My questionnaire is consists of ten 5-point scaled likert items. If the likert items have a good chronbach alpha, is it ok to take the average score of 10 items?
Technically, no. Likert scales are ordinal and, therefore, adding them isn't right.
Practically, yes. People do this all the time.
But a better method would be to do a factor analysis (accounting for the fact that the data are ordinal) and using that, rather than a simple sum. In my experience, though, it winds up not making that much difference.