No?
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There are no quadrilaterals in the Star of David- as in zero, none. The reason for this lies in the geometrical definition of the term quadrilateral. In the simplest terms, a quadrilateral is any polygon with four sides and four corners. Because the Star of David is a star polygon known as a hexagram, it is based almost exclusively on a trilateral (as in 3) system. A three-dimensional Star of David might actually have a quadrilateral, but the everyday commonly known two-dimensional variety has NO quadrilateral.
The star is a polygon, but not a simple one. It is not simple because parts of the lines forming its boundaries are inside the star. The outline of a star is a simple polygon.
No. A quadrilateral is any four-sided figure. A trapezoid is one example of a quadrilateral.
Yes, such a quadrilateral can exist.
A quadrilateral is a polygon with four sides. A parallelogram is a quadrilateral in which the opposite sides are parallel.