More interesting tilings are created if you forbid the combination of kite and dart which makes a rhombus. The following collection illustrates the superkite, the angel, the paper boat and the paper hat; and below the bird, the crab, the crown and one assymmetrical one.
In addition to these, there are (I think) 16 ways of placing five 72° angles round a point. Here are four of the more interesting: the decagon, the star, the priest and the folding chair (!)