To what extent is Angelo's fall tragic in Measure for Measure?
What exactly do you mean by "tragic"? Certainly he does suffer a massive fall from grace. At the beginning of the play he is presented as if he were characterised by self-control. Others consider that he adheres to a strict moral code. However, when the Duke gives him an opportunity to have absolute power, he shows himself to be just as corruptible and open to temptation as anyone else. Note how Isabella foreshadows his fall when protesting against the sentence imposed on her brother:
If he had been as you, and you as he,
You would have slipp'd like him;
But he, like you, would not have been so stern.
Of course, when he does fall through sleeping with Mariana, he then tries to cover his crime with hypocrisy, leading Isabella to denounce him as a "devil" inspite of the fact that he is "outward-sainted."
Thus he is not a character that excites much sympathy during the course of the play. In particular, once he has decided to proposition Isabella, it seems in his own mind as if there is no turning back. He says "I have begun, and now I give my sensual race the rein" as he goads himself on into ever-further acts of wrongdoing against Isabella and her brother to gain her virtue.
Some argue that the tragic flaw of Angelo is his very lack of self-knowledge. By considering himself to be morally above the people that he rules he is unaware of how he is related to them through the same temptations that they all suffer, and also this ignorance leaves him more open to those temptations when they do come, precisely because he believes he is above them. This is a position that is reversed when he is forced to confess and acknowledge his guilt.
Thus Angelo does have his tragic flaw of believing himself to be morally superior to everyone else, wherein lies the seeds of his destruction. However, whether we feel sorrow and pity for his fall is another matter. I certainly feel he got what he deserved, thus he is exempted from being seen as tragic.