Place six toothpicks on the table with their ends all meeting together to form a hexagon.
Place six more toothpicks inside the hexagon shape, with one point of each one between the joined ends of the first six toothpicks. Have the other point of each toothpick meet in the middle of the shape. You now have a hexagon with six equilateral triangles inside it.
Remove the topmost toothpick from the outside of the hexagon. Remove the toothpick on the very bottom of the hexagon as well. The toothpicks now form four congruent triangles.
Replace the two toothpicks you removed to form a hexagon once again. Remove any two adjacent toothpicks coming from the center of the hexagon. You now have one trapezoid and three congruent triangles.
Replace the toothpicks you removed in step 4 to restore the original hexagram. Remove one of the toothpicks coming from the center, and then remove the toothpick that lies directly opposite it, so that the remaining four toothpicks in the center create an "X" shape. You have now created two congruent parallelograms and two congruent triangles.