Label the sides of the quadrilateral a b c and d, clockwise. Place the first piece, with side a up. All subsequent pieces will be translates of this piece, or translates of its 180° rotation. There is no need to reflect the piece. Place a rotated copy of this piece above the first, side a to side a. Place the third piece to the right of the second, side d to side d. Place the fourth piece in the corner formed by the first and third, side c to side c and side b to side b. The four pieces surround a central vertex, their angles summing to 360°. Repeat this throughout the plane. Always put like sides together, and rotate every other piece.
As a corollary, every triangle tiles the plane. Two triangles form a parallelogram, and that's a quadrilateral.