hängend
hanging
Describes something that hangs down from above or has a downward-tilted position.
Die Weide am Ufer hat lange, hängende Zweige.
The willow by the bank has long, hanging branches.

hanging / pending
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hanging
Describes something that hangs down from above or has a downward-tilted position.
Die Weide am Ufer hat lange, hängende Zweige.
The willow by the bank has long, hanging branches.

The word 'hängend' (hanging) is the present participle of the verb 'hängen' (to hang). This form describes an ongoing state or an ongoing action and is used like an adjective.
Die hängenden Pflanzen auf dem Balkon müssen gegossen werden.
The hanging plants on the balcony need to be watered.
You can recognize the meaning from the context. If 'hängend' (hanging) refers to physical objects such as branches or shoulders, it means 'hanging down'. In legal or administrative texts in connection with 'Verfahren' (proceedings) or 'Klage' (lawsuit), it means 'pending' or 'being processed'.
Mit hängenden Schultern ging er nach Hause.
He went home with drooping shoulders.
In its literal meaning, 'hängend' (hanging) is often used to describe body parts or parts of plants. Typical combinations are 'hängende Schultern' (drooping shoulders), 'hängende Mundwinkel' (downturned corners of the mouth), or 'hängende Geranien' (hanging geraniums).
Der alte Baum hat tief hängende Äste.
The old tree has low-hanging branches.
The meaning 'in Bearbeitung' (being processed) belongs to legal and administrative language. This usage is not used in normal everyday speech.
Die Klage ist seit zwei Jahren bei diesem Gericht hängend.
The lawsuit has been pending at this court for two years.
open
'offen' and 'hängend' can both mean that something has not yet been decided or completed. 'offen' is the normal everyday word and fits questions, tasks, bills, or decisions. 'hängend' in this meaning is more technical and occurs mainly in legal and administrative language. Learners may confuse the words because both talk about unresolved cases. In everyday language, 'offen' almost always sounds more natural; 'hängend' is more suitable for proceedings, applications, or official cases.

Der Anwalt prüft noch ein hängendes Verfahren.
The lawyer is still reviewing a pending case.
Beim Elternabend blieb eine wichtige Frage offen.
At the parents' evening, one important question remained open.