Yes. Pick one side of a kite. Swap an adjacent with an opposite side and you will have a parallelogram!
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The diagonals of a square for example divides it into 4 isosceles triangles
Use different colours and tessalation. Works for me.
Quadrilaterals are four-sided shapes that have a lot of use in math, like using them to determine the area of more complicated shapes.
As long as the two quadrilaterals are congruent, yes. (Congruency ignores position, including rotation and reflection.)
Draw a line touching one side to the opposite side of the square, which is not parallel to any side, and does not touch any vertex, so you will have two new quadrilaterals, each one has one pair of parallel sides and one pair of nonparallel sides.