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An angle is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. Angles provide a means of expressing the difference in slope between two rays meeting at a vertex without the need to explicitly define the slopes of the two rays. Angles are studied in geometry and trigonometry. 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 definitions of right, acute, and obtuse angles are certainly quantitative. 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 are the Latin angere, meaning "to compress into a bend" or "to strangle", and the Greek (angulοs), meaning "crooked, curved"; both are connected with the PIE root Units of measure for angles In order to measure an angle , a circular arc centered at the vertex of the angle is drawn, e.g. with a compass. The length of the arc s is then divided by the radius of the circle r, and possibly multiplied by a constant k: Since the circumference of a circle is proportional to its radius, and an arc in an angle is a fixed fraction of the circumference, the measure of the angle is independent of the size of the circle. Angles are considered dimensionless, since they are defined as the ratio of lengths. There are, however, several units used to measure angles, depending on the choice of the constant k. Decimal subunits are standard, but both binary and sexagesimal subunits are still in use. Approximations Conventions on measurement A convention universally adopted in mathematical writing is that angles given a sign are positive angles if measured counterclockwise, and negative angles if measured clockwise, from a given line. If no line is specified, it can be assumed to be the x-axis in the Cartesian plane. In navigation, bearings are measured from north, increasing clockwise, so a bearing of 45 is north-east. Negative bearings are not used in navigation, so north-west is 315. In mathematics radians are assumed unless specified otherwise because this removes the arbitrariness of the number 360 in the degree system and because the trigonometric functions can be developed into particularly simple Taylor series if their arguments are specified in radians. Types of angles An angle of π/2 radians or 90°, one-quarter of the full circle is called a right angle. Two line segments, rays, or lines (or any combination) which form a right angle are said to be either perpendicular or orthogonal: Some facts In Euclidean geometry, the inner angles of a triangle add up to π radians or 180°; the inner angles of a simple quadrilateral add up to 2π radians or 360°. In general, the inner angles of a simple polygon with n sides add up to (n − 2) × π radians or (n − 2) × 180°. If two straight lines intersect, four angles are formed. Each one has an equal measure to the angle across from it; these congruent angles are called vertical angles. If a straight transversal line intersects two parallel lines, corresponding (alternate) angles at the two points of intersection are equal; adjacent angles are supplementary, that is they add to π radians or 180°. If the sum of two angles equals 90 degrees, they form a complementary angle. If the sum of two angles equals 180 degrees, they form a supplementary angle. Using trigonometric functions A Euclidean angle is completely determined by the corresponding right triangle. In particular, if is a Euclidean angle, it is true that and for two numbers and . So an angle can be legitimately given by two numbers and . To the ratio there correspond two angles in the geometric range , since Using rotations Suppose we have two unit vectors and in the euclidean plane . Then there exists one positive isometry (a rotation), and one only, from to that maps onto . Let r be such a rotation. Then the relation defined by is an equivalence relation and we call angle of the rotation r the equivalence class , where denotes the unit circle of . The angle between two vectors will simply be the angle of the rotation that maps one onto the other. We have no numerical way of determining an angle yet. To do this, we choose the vector , then for any point M on at distance from (on the circle), let . If we call the rotation that transforms into , then is a bijection, which means we can identify any angle with a number between 0 and . Angles in different contexts In the Euclidean plane, the angle θ between two vectors u and v is related to their dot product and their lengths by the formula This allows one to define angles in any real inner product space, replacing the Euclidean dot product · by the Hilbert space inner product <·,·>. 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. Various names (now rarely, if ever, used) have been given to particular cases:—amphicyrtic (Gr. ἀμφί, on both sides, κυρτόσ, convex) or cissoidal (Gr. κισσόσ, ivy), biconvex; xystroidal or sistroidal (Gr. ξυστρίσ, a tool for scraping), concavo-convex; amphicoelic (Gr. κοίλη, a hollow) or angulus lunularis, biconcave. Two intersecting planes form an angle, called their dihedral angle. It is defined as the acute angle between two lines normal to the planes. Also a plane and an intersecting line form an angle. This angle is equal to ninety degrees minus the angle between the intersecting line and the line that goes through the point of intersection and is normal to the plane. Angles in Riemannian geometry In Riemannian geometry, the metric tensor is used to define the angle between two tangents. Where U and V are tangent vectors and gij are the components of the metric tensor G, cos heta = rac . Angles in astronomy In astronomy, one can measure the angular separation of two stars by imagining two lines through the Earth, each one intersecting one of the stars. Then the angle between those lines can be measured; this is the angular separation between the two stars. Astronomers also measure the apparent size of objects. For example, the full moon has an angular measurement of approximately 0.5°, when viewed from Earth. One could say, "The Moon subtends an angle of half a degree." The small-angle formula can be used to convert such an angular measurement into a distance/size ratio. See also Notes | |||||||
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