Angles and Measurement of Angles

                          ANGLES AND MEASUREMENT OF ANGLES

Generation of angles

In plane geometry, an angle is usually said to be formed by two rays (half-lines), radiating from the same end point. The common point is called the vertex of the angle. All angles-acute, obtuse or reflex, are positive and their measures are less than 360 degree. There is no place for negative angles in geometry. This definition of an angle is not general for the purpose of trigonometry, where an angle may be positive or negative of any magnitude (size).

An angle_, in trigonometry, is formed by the rotation of a ray in a plane, around the end point.

The revolving ray is called the generating line of the angle. The initial position and the final position are respectively known as the initial side (arm) and the terminal side (arm). The end point, around which the revolving ray turns, is called the vertex of the angle and the amount of turn is the measure of the angle. The rotating ray is sometimes called the radius vector. If a revolving ray starts from the initial position OX and rotates in a plane, and gradually comes to the positions OA, OB, OC, etc. then the angles ANGLE  XOA, ANGLE XOB, ANGLE XOC, etc. are respectively formed.

POSITIVE AND NEGATIVE ANGLES

By convention, an angle is considered positive, if it is generated by the counter-clockwise motion of the revolving ray and the angle is regarded as negative if the motion is in clockwise direction. The rotating line may revolve several times around a point on it, in anti- clockwise or clockwise sense, before arriving to the terminal position. Thus, we may have a positive or a negative angle of any magnitude (size).

NOTE-Only the Initial position OA and the final position OB are not enough to know the direction and the amount of rotation of an angle.

Curved arrowhead is necessary to indicate completely the direction and the size (magnitude) of the angle.

Let be the measurement of the acute angel AOB. The rotating ray, from its initial position OA, revolving counterclockwisely, arrives to the terminal position OB, the measure of the angle AOB is now if the rotating ray arrives at the position OB after one complete revolution, then the measure of angle  AOB is 360 + .  if it arrives at OB after two complete revolutions,  the measure of angle AOB is n x 360 + . 

If the rotating ray, revolving clockwise comes to the position OB, the measures of the corresponding angles are -360 + , -2 x 360 + , -3 x 360 + , etc.

Thus the same terminal side may  be the bounding line of unlimited number of angles of different measures.