Is a set of points consisting of a vertex point and two rays extending from the vertex point

Introduction to Geometry

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Of a polytope
Of a plane tiling

ANGLES An angle is a set of points consisting of two rays, with a common endpoint called THE VERTEX of the angle. The rays are callled SIDES or LEGS of the angle An angle can be classified according its measurement:  Acute angle:Measure between 0 to 90 degrees  Right angle:Measure 90 degrees  Obtuse angle:Measure between 90 to180 degrees  Straight angle:Measure 180 degrees  Reflex angle:Measure between180 to 360 degrees

In addition, an angle can be classified according its characteristics and relationship:  Adjacent angles: They have the same vertex and a common side, but they don’t share any interior point.  Linear pair angles: Two adjacent angles which sum 180 degrees ANGLES

 Vertical angles: Are the opposite pair of angles formed when two lines intersect. They are always congruent  Complementary angles: Angles adjacents or not, which sum 90 degrees ANGLES

 Supplementary angles: Are angles which sum is 180 degrees ANGLES

Angles between parallel lines crossed by a transversal -) Co-interior angles:2-5, 3-8 -) Co-exterior angles:1-6, 4-7 -) Vertical angles:1-3, 2-4, 5-7, 6-8 -) Corresponding angles:1-5, 2-6, 4-8, 3-7 -) Alternate interior angles:2-8, 3-5 -) Alternate exterior angles:1-7, 4-6 ANGLES

TRIANGLE Definition: 3-sides geometric figure. The points of the intersection of the sides are called VERTEX.

TRIANGLES CLASSIFICATION: Sides EQUILATERAL TRIANGLE The Equilateral triangle has three equal sides and three equal angles. Each angle is 60°

TRIANGLES CLASSIFICATION : Sides ISOSCELES TRIANGLE The Isosceles has two equal sides forming two equal angles with the base.

TRIANGLES CLASSIFICATION : Sides SCALENE TRIANGLE The Scalene Triangle has no congruent sides. In other words, each side must have a different length.

TRIANGLES CLASSIFICATION : Angles ACUTE TRIANGLE The Acute Triangle has three acute angles (an acute angle measures l less than 90°)

TRIANGLES CLASSIFICATION : Angles OBTUSE TRIANGLE The Obtuse Triangle has an obtuse angle (an obtuse angle has more than 90°). In the picture the shaded angle is the obtuse angle that distinguishes this triangle Since the total degrees in any triangle is 180°, an obtuse triangle can only have one angle that measures more than 90°.

TRIANGLES CLASSIFICATION : Angles RIGHT TRIANGLE The Right Triangle has one 90° angle.

TRIANGLES CLASSIFICATION

TRIANGLES Some properties: A.The sum of the interior angles of a triangle is 180 degrees. B.An exterior angle is the angle formed by a side and the extension of one of its adjacent sides In the graphic, 120 degree is an external angle

TRIANGLES B.The Sum of an exterior and an interior angle of any triangle is 180 degrees; so they are supplementary angles C.The measure of an exterior angle of a triangle is equal to the sum of the two interior angles that are not adjacent to it. In the graphic, 120 = x + 45 In the graphic, <y = 180degree => <y = 180 – 120 = 60 degree where, <y is an internal angle and 120 degree is an external angle

TRIANGLES D.The shortest side is opposite to the smallest angle, and the longest side is opposite to the longest angle D.Any side of a triangle is shorter than the sum of the measure of the length of the other two sides

CONGRUENT TRIANGLES Triangles are congruent when they have exactly the same three sides and exactly the same three angles. These triangles are congruent:

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In geometry, a vertex (in plural form: vertices or vertexes), often denoted by letters such as $S {\displaystyle S}$

P {\displaystyle P}

Q {\displaystyle Q}

R {\displaystyle R}

, 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.[1][2][3]

A vertex of an angle is the endpoint where two line 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.[3][4]

Of a polytope

A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object.[4]

In a polygon, a vertex is called "convex" if the internal angle of the polygon (i.e., the angle formed by the two edges at the vertex with the polygon inside the angle) is less than π radians (180°, two right angles); otherwise, it is called "concave" or "reflex".[5] More generally, a vertex of a polyhedron or polytope is convex, if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, and is concave otherwise.[citation needed]

