My cabinet door is not holding in place any more because of a plastic broken part.
I have also attached a pic of the same part that is intact for the other door.
I am not sure what is the name of the part that is broken and needs replacement.. Could anyone help please?
I don't think you'll find those anymore, they're pretty old. But, they're called Cabinet Brackets/Braces, Plastic Brackets/Braces, Screw Blocks, Angle Brackets/Braces, Modesty Blocks.
I'd suggest the left Metal L-bracket below & screw in a new location, just to the left or right. Unless, the old holes still hold strong & can actually tighten down very well with no slipping or spinning.
Those are called friction catches.
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Double and single roller catches:
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Magnetic catches are probably the easiest to install, as they can be more forgiving on placement than the other types but if the doors are exceptionally heavy these might not hold.
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I think yours are called bullet catches (pin?) but you don't need to find that specific other half. You remove the other old part and use parts A and B from a new set.
Except yours look like they serve a structural purpose as well. Do not remove them, just mount your new hardware next to them. Also do both doors now, or you'll be back at it later.
Note that unless designated as a push to open latch, you'll need a handle to easily open it.
Magnetic Push to Open dual door latch:
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