We paid Mr. White a compliment on his child's success.
Based on the above sentence, I wonder if I could make a similar one--We paid Mr. White a congratulation on his child's success.
Correct me if I am wrong. Thanks.
Compliment is used for a lot of different nuances of meaning. Another example is when you call on someone in the form of a short social visit; you might describe that as paying your compliments, as well.