My professor tells me that the word
i.e should be written with a brace and quotations outside it For example:
is that the correct way?
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It is not required, perhaps it is your professor's style.
There seems to be at least one professor at Bristol University who would disagree with both the OP's prof and cowls:
Larry Trask, at the University of Sussex, has another ersatz-rule:
e.g. for example
i.e. in other words
etc. and so forth
sc. which means
et al. and other people
The correct way is the one I use, but I'm not telling you what that is (it may change anyway); you'd better do what your own prof tells you - he marks your papers, after all. (I will say that I've never seen his preferred style elsewhere though.)
The Wikipedia article makes sense: