noun
DUBIOUS : At the front of the queue a man and a lady were arguing with the cashier.
GOOD : At the front of the queue a man and a woman were arguing with the cashier.
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Lady is used when you need to be polite, especially in formal styles: 'Ladies and gentlemen, may I have your attention please?' 'Please show these ladies the way to the cloakroom.'
The usual word is woman : 'Isn't that the woman who teaches at the International School?'
Note however that old lady is generally preferred to old woman , which sounds impolite: 'Can you help that old lady across the road?'