sich lehnen
to lean
To place one's body at an angle against something in order to find support and prop oneself up.
Er lehnte sich erschöpft gegen die Wand.
Exhausted, he leaned against the wall.

to lean
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to lean
To place one's body at an angle against something in order to find support and prop oneself up.
Er lehnte sich erschöpft gegen die Wand.
Exhausted, he leaned against the wall.

Pay attention to the difference: 'sich lehnen' (to lean oneself; reflexive) means that a person supports their own body against something. 'Etwas lehnen' (to lean something; transitive) means that you place an object against something.
Sie lehnte sich müde an meine Schulter.
She leaned tiredly on my shoulder.
The verb 'lehnen' (to lean) is almost always used with a preposition that indicates the supporting surface. The most common are 'an' or 'gegen', followed by the accusative.
Lehn dich nicht gegen die frisch gestrichene Wand!
Don't lean against the freshly painted wall!
The idiom 'sich zu weit aus dem Fenster lehnen' (to lean too far out of the window) is a fixed unit. It is used metaphorically and means that someone says or does something too risky.
Mit dieser gewagten Prognose lehnst du dich wirklich weit aus dem Fenster.
With this bold forecast, you're really going out on a limb.
to tend / to tilt
'lehnen' and 'neigen' both have to do with an angled direction. With 'lehnen', however, there is usually a support: a person or object is at an angle against something, for example against a wall. 'neigen' is more about bending or tilting something, often the head or upper body. You do not have to support yourself when doing this. So you can 'den Kopf neigen' (tilt your head), but you 'lehnt sich an eine Wand' (lean against a wall). The words are easy to confuse because both describe a movement away from a straight position.

Nach dem langen Spaziergang lehnt sich Tom müde an den Zaun.
After the long walk, Tom leans tiredly against the fence.
Die Ärztin neigt den Kopf zur Seite und hört genau zu.
The doctor tilts her head to the side and listens carefully.