The applet below allows you to try to create a counterexample to the "AA~" similarity theorem.
You can move any of the points on the left triangle. You can also move any points on the right, with one big rule: angle A' will always be forced to stay congruent to angle A, and angle B' will always be forced to stay congruent to B.
Can you connect the dots on the right triangle in a way that makes a triangle that is NOT similar to ABC? If you can, it will glow red! If you create a triangle that is similar to ABC, it will glow green.
You can see the ratios of the sides on the left whenever you successfully create a triangle. Can you make them be not quite equal? Why do you think that is, if they are similar?