Puzzle as theologian's overcome by anger (6)
I believe the answer is:
riddle
'puzzle' is the definition.
(riddling is a kind of puzzling)
'theologian's overcome by anger' is the wordplay.
'theologian' becomes 'dd' (Doctor of Divinity degree).
'overcome by' is an insertion indicator.
'anger' becomes 'rile' (I've seen this in other clues).
'dd' inserted into 'rile' is 'RIDDLE'.
'as' acts as a link.
(Other definitions for riddle that I've seen before include "Gradual assimilation of ideas or knowledge" , "It may have an answer" , "Puzzle or strainer" , "mystery" , "Run through" .)