When I heard 'Are you mad at me?', at first I didn't understand what's meaning of this. I usually use anger and upset. I thought that mad means insane. What is difference between 'Are you mad at me?' and ' are you angry with me?'
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There is no difference between "mad at" and "angry with".

In the US, you'll be more likely to hear "mad at" in informal conversation.

John
This is interesting. As a father I tell my children I'm not angry with them, but I'm mad at what they did. Maybe it is my pedantic data architect behaviour revealing itself but the person and behaviour are two separate things. I never realised, until now, that I use "angry with" for a person, but "mad at" a behaviour. Hmm...

I'm not saying either is right or wrong; just that I found the realisation interesting. Thanks for posting and allowing me to know myself a bit better.
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A Ram Hwang I thought that mad means insane.
Yes, it does. However, you can be mad (insane) with anger. The expression is more North American than British; a Brit would probably frown at the expression and have a similar reaction to yours. I know that was mine when I first came to Canada.
We Americans are sometimes told in our high school English classes that we should say "Are you angry at me?" and not "Are you mad at me?"

Though this advice has been dispensed for many generations, it has never stuck. We almost universally say "Are you mad at me?" for "Are you angry at me?" Emotion: smile

CJ
You mean that you use "mad at" and "angry with" in different situation?

Is it general? or is it your own pattern?

Actually, "mad at" sounds like an informal words.

I think you used these words without any other consideration.

I also glad that you found realisation because of my question.
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A Ram HwangYou mean that you use "mad at" and "angry with" in different situation?
Is it general? or is it your own pattern?
Actually, "mad at" sounds like an informal words.
I think it is my own pattern. My family tells me I use ‘mad at’ and ‘angry with’ interchangeably, and also ‘mad with’ too, but not ‘angry at’.
Emotion: big smile
Perhaps this distinction is no longer observed, but here is the formal explanation:
"Mad" suggests insanity.
One is "angry at" things, but "angry with" persons.
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