mad is very angry angry is the state of one being furious or utterly dissatisfied with sth or sb usually shown by a red face like when sb hits your car with a hammer on purpose in front of you bothered or annoyed is when sth is at your back or basically on your nerves like neighbour's loud music upset is the state of being sad and unhappy like when you lose your pet or sth/sb dear the point is a person can feel all the above simultaneously. like in a war people will feel many emotions together.

mad, angry, bothered, upset can be interchangeable in many situations ( He was mad / angry / bothered / upset by the bad news.