starr
rigid
Immobile, stiff, and not flexible.
Nach dem langen Sitzen fühlten sich seine Glieder ganz starr an.
After sitting for so long, his limbs felt completely rigid.

rigid / fixed
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rigid
Immobile, stiff, and not flexible.
Nach dem langen Sitzen fühlten sich seine Glieder ganz starr an.
After sitting for so long, his limbs felt completely rigid.

The adjective 'starr' (rigid/fixed) is closely related to the verb 'erstarren' (to freeze/stiffen). 'Erstarren' means to become stiff or immobile, which explains the basic meaning of 'starr' very well.
Als er die schreckliche Nachricht hörte, erstarrte er.
When he heard the terrible news, he froze.
In the meaning 'fixiert' (fixed), 'starr' is very often combined with verbs of seeing. Typical phrases are 'starr blicken' (to stare fixedly), 'starr schauen' (to stare fixedly), or 'jemanden starr ansehen' (to stare fixedly at someone).
Die Katze schaute starr auf den Vogel im Baum.
The cat stared fixedly at the bird in the tree.
To express the cause of the stiffness, the preposition 'vor' with the dative is often used. The most common combinations are 'starr vor Kälte' (stiff from cold) and 'starr vor Schreck' (frozen with fright).
Nach dem Sturz ins kalte Wasser war er starr vor Kälte.
After falling into the cold water, he was stiff from the cold.
You can recognize the figurative meaning by the fact that 'starr' does not refer to a body or body part, but to abstract concepts. Examples are 'starre Regeln' (rigid rules), 'ein starrer Plan' (a rigid plan), or 'eine starre Meinung' (a rigid opinion).
Das Unternehmen leidet unter seinen starren Hierarchien.
The company suffers from its rigid hierarchies.
flexible
'flexibel' and 'starr' are often opposites. 'flexibel' fits when something is bendable or when someone reacts well to new situations. 'starr', on the other hand, means that something is immobile, or that someone holds on to fixed rules and opinions. Learners may confuse the words because both can be used for things and for behavior. Remember: 'flexibel' opens up possibilities; 'starr' allows hardly any change.

Bei der Arbeit hält er starr an seinem alten Plan fest.
At work, he rigidly sticks to his old plan.
Nach der Arbeit bin ich flexibel und kann auch später zum Training kommen.
After work, I'm flexible and can also come to training later.