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late 14c., from anger (n.) + -y (2). Originally "full of trouble, vexatious;" sense of "enraged, irate" also is from late 14c. The Old Norse adjective was ongrfullr "sorrowful," and Middle English had angerful "anxious, eager" (mid-13c.). The phrase angry young man dates to 1941 but was popularized in reference to the play "Look Back in Anger" (produced 1956) though it does not occur in that work.
"There are three words in the English language that end in -gry. Two of them are angry and hungry. What is the third?" There is no third (except some extremely obscure ones). Richard Lederer calls this "one of the most outrageous and time-wasting linguistic hoaxes in our nation's history" and traces it to a New York TV quiz show from early 1975.