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Triangle
A TRIANGLE is a polygon with three edges and three vertices . It is one of the basic shapes in geometry . A triangle with vertices A, B, and C is denoted A B C {displaystyle triangle ABC} . In Euclidean geometry any three points, when non-collinear, determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space ). This article is about triangles in Euclidean geometry except where otherwise noted
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Tally Marks
TALLY MARKS, also called HASH MARKS, are a unary numeral system . They are a form of numeral used for counting . They are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. However, because of the length of large numbers, tallies are not commonly used for static text. Notched sticks, known as tally sticks , were also historically used for this purpose. CONTENTS * 1 Early history * 2 Clustering * 3 Writing systems * 4 See also * 5 References EARLY HISTORY Main article: Prehistoric numerals Counting
Counting
aids other than body parts appear in the Upper Paleolithic . The oldest tally sticks date to between 35,000 and 25,000 years ago, in the form of notched bones found in the context of the European Aurignacian to Gravettian and in Africa's Late Stone Age
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If And Only If
In logic and related fields such as mathematics and philosophy , IF AND ONLY IF (shortened IFF) is a biconditional logical connective between statements. In that it is biconditional , the connective can be likened to the standard material conditional ("only if", equal to "if ... then") combined with its reverse ("if"); hence the name. The result is that the truth of either one of the connected statements requires the truth of the other (i.e. either both statements are true, or both are false). It is controversial whether the connective thus defined is properly rendered by the English "if and only if", with its pre-existing meaning. There is nothing to stop one from stipulating that we may read this connective as "only if and if", although this may lead to confusion
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Plane (mathematics)
In mathematics , a PLANE is a flat, two-dimensional surface that extends infinitely far. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space . Planes can arise as subspaces of some higher-dimensional space, as with a room's walls extended infinitely far, or they may enjoy an independent existence in their own right, as in the setting of Euclidean geometry
Euclidean geometry
. When working exclusively in two-dimensional Euclidean space
Euclidean space
, the definite article is used, so, the plane refers to the whole space. Many fundamental tasks in mathematics, geometry , trigonometry , graph theory , and graphing are performed in a two-dimensional space, or, in other words, in the plane
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Euclid
EUCLID (/ˈjuːklᵻd/ ; Greek : Εὐκλείδης, Eukleidēs Ancient Greek: ; fl. 300 BCE), sometimes called EUCLID OF ALEXANDRIA to distinguish him from Euclides of Megara , was a Greek mathematician , often referred to as the "father of geometry". He was active in Alexandria
Alexandria
during the reign of Ptolemy I (323–283 BCE). His Elements is one of the most influential works in the history of mathematics , serving as the main textbook for teaching mathematics (especially geometry ) from the time of its publication until the late 19th or early 20th century. In the Elements, Euclid
Euclid
deduced the principles of what is now called Euclidean geometry from a small set of axioms . Euclid
Euclid
also wrote works on perspective , conic sections , spherical geometry , number theory , and rigor
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Euclid's Elements
The ELEMENTS ( Ancient Greek
Ancient Greek
: Στοιχεῖα Stoicheia) is a mathematical treatise consisting of 13 books attributed to the ancient Greek mathematician Euclid in Alexandria
Alexandria
, Ptolemaic Egypt circa 300 BC. It is a collection of definitions, postulates, propositions (theorems and constructions ), and mathematical proofs of the propositions. The books cover plane and solid Euclidean geometry , elementary number theory , and incommensurable lines. Elements is the oldest extant large-scale deductive treatment of mathematics . It has proven instrumental in the development of logic and modern science , and its logical rigor was not surpassed until the 19th century. Euclid's Elements
Euclid's Elements
has been referred to as the most successful and influential textbook ever written
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Theorem
In mathematics , a THEOREM is a statement that has been proved on the basis of previously established statements, such as other theorems, and generally accepted statements, such as axioms . A theorem is a logical consequence of the axioms. The proof of a mathematical theorem is a logical argument for the theorem statement given in accord with the rules of a deductive system . The proof of a theorem is often interpreted as justification of the truth of the theorem statement. In light of the requirement that theorems be proved, the concept of a theorem is fundamentally deductive , in contrast to the notion of a scientific law , which is experimental . Many mathematical theorems are conditional statements. In this case, the proof deduces the conclusion from conditions called HYPOTHESES or premises
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Degree (angle)
A DEGREE (in full, a DEGREE OF ARC, ARC DEGREE, or ARCDEGREE), usually denoted by ° (the degree symbol ), is a measurement of a plane angle , defined so that a full rotation is 360 degrees. It is not an SI unit
SI unit
, as the SI unit
SI unit
of angular measure is the radian , but it is mentioned in the SI brochure as an accepted unit . Because a full rotation equals 2π radians, one degree is equivalent to π/180 radians. CONTENTS * 1 History * 2 Subdivisions * 3 Alternative units * 4 See also * 5 Notes * 6 References * 7 External links HISTORY A circle with an equilateral chord (red). One sixtieth of this arc is a degree. Six such chords complete the circle. The original motivation for choosing the degree as a unit of rotations and angles is unknown. One theory states that it is related to the fact that 360 is approximately the number of days in a year
