verb
1
BAD : Excuse me. I didn't know that you wanted to speak to me.
GOOD : I'm sorry. I didn't know that you wanted to speak to me.
BAD : Excuse me. I didn't mean to hurt you.
GOOD : I'm sorry. I didn't mean to hurt you.
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Use excuse me (1) when you interrupt someone (2) when you are trying to get past someone (3) when you speak to someone you don't know: 'Excuse me but there's a long distance phone call for you.' 'Excuse me. Do you happen to know the way to the station?'
To apologize to someone, use I'm sorry (or just sorry ): 'I'm terribly sorry. I forgot we'd arranged a meeting.'
2
BAD : When he got back home he excused himself and promised that he would never run away again.
GOOD : When he got back home he apologized and promised that he would never run away again.
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excuse oneself = give a reason for one's actions: 'She excused herself from the meeting, saying that she was not feeling well.'
apologize = say you are sorry for (doing) something: 'The bank wrote to apologize for overcharging me.'
3
BAD : He was sorry that he had lied to her and asked her to excuse him.
GOOD : He was sorry that he had lied to her and asked her to forgive him.
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Excuse is used only for minor faults and offences: 'I hope you'll excuse my untidy handwriting. I am trying to write this letter on a train.'
Forgive is used for all faults and offences, both minor and major: 'I don't think he'll ever forgive me for calling off the wedding.' 'Don't forget her birthday or she'll never forgive you.'