At 10o' 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 angle and the reflex angle made by the hands of a clock at 10 o'clock, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Clock Positions**:
- At 10 o'clock, the hour hand points at 10 and the minute hand points at 12.

2. **Calculating the Angle for Each Hour**:
- The total degrees in a clock is 360 degrees.
- Since there are 12 hours on a clock, each hour represents an angle of:
\[ \text{Angle per hour} = \frac{360 \text{ degrees}}{12} = 30 \text{ degrees} \]

3. **Finding the Angle Between the Hour and Minute Hands**:
- The hour hand is at 10, and the minute hand is at 12.
- The number of hours between 10 and 12 is 2 hours.
- Therefore, the angle between the hour hand and the minute hand is:
\[ \text{Angle} = 2 \text{ hours} \times 30 \text{ degrees/hour} = 60 \text{ degrees} \]

4. **Identifying the Acute Angle**:
- The angle calculated (60 degrees) is the acute angle.

5. **Calculating the Reflex Angle**:
- The reflex angle is the larger angle formed by the hands of the clock.
- Since the total angle around a point is 360 degrees, the reflex angle can be calculated as:
\[ \text{Reflex Angle} = 360 \text{ degrees} - \text{Acute Angle} \]
- Substituting the acute angle:
\[ \text{Reflex Angle} = 360 \text{ degrees} - 60 \text{ degrees} = 300 \text{ degrees} \]

### Final Answers:
- **Acute Angle**: 60 degrees
- **Reflex Angle**: 300 degrees

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