Here are the corrections for using articles (a, an, the) in the sentences:
Ismail spoke to the woman who had waved to him
My mother bought me an expensive watch
Cheryl borrowed a book from the library.
Ali wore a new uniform to school.
Did you see a new car which my father bought?
Kiran is an Indian girl.
Singapore lies quite near to the equator.
Suman has travelled across the Atlantic Ocean.