### Instruction: At $2\!:\!48$ what is the degree measure of the smaller angle formed by the hour and minute hands of a $12$-hour clock?### Response: I know that the hour hand moves $30$ degrees every hour, and the minute hand moves $360$ degrees every hour.
So at $2\!:\!48$, the hour hand has moved $2 \times 30 + 30 \times \frac{48}{60} = 60 + 24 = 84$ degrees from the $12$ o'clock position.
And the minute hand has moved $360 \times \frac{48}{60} = 288$ degrees from the $12$ o'clock position.
The difference between these two angles is $288 - 84 = 204$ degrees.
But this is the larger angle formed by the hour and minute hands, not the smaller one.
To find the smaller angle, I can subtract $204$ from $360$, since the sum of the angles in a circle is $360$ degrees.
So the smaller angle is $360 - 204 = 156$ degrees.