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Lines and Angles - Making Rotating Arms

Grade 6CBSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The rotating arm concept helps visualize how an angle is formed by the rotation of a ray from its initial position to its terminal position. A full circle rotation is equivalent to one complete revolution or 360∘360^\circ.

Rotating arm showing a 90 degree movement from initial to terminal position.
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Direction of rotation matters: moving clockwise or anti-clockwise from a reference point like North, South, East, or West creates specific angles. For example, moving from North to South is a half-turn or 180∘180^\circ.

Direction compass showing North, South, East, and West.
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A right angle (90∘90^\circ) represents 14\frac{1}{4} of a revolution. A straight angle (180∘180^\circ) represents 12\frac{1}{2} of a revolution. A complete turn (360∘360^\circ) is 11 whole revolution.

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Angles between 0∘0^\circ and 90∘90^\circ are acute, between 90∘90^\circ and 180∘180^\circ are obtuse, and angles greater than 180∘180^\circ but less than 360∘360^\circ are reflex angles.

📐Formulae

1 Complete Revolution=360∘1 \text{ Complete Revolution} = 360^\circ

Right Angle=14×360∘=90∘\text{Right Angle} = \frac{1}{4} \times 360^\circ = 90^\circ

Straight Angle=12×360∘=180∘\text{Straight Angle} = \frac{1}{2} \times 360^\circ = 180^\circ

Reflex Angle=360∘−Acute/Obtuse Angle\text{Reflex Angle} = 360^\circ - \text{Acute/Obtuse Angle}

💡Examples

Problem 1:

If a rotating arm moves from North to East in a clockwise direction, what fraction of a revolution has it completed and what is the angle formed?

Solution:

Fraction = 14\frac{1}{4}, Angle = 90∘90^\circ

Explanation:

The four main directions (North, East, South, West) divide a circle into four equal parts. Moving from North to East is one part out of four, which is 14\frac{1}{4} of a revolution. Since a full revolution is 360∘360^\circ, the angle is 14×360∘=90∘\frac{1}{4} \times 360^\circ = 90^\circ.

Problem 2:

What is the measure of the angle formed by the hands of a clock when it moves from 1212 to 66?

Solution:

180∘180^\circ

Explanation:

The movement from 1212 to 66 on a clock face represents exactly half of the circular face. This is 12\frac{1}{2} of a revolution. 12×360∘=180∘\frac{1}{2} \times 360^\circ = 180^\circ, which is also known as a straight angle.

Problem 3:

Classify an angle that measures 245∘245^\circ.

Solution:

Reflex Angle

Explanation:

Any angle that is greater than 180∘180^\circ but less than 360∘360^\circ is classified as a reflex angle. Since 180∘<245∘<360∘180^\circ < 245^\circ < 360^\circ, it is a reflex angle.

Problem 4:

A rotating arm starts at North and turns clockwise to reach South-West. What is the angle of rotation and what fraction of a revolution has it covered?

Angle of 225 degrees shown from North to South-West.

Solution:

  1. From North to South (clockwise) is 180∘180^\circ (12\frac{1}{2} revolution).
  2. From South to South-West is half of a right angle, which is 45∘45^\circ (18\frac{1}{8} revolution).
  3. Total angle = 180∘+45∘=225∘180^\circ + 45^\circ = 225^\circ.
  4. Fraction of revolution = 225360=58\frac{225}{360} = \frac{5}{8}.

Explanation:

Moving from North to South is a straight line (180∘180^\circ). Adding the movement to the midway point between South and West adds another 45∘45^\circ, totaling 225∘225^\circ.

Problem 5:

Determine the angle and the fraction of the revolution when a clock hand moves from 22 to 55.

Clock face showing a 90 degree angle between the 2 and 5 hour marks.

Solution:

  1. Each hour mark on a clock represents 360∘12=30∘\frac{360^\circ}{12} = 30^\circ.
  2. The hand moves from 22 to 55, which is 5−2=35 - 2 = 3 hour spaces.
  3. Angle = 3×30∘=90∘3 \times 30^\circ = 90^\circ.
  4. Fraction = 90360=14\frac{90}{360} = \frac{1}{4} revolution.

Explanation:

Since the clock is divided into 12 equal sectors, moving through 3 sectors constitutes a quarter of the total rotation.