11.1 Parts of an angle.

∠

angle = two sides + the vertex

11.2 Types of angles.

                                                             A) acute  angle < 90˚

                                                             B) right angle = 90˚

                                                             C) obtuse angle > 90˚

                                                             D) straight angle = 180˚

                                                             E) reflex angle > 180˚

                                                             F) one turn = 360 ˚

     right angle = 90˚                      acute (a), obtuse (b) and straight (c) angles                    reflex angle

11.3 Combine angle pairs.

A) complementary angles

Two angles that sum to one right angle (90˚) are called complementary angles.

The difference between an angle and a right angle is termed the complement of the angle.  

The complementary angles a and b (b is the complement of a and a is the complement of b).  

B) supplementary angles

Two angles that sum to a straight angle (180˚) are called supplementary angles.

The difference between an angle and a straight angle (180˚) is termed the supplement of the angle.

Here a and b are supplementary angles.

C) explementary (conjugate) angles

Two angles that sum to one turn (360˚) are called explementary angles or conjugate angles.

Here the sum of the reflex angle and the acute angle makes an explementary angle. 

11.4 Angles between curves.

(mixed angles and curvilinear angles)

The angle between a line and a curve (mixed angle) or between two intersecting curves (curvilinear angle) is defined to be the angle between the tangents at the point of intersection.

In geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Angles are usually presumed to be a Euclidean plane or in the Euclidean space, but are also defined in non-Euclidean geometries. In particular, in spherical geometry, the spherical angles are defined, using arcs of great circles instead of rays.

Angle is also used to designate the measure of an angle or of a rotation. This measure is the ratio of the length of a circular arc to its radius. In the case of a geometric angle, the arc is centered at the vertex and delimited by the sides. In the case of rotation, the arc is centered at the center of the rotation and delimited by any other point and its image by the rotation.

The word angle comes from the Latin word angulus meaning “a corner”. The word angulus is a diminutive, of which the primitive form, angus, does not occur in Latin. Cognate words are the Greek ἀγκύλος  (ankylos), meaning “crooked, curved”, and the English word “ankle”.

Euclid defines a plane angle as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other. According to Proclus an angle must be either a quality or a quantity, or a relationship. The first concept was used by Eudemus, who regarded an angle as a deviation from a straight line; The second by Carpus of Antioch, who regarded it as the interval or space between the intersecting lines; Euclid adopted the third concept, although his definition of right, acute and obtuse angles are certainly quantitative.

Measuring angles.

The size of a geometric angle is usually characterised by the magnitude of the smallest rotation that maps one of the rays into the other. Angles that have the same size are sometimes called congruent angles. In some contexts, such as identifying a point on a circle or describing the orientation of an object in two dimensions relative to a reference orientation, angles that differ by an exact multiple of a full turn are effectively equivalent. In other contexts, such as identifying a point on a spiral curve or describing the cumalative rotation of an object in two dimensions relative to a reference orientation, angles that differ by a non-zero multiple of a full turn are not equivalent.

                 The measurement of angle  θ  is the quotient of s and r. 

In order to measure an angle θ, a circular arc centered at the vertex of the angle is drawn, e.g.with a pair of compasses. The length of the arc s is then divided by the radius of the arc r, and possibly multipied by a scaling constant (which depends on the units of measurement that are chosen):

The value of θ thus defined is independent of the size of the circle: if the length of the radius is changed then the arc length changes in the same proportion, so the ratio s/r is unaltered.