5. (20 points) The clock shows the time as fifteen minutes past five. Determine the angle between the minute and hour hands at this moment.

---

Note: The translation maintains the original text's formatting and structure.
Answer: $67.5^{\circ}$

Solution. Five minutes is $\frac{5}{60}$ of the circumference, which is $30^{\circ}$. The minute hand shows fifteen minutes, which is $90^{\circ}$. The hour hand has moved a quarter of the distance between five ($150^{\circ}$) and six ($180^{\circ}$) hours in fifteen minutes, so the hour hand shows $157.5^{\circ}$. The angle between the minute and hour hands at this moment is: $157.5^{\circ}-90^{\circ}=67.5^{\circ}$.