In "Five By Five", Angel goes into the man's apartment to rescue Wesley and confront Faith. The man is not dead. How did he get in there without an invite?

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Most likely it's just a writing fluke.

probly because hes part of the detective angency

Maybe The Powers That Be intervened to let Angel in to fight/help faith like they (presumably) did when Kate tried to kill herself.

The guy either was just renting the place, so that was not technically his home, or there could have been welcome mat ouside of the apartment building/door!

I remember wondering that too!

The other possible explanation, I guess, is that the guy that Faith beat up died in hospital.

There was a previous ep (can't remember which...) where Angel was leaning against nothing in the doorway of an apartment, then he falls through it because the owner had just died and the 'magic' that kept him from entering was lifted.

There are 3 options:
1. The man who owned the apartment had died in the time it took Angel to travel there.
2. When they visited him in the hospital, the man gave Angel an invitation.
3. The writers made a mistake and forgot the invitation rule.