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Measuring angles is a fundamental concept in geometry, crucial for calculating the positions of clock hands. Angles on a clock are measured in degrees. One full rotation of a clock's hands represents 360 degrees. To understand clock angles better, consider this:

• The entire circumference of the clock face is divided into 360 degrees.
• As the hands move, they form angles with the vertical line from the center at 12 o'clock.
• Angles are typically measured from this 12 o'clock position clockwise to the position of the hand.

Knowing how to measure angles helps you solve problems such as the one given. It's key to understanding the relationship between the positions of the hour and minute hands at various times.

The hour and minute hands of a clock move at different rates, contributing to the unique angles they form.

• The minute hand completes a full circle (360 degrees) every 60 minutes.
• It moves 6 degrees per minute because 360 degrees divided by 60 minutes equals 6 degrees per minute.
• The hour hand, meanwhile, moves 30 degrees per hour or 0.5 degrees every minute, this is because 360 degrees divided by 12 hours equals 30 degrees per hour.

Understanding the movement of both hands is necessary to calculate the angle between them at any given time. For example, by knowing these per-minute movements, we can determine that at 12:20: - The minute hand is at 120 degrees from 12. - The hour hand is at 10 degrees from 12.

Calculating the angle between the hour and minute hands involves determining the angle each hand has moved from the 12 o'clock position and finding their difference.Here's the step-by-step process:- First, calculate the angle for the minute hand, which moves 6 degrees per minute. At 20 minutes past, it has moved to: \[ \text{Angle of minute hand} = 6 \times 20 = 120 \text{ degrees} \] - Next, calculate the hour hand's angle, which moves 0.5 degrees per minute from its last hour position. At 12:20, it has moved: \[ \text{Angle of hour hand} = (12 \times 30) + (20 \times 0.5) = 0 + 10 = 10 \text{ degrees} \] - Finally, find the difference between these two angles to get the angle between the hands: \[ \text{Angle between hands} = |120 - 10| = 110 \text{ degrees} \]This difference gives us the smaller angle when both calculations provide a result less than 180 degrees. If the calculation results in a larger angle, you would subtract it from 360 degrees to find the smaller angle. However, for 12:20, 110 degrees is already the smaller angle.