At 2o'clock, the hands of a clock make an acute angle and a reflex angle. Calculate the size, in degrees, of each of these angles

Answer

To find the acute and reflex angles formed by the hands of a clock at 2 o'clock, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Position of the Hands:**
At 2 o'clock, the hour hand points at 2 and the minute hand points at 12.

2. **Calculating the Angle of the Hour Hand:**
The clock is divided into 12 hours, and the full circle of the clock is 360 degrees. Therefore, each hour represents:
\[ \text{Angle per hour} = \frac{360 \text{ degrees}}{12 \text{ hours}} = 30 \text{ degrees} \]
At 2 o'clock, the hour hand has moved:
\[ \text{Angle of hour hand} = 2 \text{ hours} \times 30 \text{ degrees/hour} = 60 \text{ degrees} \]

3. **Calculating the Angle of the Minute Hand:**
At 2 o'clock, the minute hand is at the 12, which corresponds to:
\[ \text{Angle of minute hand} = 0 \text{ degrees} \]

4. **Finding the Angle Between the Hour Hand and the Minute Hand:**
The angle between the hour hand and the minute hand is given by:
\[ \text{Angle between hands} = |\text{Angle of hour hand} - \text{Angle of minute hand}| \]
Substituting the values:
\[ \text{Angle between hands} = |60 \text{ degrees} - 0 \text{ degrees}| = 60 \text{ degrees} \]

5. **Determining the Reflex Angle:**
The reflex angle is the angle that is not the acute angle. It can be calculated by subtracting the acute angle from 360 degrees:
\[ \text{Reflex angle} = 360 \text{ degrees} - \text{Acute angle} \]
Thus:
\[ \text{Reflex angle} = 360 \text{ degrees} - 60 \text{ degrees} = 300 \text{ degrees} \]

### Final Answers:
- The acute angle at 2 o'clock is **60 degrees**.
- The reflex angle at 2 o'clock is **300 degrees**.