Kind of true and kind of false : Sometimes they are, but there is nothing that force two acture triangles to be similar, even if they have the same measurement of the acute angle.
For that you would need to have 3 angles the same.
For example, if the angles were 40-60-80, you have an acute triangle. But if you had a 45-55-80 triangle, that is also acute, but not similar even though it shares an angle of 80 degrees.