it's impossible to have a quadrilateral that can't be cut into triangles.
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Yes
Two triangles
You cannot make four triangles our of a quadrilateral by ONE line through it. One line can give you at most two triangles. Two lines, each running from one vertex of the quadrilateral to the opposite vertex, will give 4 triangles.
A pair of triangles can form a 4 sided quadrilateral
Yes