She was obviously upset that members of her family were not taking her side and boycotting our friendship.
Mind-reading.
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She was also upset that I had made some plans for a group of friends to get together to see "rock of ages" at a local theater, a couple weeks from now.
Her problem, not yours. I think you handled it appropriately by just telling her to consult with her cousin and then letting her go.
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"I understand your feelings. Actually, I was going to ask if you would like to go to the show with us".
That sounds like you're trying to backtrack. Next time just say "you're welcome to join us if you wish."
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I ended the conversation with....
It will be fun. If you change your mind, the offer is out there. Have a good night"
I wouldn't have replied back at all. Tell her she can come along, if she says no then shrug your shoulders and move on like you don't care whether she goes or not.