Angle
Is an angle in the geometry of a part of the plane of the two in-plane rays ( half lines ) is defined with a common starting point.
The common starting point of the two beams is called apex of the angle, the apex angle or simply " peak "; the rays of the hot leg of the angle. An angle can be defined by three points, one of which is located in the vertex of the angle and the other two on each side of the angle a.
The physical quantity, which describes the relative position of the beam to each other is referred to as angular width, rotational angle or angular distance (angle distance), normally also for short as "Angle", when a distinction between the geometric object is not necessary, for example, in physics. The size of the angle is marked with a square.
The angular width can also be defined as a measure of a planar rotation.
To distinguish it from the solid angle of the angle defined here is also referred as a flat angle.
- 4.1 complementary angle or complementary angle
- 4.2 Supplement angle or supplementary angles
- 4.3 supplementary angles
- 4.4 apex angle or angle counter
- 4.5 step angle or F- angle
- 4.6 alternate angles or Z angle
- 4.7 neighbor angle or e- angle
- 4.8 angle with pairwise orthogonal legs
- 5.1 Construction of the 90-degree angle ( right angle ) 5.1.1 Corollary ( line bisection perpendicular bisector )
- 5.2.1 Corollary ( construction equilateral triangles)
- 5.2.2 Corollary ( construction of hexagons )
- 5.5.1 bisector
- 5.5.2 trisection
- 5.5.3 Any division
Definition
In geometry, different approaches are possible to define the angle of an object. Here, two types can be distinguished:
- The non-directional angle, which is characterized by an unsigned angular width.
- The directional angle of the orientation features, and is measured as the angle of rotation or angular distance.
View as beam pair
The above- mentioned definition of one point two outgoing beams is included in the applications such as the coordinate systems and their axes.
View as half-line pair
The angle is a geometric structure of two half-lines.
About the " original " line this representation allows some considerations about the various pairs of angles.
View as part of the plane
The definition is used in the classroom and emphasizes the " corporeal " of the structure, and is used - on the establishment of an indoor and outdoor space - an introduction to the triangle geometry: The triangle can be defined as the intersection of two angles with a common leg.
Ad hoc is in these three approaches, the angle an undirected angle, only an additional award one of the two half lines or half-lines as the "first " allows the specification of a directed angle.
View as rotation
One can also say that an angle is formed by rotating a ray or a half-line in a plane to his or her starting point.
Since there are two different ways to turn the beam, the direction of rotation must be specified in addition:
- Left rotation, called counterclockwise mathematically positive direction ( angle is positive ); shown in green in the image.
- Right rotation, called clockwise, also mathematically negative sense of rotation (angle is negative); purple in the picture shown.
In mathematics, it is common to the counterclockwise rotation - ie, in the mathematically positive sense - run. When the rotation is to be the other way round, it should be explicitly stated.
In geodesy ( surveying ) the clockwise angle, ie clockwise from 0 to 400 Gon Gon is counted. Since, by definition, are no negative angle in geodesy, the sense of rotation is positive. Analogous to the clock, also here is from 0 to 24 h positive, clockwise counted. All geodetic measuring instruments are rotated to the direction or angle measurement to the right.
Designation of angles
The specification of an angle according to DIN 1302 or ISO 31-11, more recently, according to ISO 80000-2.
- Angles are usually denoted by small Greek letters, such as α or β.
- An angle is an angle between two half- rays, lines, edges, and the like. He is then starting direction of f g counted.
- Alternatively, you can specify the three points that define the angle, the vertex is always in the middle, such as angle ABC, or outdated. This is the angle between [BA ] and [BC ], where [ BA] is rotated in the mathematically positive direction on [BC ].
- In the English -speaking world is only an indication of the vertex or common.
