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Tag questions are used when seeking confirmation of what one believes to be true. They restate, in question form, the previously spoken sentence. For example,

He is an engineer. Isn't he? They're not from Colorado. Are they? Zack is really tired. Isn't he? She is going to Taiwan, isn't she? You have never been to Las Vegas. Have you? The Rays are playing the Jays. Aren't they? You went to the supermarket. Didn't you? Zane doesn't live in Missouri. Does he? Ms. Zeller has a new car. Doesn't she?

(Positive statement, negative tag) (Negative statement, positive tag) (Positive statement, negative tag) (Positive statement, negative tag) (Negative statement, positive tag) (Positive statement, negative tag) (Positive statement, negative tag) (Negative statement, positive tag) (Positive statement, negative tag)

In tag questions, the corresponding pronoun and the first verb of the corresponding yes/no question are used. Also, notice that with positive statements, negative tag questions are used and vice versa.
See also: Grammar: Yes/No Questions
Common mistakes:

You are the zookeeper. Yes? You're the zookeeper. Aren't you? You come from Canada. No? You come from Canada. Don't you? You're the boss. Are you? You're the boss. Aren't you? Simon is from Singapore. Isn't it? Simon is from Singapore. Isn't he?

(Incorrect) (Correct) (Incorrect) (Correct) (Incorrect--in most situations) (Correct) (Incorrect) (Correct)

Some modals can be used in tag questions: can, will, would, could, should and must.

Mr. James will be at the ceremony. Won't he? They couldn't do it. Could they?

Note: When the subject is ?I? and the statement is in present tense, aren't is commonly used for tag questions.

I'm the winner. Aren't I? I'm the winner. Am I not?

(Common usage) (Formal)