1 and 2 are fine, and yes they mean the same thing.
The word "with" is part of the idiom, to be happy with something (that is, to be happy about something). I'm not sure I can explain it!
3 and 4 are awkward, and I doubt they'd be used.
1. Are you happy with your school uniform?
2. Are you satisfied with your school uniform?
3. Are you happy related to your school uniform?
4. Are you happy because of your school uniform?
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Does #1 mean #2? What about the other sentences? The meaning of 'with' is _____.
1 answer