For the set of formulas is the character " ∠ " (HTML ∠ / ∠, TeX \ angle, Unicode U 2221 ) are available for the directed angle also " ∡ " ( TeX \ measuredangle, U 2220 MEASURED ANGLE, no HTML Entity ), which are both found in the Mathematical operators Unicode block. The caret is located corresponds to the Anglo-American habits in the European Formula set a sign is common that the American " ∢ " U 2222 for the solid angle for the spitting image. " ∠ " is also for slope and angularity ( position tolerance, DIN EN ISO 1101) use. Especially for the right angle one uses " ∟ ", a punctured angle in the art as a square or.
Angular dimensions and units of measurement for angles
Detailed information can be main article angular, conversions can be found in the individual measurements.
Other forms of expression of an angle:
- The tangent of the angular width of the pitch angle (also called pitch dimension corresponds to the readout in percent)
- A pair (x, y) with cosine and sine (corresponding to the Cartesian coordinates of the point on the unit circle )
Types of angles
Between two intersecting straight lines, there are four angles. Each two adjacent sum up this to 180 °. The right angle has the peculiarity that these two angles are exactly the same. Two opposite angles are equal. The full angle has the particularity that two of the angles are zero.
Two straight lines or lines that intersect at right angles, is called orthogonal. In a drawing, the right angle is represented by a quarter circle with a dot or a square.
The full angle is in Germany a legal unit of measurement, it has no unit symbol.
Special pairs of angles
The geometry knows special names for pairs of angles that are in a special relationship. The rules governing such angle laws help in the investigation of complex geometric objects.
Complementary angle or complementary angle
Two angles are called complementary angle or complementary angles if they add up to a right angle ( 90 °).
Supplementary angle or supplementary angles
Two angles are called supplementary angle (also: Supplementärwinkel ), supplementary angles or short e- angle if they add up to 180 °.
Supplementary angles
Intersect two lines, one is a pair of adjacent angles as a side angle.
So you are supplementary angle.
Vertex angle or angle counter
Intersect two straight lines, one refers to the pair of opposite angle as the vertex angle or opposite angle.
The term vertex angle is because the two angles are represented by point reflection at the apex of each other.
Step angle or F- angle
Cuts a straight line and two lines, so the angles of each other and on other pages of this and are on the same page, stage or F- angle hot. For the case, that the straight lines and are parallel, the following applies:
From the angular equality can be reversed closed on the parallelism of the line: if a pair of straight lines cut by a further straight line such that the cutting angle is on the same side of and on each respective side of, and equal, then the straight line and parallel.
Alternate angles or Z angle
Cuts a straight line and two lines, the angles that lie on opposite sides of and opposite sides of and, changing, or Z angle hot. For the case, that the straight lines and are parallel, the following applies:
From the angular equality can be reversed closed on the parallelism of the line: if a pair of straight lines cut by a further straight line such that the average angles are on opposite sides of and different sites or the same size as are the straight and parallel.
Neighbor angle or e- angle
Intersects a straight line, two further parallel lines and is referred to as the angles on the same side of, but on different sides of, and, as a neighbor or E- angle.
From the addition to the angle of 180 ° can be reversed closed on the parallelism of the line: if a pair of straight lines cut by a further straight line in such a way that the intersection angle, which lie on the same side of, but in each case on different sides of, and to supplement to 180 °, then the straight line and parallel.
The property that complement neighbor angle to 180 °, it follows directly from the parallel postulate of Euclidean geometry. The above characteristics of stages and alternating angles can be derived from consideration of secondary and apex angles of adjacent angles.
Angle with pairwise orthogonal legs
Two angles whose legs are perpendicular to each other, are equal or add up to 180 °. Compare the following illustrations.
Angle construction
Some angle you can only construct with ruler and compass. These include the 90 degree, 60 degree, 72 degree and 54 degree angle, and all angles (see below), this angle caused by doubling, halving, addition or subtraction.
Not every angle can be divided into thirds with ruler and compass.
Construction of the 90-degree angle ( right angle )
We designed more specifically the perpendicular to a given line already.
