Find the measure of the smaller angle formed by the hands of a clock at the following time. $$ 1: 15 $$ The angle is $$ \square^{\circ} \square^{\prime} $$. (Simplify your answers.)

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To determine the smaller angle between the hour and minute hands at **1:15**, follow these steps:

1. **Minute Hand Position:**
   - Each minute represents **6°** (since $$360° \div 60 = 6°$$).
   - At **15 minutes**, the minute hand is at:
     $$
     15 \times 6° = 90°
     $$

2. **Hour Hand Position:**
   - Each hour represents **30°** (since $$360° \div 12 = 30°$$).
   - At **1:00**, the hour hand is at:
     $$
     1 \times 30° = 30°
     $$
   - Additionally, the hour hand moves **0.5° per minute** (since $$30° \div 60 = 0.5°$$).
   - In **15 minutes**, it moves:
     $$
     15 \times 0.5° = 7.5°
     $$
   - Therefore, at **1:15**, the hour hand is at:
     $$
     30° + 7.5° = 37.5°
     $$

3. **Calculating the Angle Between the Hands:**
   - The absolute difference between the minute and hour hands is:
     $$
     |90° - 37.5°| = 52.5°
     $$
   - Converting **52.5°** to degrees and minutes:
     $$
     52° \, 30′
     $$

**Answer:**  
The angle is $$52^{\circ}\, 30′$$.
The smaller angle between the clock hands at 1:15 is $$52^{\circ}\, 30′$$.

Find the measure of the smaller angle formed by the hands of a clock at the following time. \( 1: 15 \) The angle is \( \square^{\circ} \square^{\prime} \). (Simplify your answers.)

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