Anger involving theologian's conundrum (6)
I believe the answer is:
riddle
'conundrum' is the definition.
(synonyms)
'anger involving theologian's' is the wordplay.
'anger' becomes 'rile' (I've seen this before).
'involving' is an insertion indicator.
'theologian' becomes 'dd' (Doctor of Divinity degree).
'rile' enclosing 'dd' is 'RIDDLE'.
(Other definitions for riddle that I've seen before include "mystery" , "Ingenious puzzle - sieve" , "Perforate with gunshot - poser" , "The large sieve is a puzzle" , "Strain" .)