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happen有意外发生的意思，相当于汉语中的“碰巧发生”，例句：I don't know what happened.我不知道发生了什么（碰巧）。appear的意思就是“出现；显现”，没有什么特别的强调碰巧，就是正常使用就行了。
happen的相关例句
All these changes happened over a period of time.
所有这些变化都是在一段时间内发生的。
You must on no condition tell them what happened.
你决不能告诉他们所发生的事。
This is the second time it's happened.
这已是第二次了。
You don't happen to know his name, do you?
你不会碰巧知道他的名字吧？
He gave me a guarantee that it would never happen again.
他向我保证这种事情绝不会再发生。
appear的相关例句
This problem first appeared in the inner cities.
这个问题最初在市区中心出现。
Your name will appear at the front of the book.
你的名字将出现在书的前页。
It would appear that this was a major problem.
看来这是个主要问题。
The magazine will appear in a new design from next month.
从下月起这本杂志将以新的设计问世。
The president appears likely to change course on some key issues.
总统看起来可能要在某些重要问题上改变方针。