My German teacher has told me that when writing a letter, in the sentence following "Dear xxx," I shouldn't use a capital letter, and that this rule also holds in English. (I'm not a native English speaker.) I'm wondering if both of her statements are true, seeing as I found this: https://english.stackexchange.com/questions/11906/should-i-capitalize-the-starting-sentence-after-a-greeting-that-ends-in-a-comma
The rules of English and German are different:
In German you use NO capital, unless the first word after adress ist a noun. See also this page.
Sehr geehrter Herr Müller,
ich entschuldige mich für...
In English you use capital. You needn’t set a comma, but you can. In American Englisch you use a colon.
Dear Mrs Fisher,
My sincere apologies for....
If you use a comma ("Liebe Anna,"), the following part has to start like the rules of the german language are providing. So if a verb is following after the comma, it has to be written without a capital letter; if a substantive is following, you have to write it capitalized.
A different case is the usage of an exclamation mark ("Liebe Anna!") - in this case, the greeting is an independent sentence and the following part has to start capitalized.
In German, there are different rules depending on the variety.
In Germany standard German, the salutation is separated by a comma from the first sentence, which starts with a small letter (as if the salutation were part of the first sentence):
in Deutschland schreibt man klein.
In Swiss standard German, the salutation has no punctuation mark and the first sentence starts with a capital letter (as if the salutation were a title):
In der Schweiz schreibt man gross.