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Angle

The noun ‘angle’ refers to a shape created by two rays emerging from a common point called as vertex of the angle.
The rays are called legs, edges or sides of the angle.

In other words, angle is defined as the measure of rotation of a radius on a circular motion with reference to a given axis.

In practice, the radius or the circular motion is not visualized and the measure is just the angular displacement between two sides. Mostly the latter terminology is often used in practice.

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Unit of measurement of Angles

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The two units that are most common for measuring angles are degrees and radians.
The unit of degree is denoted by a superscript "$^{\circ}$".
For example, a measure of 45 degrees is denoted as 45$^{\circ}$.
As mentioned earlier angle measures are related to rotations on a circular path. When a point is rotated its measures in degrees at various positions are,
  • 90$^{\circ}$, when the rotating radius is at a perpendicular position to the initial position. This measure is called as perpendicular angle or right angle.
  • 180$^{\circ}$, when the rotating radius is at the same straight line as the initial position (but the tip of the radius on the opposite side). This measure is called as straight line angle.
  • 360$^{\circ}$ when the rotating radius comes back to the initial position. This measure is called as full angle.
The unit of radian is denoted as rad. This unit is a bit easier to define. It is a ratio of the circular arc generated in the rotation to the radius of the circular path. We all know that the total length of the circular arc (the circumference) of a complete circle is $2\pi r$ units and hence a full angle is $\frac{2\pi r}{r}$ = $2\pi$ rad. Applying the same concept, a right angle is $\frac{\pi}{2}$ rad and a straight angle is $\pi $ rad.

Types of Angles

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Angles are classified in several ways.
Types of Angles
  • If the angle is referred to the measure that is interior of two sides of a shape, then it is called as interior angle and the angle that is exterior of the same two sides is named as exterior angle. One can easily conclude that the sum of interior and exterior angles for the same set of sides is a straight angle.
  • A set of two angles that are formed on the same vertex but have different interior areas (not sharing a common area) are called adjacent angles. If the sum of two adjacent angles is a right angle, then the set of angles are called as complementary angles. On the other hand, if the sum of two adjacent angles is a 180$^{\circ}$, then the set of angles are called as supplementary angles. The set is also called as linear pair.
  • An acute angle is an angle measuring < 90$^{\circ}$, an obtuse angle measures > 90$^{\circ}$ but < 180$^{\circ}$ and the measure of a reflex angle lies between 180$^{\circ}$ and 360$^{\circ}$.
  • A set of four angles are formed when two lines intersect sharing a single vertex. The set angles that are opposite (meaning which are not adjacent angles) are called as vertical angles. Vertical angles are always congruent.

More topics in  Angle
Angle between Two Planes Angle of Inclination
Angle Between Line and Plane Vertical Angles Theorem
Angle formed by Parallel Lines and Transversals
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