Yes it will.
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Yes a quadrilateral will tessellate.
BECAUSE thats why!
Yes a square will tessellate on its own
All of them can be used.Any triangle or quadrilateral will tessellate. There are 15 classes of irregular convex pentagons which will tessellate. Regular hexagons and 32 classes of irregular convex hexagons will tessellate. There are also concave pentagonal and hexagonal shapes which will tessellate.
A square, rectangle, rhombus, parallelogram, trapezium, kite and arrowhead will tessellate. Other quadrilaterals will not.