angehören
to be a member
To be a member of a group, organization, party, or family.
Sie gehört seit vielen Jahren dem örtlichen Schachclub an.
She has belonged to the local chess club for many years.

to belong to
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to be a member
To be a member of a group, organization, party, or family.
Sie gehört seit vielen Jahren dem örtlichen Schachclub an.
She has belonged to the local chess club for many years.

'Angehören' (to belong to) is a separable verb. The prefix 'an-' comes at the end in a main clause. In a subordinate clause or when used with a modal verb, the verb stays together.
Er gehört einer politischen Partei an.
He belongs to a political party.
The verb 'angehören' (to belong to) is always used with the dative. The group or category that someone or something belongs to is in the dative.
Dieses Kunstwerk gehört der Moderne an.
This work of art belongs to the modern era.
To express membership, the verb is often combined with social, political, or family groups. Typical phrases are: einer Partei angehören (to belong to a party), einem Verein angehören (to belong to a club/association), einer Familie angehören (to belong to a family).
Seit wann gehören Sie dem Vorstand an?
How long have you been a member of the board?
In this meaning, you can recognize the verb by the fact that it refers to abstract classifications. The object is then usually an era, a genus, a species, or a style.
Der Wal gehört der Gattung der Säugetiere an.
The whale belongs to the class of mammals.
to belong
'gehören zu' and 'angehören' both mean that someone or something is part of a group or category. 'gehören zu' is very general and common in everyday language: a person belongs to a team, a topic belongs to a course. 'angehören' sounds more formal and is often used with organizations, parties, families, or fixed groups. It can also mean that something counts as part of a type or category. The words can be confused because their meanings are similar. In normal conversations, 'gehören zu' usually fits; in official or written texts, 'angehören' often fits better.

Sie gehört seit fünf Jahren dem Verein an.
She has been a member of the club for five years.
Diese Übungen gehören zum neuen Deutschkurs.
These exercises are part of the new German course.