thanks to all

'Proper' often has the idea of being an absolute standard, whereas what is 'appropriate' can vary with the circumstances. In this class, you will speak properly at all times. (not 'appropriately') Behave properly. (There is an absolute standard that everyone knows) He was an impressive world traveler. He knew how to behave appropriately in hundreds of different situations. And I think it is pretty rare to say that a math answer is 'proper'. It's much more common to say 'correct' or 'right'.

There is a good deal of overlap. "Appropriately" usually carries social overtones. It usually has to do with politeness, or courtesy, or, well, decency. Something "inappropriate" might offend somebody. "Proper" usually means the best or most correct choice of a set of choices. "How much is 2 + 2? Is it 3, 4, or 5?" A) "The proper answer is 4." B) "The appropriate answer is 4." "Proper" is the right word here. "Appropriate" is not really wrong, but it sounds odd, because it is hard to imagine anybody being offended by the wrong choice. In the case of a set of social choices, one of which might offend people, you can use either word. For example, both of these are fine: A) "For a job interview, it is proper to wear a business suit." B) "For a job interview, it is appropriate to wear a business suit." The word "appropriate" tells you not only that the business suit is the correct choice, but also that the reason why it is correct: it is socially expected. A job interview has a level of formality for which a business suit is appropriate.

appropriately - suitable for the occasion, right under the circumstances properly - done in a correct manner, as it should be done He wasn't dressed appropriately for a date. You didn't close the fridge door properly, that's why milk has gone bad.