From Longman Dictionary of Contemporary Englishoffer your hand (to somebody)offer your hand (to somebody)HELLOto hold out your hand in order to shake hands with someone → offer
Examples from the Corpus
offer your hand (to somebody)• An old woman sat veiled in black in a corner; toward whom people nodded or quickly offered their hand.• When I look down toward his waist, I see that he has covertly offered his hand.• We have offered our hand again and again.• Delia Sutherland offered her hand to a young girl in an overall.• Jotan was getting slowly to his feet, offering his hand to Arkhina. 