I sometimes include Origami folding in my geometry class for two reasons. First, it makes students pay attention to the directions and second it uses angles, bisectors, triangles, and other such geometric shapes and terms. I found a nice app called animal origami and downloaded it. It has several different animals that you can make following the step by step directions. I picked it up the day it was being offered for free. I did a quick check and there are several free to very low cost origami apps which show how to fold the paper step by step.
The ways I work origami folding into Geometry is to have students:
a. follow directions and then write a paper on all the geometric shapes and figures they used in the process.
b. unfold their creation to use a protractor to measure the angles of all the folds off of certain lines.
c. label and identify geometric shapes and figures on the paper itself.
d. Write a set of directions on making the figure.
After they've had some practice with fun shapes, I pull out the origami book I have for geometry and have them follow directions for specific items such as their own specialized angle reader.