Inference ( line bisection perpendicular bisector )
Halving a given distance by circles with the same radius, the radius of which is greater than half the distance and which intersect at two points to the end points takes this route. If you connect up the intersections that have both circles together, this connection line intersects the line exactly in the middle and at right angles. As a result, a central vertical line was constructed. ( A Streckenhalbierende nonperpendicular goes by the distance you standing obtained by appropriate design with an angle to the path provided ellipses. Ellipses not at an angle, one obtains a central vertical line again. )
Construction of a 60-degree angle
Constructing at the apex point on a given line, a circle and carrying, starting from the intersection point between the circle and line once the radius of the circle on the circuit itself. The connection between the apex and the thus constructed includes the intersection with the straight line given an 60 - degree angle.
You connect this intersection and the vertex by a line with a ruler.
Inference ( construction equilateral triangles)
If you connect in addition to the first step constructed intersection point on the given line, with the last constructed point of intersection, the result is an equilateral triangle. This consequently has three equal angles of 60 degrees.
Should we therefore construct an equilateral triangle from a given page size, so you draw a line, take the page size in the circle, and smite about an arbitrary point on the line a circle. Man stabs the point of intersection between circle and line and so carries the page length to the circle itself. Now you connect the last point constructed with two puncture points.
Inference ( construction of hexagons )
Plotting on any circle the radius, has the circle itself, with the compass off, you get when you all on the circle adjacent intersections by a straight line connecting a regular hexagon ( hexagon ).
This is because if one connects the circle center to the corners of the hexagon are each 6 equilateral triangles is replaced, be the angle at the center of the circle by 60 degrees. 6.60 ° = 360 °, ie, a circle of isosceles triangles, whose peculiarity is to be also equilateral.
Construction of a 72 - or 54 -degree angle
For something more exotic construction of the 72 ° - or 54 ° - angle to construct a regular pentagon.
Addition and subtraction of angles
Each angle can be add and subtract constructively to another angle. To the vertices of the two angles, if you want to add or subtract a first to a second angle, ie increase or decrease the second angle about the size of the first, as distinguished initially each one the same size for both angle circle. ( It is sufficient in both cases an arc that intersects both legs of the angle you want. During the second angle of the circular arc must also be so great that it even to be added to or subtracted angle, where it is removed later, include. analog ranges may also have a greater number of even smaller arcs that make only the necessary intersections marked. )
Now one uses the first angle from the distance between the two points of intersection of the circle and legs with a compass and this contributes to the circular arc of the second angle from. For the latter, it pierces the circle into those intersection of circle and legs of the second angle from the leg you want to add or subtract the first angle. Then you wear with a pair of compasses the set distance on the arc of the second angle from - either from the angle away ( in the direction of " angle appearance ") if you want to add, or to the angle towards ( towards " angle Affairs ") if you want to subtract. Thereafter, drawing a ray from the apex of the second angle by the flat construct point of the arc. This beam, together with that of the second leg angle at which they had not inserted, the legs of the now constructed sum and difference angle.
Angular pitches
Bisector
An angle always consists of two legs that meet at the vertex. If we now draw two equal circles on the forks through the vertex, the distance between the circle intersections is the bisector. Each point on the bisecting line is an equal distance from said legs.
Tripartition
The general trisection of an angle is not possible with Euclidean tools. However, there are (hand) drawing utensils (eg Tomahawk ) for this task.
Any division
The arbitrary division requires a tool with which a corner can be imaged in proportion to a distance, and vice versa, such as a mask, with a shaped edge of a spiral of Archimedes. This allows an angular division transfer it to a division into sections.
Inference ( general angle structures)
Constructed to the above angles (90 °, 60 °, 72 ° or 54 ° or their sums or differences ), as can be derived from this by bisection other angles ( 45 °, 30 °, 36 ° and 27 ° or the corresponding sums or differences ) construct, which can be halved again, and their descendants. In general, let all integer angle construct that are multiples of 3 °.