
das Mitglied
member
A person who belongs to a group, club, organization, or family.
Jedes Mitglied des Vereins muss einen Jahresbeitrag zahlen.
Every member of the club has to pay an annual fee.
member
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member
A person who belongs to a group, club, organization, or family.
Jedes Mitglied des Vereins muss einen Jahresbeitrag zahlen.
Every member of the club has to pay an annual fee.

limb
A body part, especially an arm or a leg. This meaning is outdated and is hardly used today.
Nach der langen Wanderung streckte er seine müden Mitglieder aus.
After the long hike, he stretched out his tired limbs.
The word consists of 'Mit-' and 'Glied'. 'Mit' means 'together/with', and a 'Glied' is a part of a whole (like a link in a chain). So a member is literally a 'part that belongs to it'.
Als neues Mitglied fühlte er sich schnell in der Gruppe willkommen.
As a new member, he quickly felt welcome in the group.
Although 'das Mitglied' (member) refers to people (male, female, diverse), the word is grammatically neuter. It always remains 'das Mitglied', regardless of the person's gender.
Sie ist seit zehn Jahren Mitglied in unserem Chor.
She has been a member of our choir for ten years.
You are 'Mitglied in' (a member in/of) a group in which you are active (e.g., a club), or 'Mitglied von' (a member of) a larger organization. Often both prepositions are possible.
Er ist Mitglied im Schachklub.
He is a member of the chess club.
Common verbs used with 'Mitglied' (member) are 'Mitglied werden' (to become a member), 'Mitglied sein' (to be a member), and 'als Mitglied austreten' (to resign/withdraw as a member).
Möchtest du Mitglied in unserem Fitnessstudio werden?
Would you like to become a member of our gym?
The meaning 'body part' is outdated (archaic). You will only find it in very old literature. It is no longer used in modern language.
Der Ritter schützte seine Mitglieder mit einer Rüstung.
The knight protected his limbs with armor.
relative / member
Both words can refer to a person who belongs to a family. Mitglied is the general term for belonging to a group and is often used with families in the compound Familienmitglied. Angehöriger directly describes the family relationship and is therefore usually more natural in this context.

Tom ist ein Mitglied der Familie und hilft oft im Haushalt.
Tom is a member of the family and often helps around the house.
Tom ist ein Angehöriger der Familie und hilft oft im Haushalt.
Tom is a relative and often helps around the house.