If I translate
Peter was afraid
with Google Translate, I obtain:
Peter hatte angst
Why does it use hatte instead of war?
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Look at it like this:
Afraid is a funny adjective like hungry or thirsty in German. We’ll use hungry as our example here.
For the most part, people won’t say “I am hungry” in German. They will say “Ich habe Hunger” (literally: “I have hunger”). Hunger, in this case, isn’t something you can be, but rather something you have!
Now, in your example, “Peter hatte Angst” actually means “Peter had fear”. It only means “Peter was afraid” when you adjust for grammar.
TL;DR: Because words like die Angst, der Hunger, and der Durst are all actually nouns! They are all things that, in German, you can have, not something you can be.
You have overlooked a very important piece of information that Google's translator has given to you. This does NOT come out of Google's translator:
wrong: Peter hatte angst.
This is the real output, and it is correct:
correct: Peter hatte Angst.
As you can see, »Angst« is written with an uppercase first letter, which means (when not being the first word of a sentence), that it is a noun, not an adjective.
The word by word-translation of »Peter hatte Angst.« is:
Peter had fear.
The english adjective »afraid« has no German counterpart with exactly the same meaning. But you can translate »to be afraid« into »Angst haben«, but later means literally »have fear«.
An other author compared in his answer »afraid« with »hungry« and »thirsty«. But there are German adjectives that means exactly the same as the english »hungry« and »thirsty«, which are »hungrig« and »durstig«:
engl: Peter was hungry and thirsty.
germ: Peter war hungrig und durstig.
But there is also the possibility to translate it into a sentence that is similar to »Peter hatte Angst.«:
engl: Peter was hungry and thirsty.
germ: Peter hatte Hunger and Durst.
But don't forget to write »Hunger« (engl: the hunger) and »Durst« (the thirst) with an uppercase first letter, because they are nouns. »Angst, Hunger« and »Durst« are nouns, the are the names of feelings you can have. They are not adjectives that describe what you are.
As has already been pointed out, Angst here is a noun, which you may already know, because it has made its way into the English language via Freud. Since Peter is not fear, it would not make sense to say that he “war Angst”. In German however, if you have a feeling of “Angst” you say “ich habe Angst”.
Incidentally, there is also an adjective angst in German, and so
Peter war angst.
is correct German. There are some caveats, though. First, Peter here is is the dative case, not in the nominative case, so it would not be
*Er war angst.
Ihm war angst.
Second, the adjective angst is obsolescent and mostly survives in the expression angst und bange. So from a German learner's perspective it is probably best to forget about the adjective for now, I only mentioned it for the sake of completeness.
Because "Peter hatte Angst" means "Peter was scared".
There is a version of that sentence with an adjective, but it means something different. "Peter war ängstlich" means "Peter was easily scared".
As for why German expresses this and a few other emotions via 'having' vs. 'being', there is no particular reason. They are consistent with each other, but there is no grand plan or conspiracy behind German verb forms generally (other than to confuse foreign learners).