True or false Two right triangles that are not similar still may have one congruent acute angle in common?

False.

Assume that you had a two right triangles with one congruent acute (<90 degrees) angle in common. Let x represent the number of degrees in this angle in both triangles (which we can do since the angles are congruent). Let y represent the degree of the other angle in the first triangle and let z represent the degree of the other angle in the second triangle. We know that the sum of the degrees of the angles in a triangle is 180. So for the first triangle we have,

90+x+y = 180

For the second triangle,

90+x+z=180

Therefore,

90+x+y=90+x+z

Subtract the 90+x from each side:

y=z

Therefore the degrees of the angles of the two triangles both are 90 [because they are both right triangles], x [because we said that this is the number of degrees of the congruent angles given in the problem], and y [because y=z]. Because the three angles of both triangles have the same measurement, the triangles must be similar.