Polytope vertices are related to vertices of graphs, in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the polytope,[6] and in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices.[citation needed]

However, in graph theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. There is also a connection between geometric vertices and the vertices of a curve, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, and if a polygon is approximated by a smooth curve, there will be a point of extreme curvature near each polygon vertex.[7] However, a smooth curve approximation to a polygon will also have additional vertices, at the points where its curvature is minimal.[citation needed]

Of a plane tiling

A vertex of a plane tiling or tessellation is a point where three or more tiles meet;[8] generally, but not always, the tiles of a tessellation are polygons and the vertices of the tessellation are also vertices of its tiles. More generally, a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope; the vertices of other kinds of complexes such as simplicial complexes are its zero-dimensional faces.[citation needed]

Vertex B is an ear, because the open line segment between C and D is entirely inside the polygon. Vertex C is a mouth, because the open line segment between A and B is entirely outside the polygon.

A polygon vertex $xi$ of a simple polygon $P$ is a principal polygon vertex if the diagonal $[x(i - 1), x(i + 1)]$ intersects the boundary of $P$ only at $x(i - 1)$ and $x(i + 1)$ . There are two types of principal vertices: ears and mouths.[9]

Ears

A principal vertex $xi$ of a simple polygon $P$ is called an ear if the diagonal $[x(i - 1), x(i + 1)]$ that bridges $xi$ lies entirely in $P$ . (see also convex polygon) According to the two ears theorem, every simple polygon has at least two ears.[10]

Mouths

A principal vertex $xi$ of a simple polygon $P$ is called a mouth if the diagonal $[x(i - 1), x(i + 1)]$ lies outside the boundary of $P$ .

Any convex polyhedron's surface has Euler characteristic

V − E + F = 2 , {\displaystyle V-E+F=2,}

where $V$ is the number of vertices, $E$ is the number of edges, and $F$ is the number of faces. This equation is known as Euler's polyhedron formula. Thus the number of vertices is 2 more than the excess of the number of edges over the number of faces. For example, since a cube has 12 edges and 6 faces, the formula implies that it has eight vertices.[citation needed]

In computer graphics, objects are often represented as triangulated polyhedra in which the object vertices are associated not only with three spatial coordinates but also with other graphical information necessary to render the object correctly, such as colors, reflectance properties, textures, and surface normal.[11] These properties are used in rendering by a vertex shader, part of the vertex pipeline.

Vertex arrangement
Vertex figure

^ Weisstein, Eric W. "Vertex". MathWorld.
^ "Vertices, Edges and Faces". www.mathsisfun.com. Retrieved 2020-08-16.
^ a b "What Are Vertices in Math?". Sciencing. Retrieved 2020-08-16.
^ a b Heath, Thomas L. (1956). The Thirteen Books of Euclid's Elements (2nd ed. [Facsimile. Original publication: Cambridge University Press, 1925] ed.). New York: Dover Publications. (3 vols.): ISBN 0-486-60088-2 (vol. 1), ISBN 0-486-60089-0 (vol. 2), ISBN 0-486-60090-4 (vol. 3).
^ Jing, Lanru; Stephansson, Ove (2007). Fundamentals of Discrete Element Methods for Rock Engineering: Theory and Applications. Elsevier Science.
^ Peter McMullen, Egon Schulte, Abstract Regular Polytopes, Cambridge University Press, 2002. ISBN 0-521-81496-0 (Page 29)
^ Bobenko, Alexander I.; Schröder, Peter; Sullivan, John M.; Ziegler, Günter M. (2008). Discrete differential geometry. Birkhäuser Verlag AG. ISBN 978-3-7643-8620-7.
^ M.V. Jaric, ed, Introduction to the Mathematics of Quasicrystals (Aperiodicity and Order, Vol 2) ISBN 0-12-040602-0, Academic Press, 1989.
^ Devadoss, Satyan; O'Rourke, Joseph (2011). Discrete and Computational Geometry. Princeton University Press. ISBN 978-0-691-14553-2.
^ Meisters, G. H. (1975), "Polygons have ears", The American Mathematical Monthly, 82 (6): 648–651, doi:10.2307/2319703, JSTOR 2319703, MR 0367792.
^ Christen, Martin. "Clockworkcoders Tutorials: Vertex Attributes". Khronos Group. Archived from the original on 12 April 2019. Retrieved 26 January 2009.