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What is an Angle?

Angle is a term used in geometry and trigonometry and is defined as the figure which is formed by two intersecting lines. These lines are called the sides of an angle. The point where these lines (or line segments) meet is known as the “point of intersection” or vertex of the angle. The magnitude of the angle is measured in terms of how far apart these two lines are, and is termed as the amount of rotation. Angles are measured in degrees, radians, points, and grads.

Mathematically, an angle is defined as the ratio of two lengths, multiplied by a scaling constant k. Since the two lengths cancel each other out, an angle is considered dimensionless. Angles can be measured in both directions - clockwise or anticlockwise. If measured anticlockwise the angle is termed as positive, and if measured clockwise, the angle is termed as negative.

Angles are also used in geography to determine the location of any point on the Earth, and in astronomy to measure the separation of two celestial objects.

Measuring Angles

An angle is denoted by the symbol θ (Theta). Angles are measured by using a protractor, which is divided into 180 degrees and is shaped like a semi-circle. One whole revolution or circle is equal to 360 degrees, 2π radians or 400 grads. Hence, 180 degrees is equal to π radians.

In order to measure an angle, an arc is drawn with the centre at the vertex and the angle θ is then equal to,

θ = length of the arc / radius of the circle

Radius of the circle can be chosen arbitrarily since it has no effect on the size of the angle. The above ratio is multiplied by a constant, k, which is dependent on the choice of units. We know that a full rotation in degrees is equal to 360, so the value of k will be n/360 in degrees or n/2π in radians, where n represents a full circle. Depending upon these measurements, the units can now be defined as,

The degree is equal to 1/360 of a full circle and is denoted by o. The radian (abbreviated as rad) is equal to 180/π degrees. The grad is equal to 1/400 of a full circle.

Types of Angles

There are many types of angles depending on the size or magnitude of an angle. These types are acute angle, right angle, obtuse angle, straight angle, reflex angle, and full angle.

  • An acute angle is between 0 and 90 degrees. It is also defined as an angle less than a right angle.
  • A right angle is equal to 90 degrees (or a quarter of a full rotation). The lines, which make up a right angle are said to be perpendicular to each other.
  • An obtuse angle measures between 90 and 180.
  • A straight angle is defined as half of a full rotation or 180 degrees.
  • A reflex angle is greater in measurement than a straight angle but less than a full angle.
  • A full angle is equal to 360 degrees or a complete circle.

Angles are also categorized as supplementary and complementary angles. Two angles are known as supplementary (supplementing each other) if the sum of their magnitudes is equal to 180 degrees. Similarly, two angles are termed as complementary (complementing each other) if the sum of their degree measurements is equal to 90 degrees.

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