At 10 O' clock the hands of a clock make an acute angle and a reflex angle. Find the measure of each of these angles

Answer

To find the acute and reflex angles formed by the hands of a clock at 10 o'clock, we can follow these steps:

### Step 1: Understand the positions of the clock hands
At 10 o'clock, the minute hand points at 12, and the hour hand points at 10.

### Step 2: Calculate the angle between the hour and minute hands
The clock is divided into 12 hours, and each hour represents an angle of:
\[ \text{Angle per hour} = \frac{360^\circ}{12} = 30^\circ \]
The hour hand at 10 o'clock is at the 10th hour mark.

### Step 3: Determine the number of hour marks between the hour hand and the minute hand
From 10 to 12, there are 2 hour marks (10 to 11 and 11 to 12). Therefore, the angle between the hour hand and the minute hand can be calculated as:
\[ \text{Angle} = \text{Number of hour marks} \times \text{Angle per hour} = 2 \times 30^\circ = 60^\circ \]

### Step 4: Identify the acute angle
The angle we calculated (60°) is the acute angle formed between the hour and minute hands at 10 o'clock.

### Step 5: Calculate the reflex angle
The reflex angle is the larger angle formed by the hands of the clock. Since a full rotation is 360°, the reflex angle can be found by subtracting the acute angle from 360°:
\[ \text{Reflex angle} = 360^\circ - \text{Acute angle} = 360^\circ - 60^\circ = 300^\circ \]

### Final Answers:
- The acute angle at 10 o'clock is **60°**.
- The reflex angle at 10 o'clock is **300°**.

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