sich beugen
to bend down
To lean the upper body or a body part forward or downward.
Er musste sich tief beugen, um unter dem Ast hindurchzugehen.
He had to bend down low to walk under the branch.

to bend
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to bend down
To lean the upper body or a body part forward or downward.
Er musste sich tief beugen, um unter dem Ast hindurchzugehen.
He had to bend down low to walk under the branch.

The verb 'beugen' (to bend) has the prefix 'be-' and is therefore inseparable. The prefix remains firmly attached to the verb in all sentence forms and is not placed at the end of the sentence.
Er beugte den Draht zu einem Haken.
He bent the wire into a hook.
Pay attention to the difference: 'sich beugen' (to bend down / to bow) is reflexive and usually describes a body movement or giving in to a power. The non-reflexive form 'beugen' (to bend) means to make an object crooked or is used as a grammatical technical term.
Sie beugt sich zu dem Kind hinunter.
She bends down toward the child.
When 'sich beugen' means 'nachgeben' (to give in), the power or force that one gives in to is often in the dative. Typical constructions are 'sich einer Sache beugen' (to bow to something) or 'sich vor jemandem beugen' (to bow before someone).
Am Ende musste er sich der Entscheidung der Mehrheit beugen.
In the end, he had to bow to the decision of the majority.
You can easily recognize this meaning from the context. When it is about language, words, nouns, adjectives, or declension, 'beugen' always means 'flektieren' (to inflect).
In diesem Satz musst du das Adjektiv richtig beugen.
In this sentence, you have to inflect the adjective correctly.
to tend / to tilt
'neigen' and 'beugen' can both describe a movement where something does not stay straight. 'neigen' often fits when you tilt your head or upper body slightly in one direction, also as a respectful movement. 'beugen' sounds more like bending or moving forward and downward; you can also bend an arm, a knee, or a piece of metal. The words are easy to confuse because they seem similar with body movements. As a guide: 'den Kopf neigen' (to tilt one's head) is often a slight tilting movement, while 'den Rücken beugen' (to bend one's back) is more like making it clearly curved.

Beim Training beugt Max die Knie, damit sein Rücken gerade bleibt.
During training, Max bends his knees so his back stays straight.
Die Ärztin neigte den Kopf zur Seite und hörte genau zu.
The doctor tilted her head to the side and listened carefully.