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Necessary And Sufficient Condition
In logic , NECESSITY and SUFFICIENCY are implicational relationships between statements . The assertion that one statement is a necessary and sufficient condition of another means that the former statement is true if and only if the latter is true. That is, the two statements must be either simultaneously true or simultaneously false. In ordinary English , "necessary" and "sufficient" indicate relations between conditions or states of affairs, not statements. Being a male sibling is a necessary and sufficient condition for being a brother
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Vertex (geometry)
In geometry , a VERTEX (plural: VERTICES or VERTEXES) is a point where two or more curves , lines , or edges meet. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices. CONTENTS* 1 Definition * 1.1 Of an angle * 1.2 Of a polytope * 1.3 Of a plane tiling * 2 Principal vertex * 2.1 Ears * 2.2 Mouths * 3 Number of vertices of a polyhedron * 4 Vertices in computer graphics * 5 References * 6 External links DEFINITIONOF AN ANGLE A vertex of an angle is the endpoint where two line segments or rays come together. The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments and lines that result in two straight "sides" meeting at one place
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Polytope
In elementary geometry , a POLYTOPE is a geometric object with "flat" sides. It is a generalisation in any number of dimensions, of the three-dimensional polyhedron . Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or N-POLYTOPE. Flat sides mean that the sides of a (k+1)-polytope consist of k-polytopes that may have (k-1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations , decompositions or tilings of curved manifolds including spherical polyhedra , and set-theoretic abstract polytopes . Polytopes in more than three dimensions were first discovered by Ludwig Schläfli
Ludwig Schläfli

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Exterior Angle
2D ANGLES * Right * Interior * Exterior-------------------------2D ANGLE PAIRS * Adjacent * Vertical * Complementary * Supplementary * Transversal -------------------------3D ANGLES * Dihedral Internal and External angles In geometry , an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex , this angle is called an INTERIOR ANGLE (or INTERNAL ANGLE) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex . If every internal angle of a simple polygon is less than 180°, the polygon is called convex . In contrast, an EXTERIOR ANGLE (or EXTERNAL ANGLE) is an angle formed by one side of a simple polygon and a line extended from an adjacent side . :pp
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Internal Angle
2D ANGLES * Right * Interior * Exterior-------------------------2D ANGLE PAIRS * Adjacent * Vertical * Complementary * Supplementary * Transversal -------------------------3D ANGLES * Dihedral Internal and External angles In geometry , an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex , this angle is called an INTERIOR ANGLE (or INTERNAL ANGLE) if a point within the angle is in the interior of the polygon. A polygon has exactly one internal angle per vertex . If every internal angle of a simple polygon is less than 180°, the polygon is called convex . In contrast, an EXTERIOR ANGLE (or EXTERNAL ANGLE) is an angle formed by one side of a simple polygon and a line extended from an adjacent side . :pp
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Tetrakis Square Tiling
In geometry , the TETRAKIS SQUARE TILING is a tiling of the Euclidean plane . It is a square tiling with each square divided into four isosceles right triangles from the center point, forming an infinite arrangement of lines . It can also be formed by subdividing each square of a grid into two triangles by a diagonal, with the diagonals alternating in direction, or by overlaying two square grids, one rotated by 45 degrees from the other and scaled by a factor of √2 . Conway calls it a KISQUADRILLE, represented by a kis operation that adds a center point and triangles to replace the faces of a square tiling (quadrille). It is also called the UNION JACK LATTICE because of the resemblance to the UK flag of the triangles surrounding its degree-8 vertices. It is labeled V4.8.8 because each isosceles triangle face has two types of vertices: one with 4 triangles, and two with 8 triangles
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Plane Figure
A GEOMETRIC SHAPE is the geometric information which remains when location , scale , orientation and reflection are removed from the description of a geometric object . That is, the result of moving a shape around, enlarging it, rotating it, or reflecting it in a mirror is the same shape as the original, and not a distinct shape. Objects that have the same shape as each other are said to be similar . If they also have the same scale as each other, they are said to be congruent . Many two-dimensional geometric shapes can be defined by a set of points or vertices and lines connecting the points in a closed chain, as well as the resulting interior points. Such shapes are called polygons and include triangles , squares , and pentagons . Other shapes may be bounded by curves such as the circle or the ellipse
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Congruence (geometry)
In geometry , two figures or objects are CONGRUENT if they have the same shape and size, or if one has the same shape and size as the mirror image of the other. More formally, two sets of points are called CONGRUENT if, and only if, one can be transformed into the other by an isometry , i.e., a combination of RIGID MOTIONS, namely a translation , a rotation , and a reflection . This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted. In elementary geometry the word congruent is often used as follows. The word equal is often used in place of congruent for these objects
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