You said that
From the tone of her voice I can tell when she just wants to vent.
If you can tell that she wants to vent then why offer her your input? This isn't what she is looking for and this is why she gets mad and hangs up. She doesn't want you to tell her what you think. She just wants you to listen.
Now from your question it seems-and correct me if I'm wrong- that you might be tired of just repeating yourself. So you are basically trapped. On one hand, if you do what she wants you to do-just listen and repeat yourself-it affects you to the point it forces you to give her your opinion. And when you do that she gets upset with you. So you are both playing a game and you probably got tired of it.
Be honest with her and ask her how she would like you to help and be supportive, otherwise don't say anything, listen to her vent or rant and when she's done, ask her if she cares to know what you think (though she seems to have reached a point where she's going on circles. She most likely knows what you think but doesn't want to accept it or act on it or change the situation).
Or change the subject or talk about you. If she continues to get mad or put you in a position where you end up offering suggestions or advice when it clearly hasn't been effective, just tell her politely either on the phone or in person that it seems that you aren't able to help her anymore because you either have to repeat yourself which drains you (if it does) or be tricked into offering advice which she clearly rejects by getting mad and hanging up. Your friend seems either to not be able to or be ready to change whatever it is she complains to you about. There isn't much you can do here. It's basically up to you to enable or not enable her behavior.
Something helpful is to use her own arguments or examples from your discussion to defend your position. You could tell her that every time she says A, it forces you to do B, which makes her do C (gets mad and so on).
Don't be afraid to address that and ask her what she would do in your situation.
From your comment to my answer:
If the fact that she won't accept your help when you seem to know how to help her, bothers you so much, ask her why. Find out why she won't. Tell her if you feel rejected or offended, be open about your feelings without being rude to her. Her answer might give you more clues as to whether this is her not really wanting to be helped, or not trusting you can help her, or there could be other reasons. I don't know if you only talk on the phone but this seems like a discussion you might need to have in person.