ALL About Angle

Definition:
An angle is a combination of two rays (half-lines) with a common endpoint. The latter is known as the vertex of the angle and the rays as the sides, sometimes as the legs and sometimes the arms of the angle.

Under this definition, two angles are congruent provided either can be made to coincide (the vertex and the sides) with the other by a rigid motion. If O is the vertex of an angle while A and B are points on the two sides, the angle may be referred to as ∠AOB or ∠BOA (and this for any selection of the two points A and B.)

In elementary geometry, the definition mostly works, altough at times a text author is forced to make excuses or skip over important details. Angles can be compared and, just as linear segments, added and subtracted. To this end, the definition alone does not suffice.

To enable comparison and addition, some texts [Hilbert, Kiselev, O'Daffer] associate with an angle one of two regions into which the two sides of the angle split the plane. One of these is termed theinterior and the other the exterior of the angle. In order to compare the angles they should be placed so their interiors intersect while some two sides and the vertices coincide. The angle whose other side is located in the interior of the other angle is declared (and naturally so) the smaller of the two. For addition, we overlap one side of one angle with a side of the other so as to insure that their interiors do not intersect. The two free sides (one from each of the addends) form an angle which is declared the sum of the two.

With some caution, we can define straight and right angles. An angle is straight when its sides form a straight line. That angle is right which, when doubled (i.e. added to itself), gives a straight angle. In Euclid's terms (Definition I.10), "When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is right." Acute and obtuse angles are smaller and respectively greater than right. Usually, obtuse angles are taken to be smaller than straight, in which case the angles that exceed the straight angle are said to be reflex. A reflex angle so big as to have its two sides overlap is full. (A dynamic illustration is available elsewhere.)