What is the number of degrees in the smaller angle formed by the hands of a clock at $8:55$?
It takes 12 hours for the hour hand to do a full revolution of 360 degrees. Moving 5 minutes before 9, the hour hand is somewhere in the sector that is $\frac{5}{60}=\frac{1}{12}$ of the sector between the 8 and 9. Therefore, the angle between the hour and 8 o'clock hand is $\frac{1}{12}\times 360^\circ=\boxed{30^\circ}$.
In addition, the minute hand of a clock goes $360^\circ$ every hour and $6^\circ$ every minute, so it makes $\left(\frac{55}{60}\right)\times 6^\circ=\boxed{330^\circ}$ with the vertical 12 o'clock position.