Where Are They Now? Tricia Romano May 20, 2013

Nothing especially surprising there, except that she kept opening the door instead of leaving it ajar while cleaning.
The Martyrdom of DSK Christopher Dickey July 4, 2011

Historical Examples

The door was ajar: I entered hastily and pushed it to, but he had followed me on the instant, and now stood with it in his hand.
Lippincott’s Magazine of Popular Literature and Science, Vol. XII. No. 31. October, 1873. Various

Then, as the door of the first reception-room was ajar, he at last ventured in.
The Three Cities Trilogy, Complete Emile Zola

Benedicta’s door was ajar, and I stood outside a long time, gazing upon the beautiful picture presented to my eyes.
The Monk and The Hangman’s Daughter Adolphe Danziger De Castro and Ambrose Bierce

It was broad daylight, and the door leading into the prim little hall was ajar.
Henry Dunbar M. E. Braddon

And when she had let Serge knock at the door twice, she softly set it ajar, and received him with an old-fashioned courtesy.
Abbe Mouret’s Transgression Emile Zola

The landing at the top was dark, but the door at the rear was ajar.
Kent Knowles: Quahaug Joseph C. Lincoln

The door was ajar, but he could see that there was some person standing out in the hall.
Harper’s Round Table, August 6, 1895 Various

adjective, adverb (postpositive)
(esp of a door or window) slightly open
adjective
(postpositive) not in harmony

1718, perhaps from Scottish dialectal a char “slightly open,” earlier on char (early 16c.), from Middle English char, from Old English cier “a turn.”