Vertically Opposite Angles: Relationship Between Acute and Obtuse Pairs

To solve this problem, we need to understand the properties of vertically opposite angles:

• Vertically opposite angles are the angles that are opposite each other when two lines intersect.
• One key property of vertically opposite angles is that they are always equal in measure.
• An acute angle is defined as an angle that is less than $90^\circ$.
• An obtuse angle is defined as an angle that is greater than $90^\circ$.

Given that vertically opposite angles are equal, if one angle is acute, the opposite angle must also be acute. This contradicts the statement in the problem that if one is acute, the other will be obtuse.

Therefore, the correct analysis of the problem reveals that the statement is incorrect.

Thus, the solution to the problem is False.