How do you draw a irregular hexagon with one line of symmetry

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Start by drawing a regular hexagon.

Now move one of the vertices either a little way towards, or away from the centre of the hexagon, and redraw the lines joining it to its neighbour vertices.

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To construct using a straight edge and a compass:

1. Construct a regular hexagon:
2. 1. Draw a circle; call its centre O.
   2. Mark a point on the circumference of the circle.
   3. Keeping the compass set to the same width (the radius of the circle), draw an arc to intercept the circle with the centre on the marked point.
   4. Repeat step 3 using the last arc-circumference intercept as the centre for the next arc until 6 marks have been made.
   5. Label the marks in the order they were drawn A, B, ..., F
   6. Draw in only the lines BC, CD, DE, EF (that is miss out lines AB and FA which would complete the regular hexagon).
      
3. Draw in the line AOD (if the vertices were marked properly in construction, these will lie on a diameter of the circle) and extend it either side of the circle.
4. Mark a point beyond A on AOD and call it G.
5. Draw in lines GB and FG.

Hexagon GBCDEF is an irregular hexagon with one line of symmetry, namely GAOD.