Minimum Angle Between Clock Hands

Angle between hands at time $H:M$ is given by $|30H - 5.5M|$. At 2:30 p.m.: $|30(2) - 5.5(30)| = |60 - 165| = 105^\circ$. As minute hand moves from 30 to 60, the hour hand moves from 2.5 to 3. At $M$ minutes past 2 p.m., angle is $|60 + 0.5M - 6M| = |60 - 5.5M|$. For $M \in [30, 60]$, $60 - 5.5M$ is strictly negative and decreasing (its absolute value increases from 105 at $M=30$ to 270 at $M=60$). So smaller angle $x = 360^\circ - (5.5M - 60)$. At 3:00 p.m., the angle is $90^\circ$. Throughout $2:30$ to $3:00$, the angle continuously decreases to a minimum of $90^\circ$ at 3:00 p.m.