No. Good question though!

Angles (like distances) can be measured in "signed" or "unsigned" form. If I measure a swimming pool from the shallow end to the deep end and find it is 50 meters long, while my friend measures it in the other direction, I expect him to tell me that it is 50 meters long, not -50 meters. And I don't expect him to tell me "we cannot hold the swimming meet here, this pool is -50 meters long and that is less than the standard Olympic length of 50 meters!"

The definition of an acute angle as "one less then 90 degrees" is for unsigned angles. If we want to define acute angles using signed measurements, then we say "an acute angle is one with absolute value less than 90 degrees (and not zero)."

Good Hunting!
RD