Die Verspätung ist ärgerlich.

Because Verspätung is a feminine noun in German, so its nominative singular article is die.

• die Verspätung = the delay
• grammatical gender in German does not always match anything logical in English, so you usually just learn the noun together with its article

A good habit is to memorize it as die Verspätung, not just Verspätung.

Verspätung means delay or lateness, depending on context.

In this sentence, Die Verspätung ist ärgerlich means something like:

• The delay is annoying
• The delay is frustrating

It often refers to things like:

• a train delay
• a flight delay
• someone arriving late

The noun comes from the verb verspäten / sich verspäten, which is related to being late or delayed.

German main clauses normally follow the V2 rule: the conjugated verb comes in the second position.

So in:

• Die Verspätung = first element
• ist = second element
• ärgerlich = rest of the sentence

This is one of the most basic German sentence patterns:

• Subject + verb + complement

Because ärgerlich is being used as a predicate adjective, not an adjective directly before a noun.

Compare:

• Die Verspätung ist ärgerlich.
  Here ärgerlich comes after ist, so it does not take an ending.

But:

• die ärgerliche Verspätung
  Here the adjective is directly before the noun, so it does take an ending.

This is similar to English:

• The delay is annoying
• the annoying delay

Ärgerlich means annoying, irritating, or frustrating.

It describes something that causes Ärger, which means annoyance, trouble, or anger.

So:

• Die Verspätung ist ärgerlich = The delay is annoying/frustrating

Depending on context, it can sound mildly annoyed or more strongly frustrated.

No. They are related, but they are used differently.

• ärgerlich = annoying
  It describes the thing that causes annoyance.
• verärgert = annoyed / upset
  It describes the person who feels annoyed.

So:

• Die Verspätung ist ärgerlich.
  = The delay is annoying.
• Ich bin verärgert.
  = I am annoyed.

That distinction is very useful.

Good question, because die can mean:

• the for feminine singular
• the for plural nouns

Here, you know it is singular because:

1. Verspätung is a singular noun form
2. the verb is ist, which is singular

If it were plural, you would expect:

• Die Verspätungen sind ärgerlich.
  = The delays are annoying.

So ist helps confirm that this sentence is singular.

Yes.

• Die Verspätung ist ärgerlich. = The delay is annoying.
• Eine Verspätung ist ärgerlich. = A delay is annoying.

The difference is the same as in English:

• die points to a specific delay
• eine means any delay / a delay in general

Yes, but the normal neutral order is:

• Die Verspätung ist ärgerlich.

You could also say:

• Ärgerlich ist die Verspätung.

That version is more marked and puts extra emphasis on ärgerlich. German allows this because the verb still stays in second position.

So both are grammatical, but the original sentence is the most natural basic version for most situations.

The usual way is:

• Die Verspätung ist nicht ärgerlich.
  = The delay is not annoying.

Here nicht negates the adjective ärgerlich.

If you wanted to negate the noun with kein, you would need an indefinite noun phrase, for example:

• Das ist keine Verspätung.
  = That is not a delay.

But for your sentence, nicht is the correct negation.

A rough pronunciation guide is:

• ärgerlich ≈ AIR-gər-likh or EHr-gər-likh

A few useful points:

• ä is not the same as plain a
• g in ärgerlich is pronounced as a hard g
• the final -ch is the soft German ich sound, not a k sound

If you are just starting out, the most important thing is not to pronounce it like English ar-ger-lick.

Because in German, all nouns are capitalized.

So:

• die Verspätung
• der Zug
• das Problem

But adjectives and verbs are usually not capitalized unless they begin the sentence or are used as nouns.

That is why you get:

• Die Verspätung ist ärgerlich.

with Verspätung capitalized and ärgerlich not capitalized.