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Topic: Congruence (geometry)

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differential geometry

Euclidean geometry

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	Congruence (geometry) - Search Results - MSN Encarta
		Congruence (geometry), the relationship between two- or three-dimensional figures having the exact same size and shape.
		Geometry may be thought of as the science of space.
		In geometry, two sets are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections.
	encarta.msn.com /Congruence_(geometry).html (193 words)

	Congruence (geometry) - Wikipedia, the free encyclopedia
		In geometry, two sets are called congruent if one can be transformed into the other by an isometry, i.e., a combination of translations, rotations and reflections.
		In a Euclidean system, congruence is fundamental; it's the counterpart of an equals sign in numerical analysis.
		In analytic geometry, congruence may be defined intuitively thus: two mappings of figures onto one Cartesian coordinate system are congruent if and only if, for any two points in the first mapping, the Euclidean distance between them is equal to the Euclidean distance between the corresponding points in the second mapping.
	en.wikipedia.org /wiki/Congruence_(geometry) (562 words)

	Congruence - Wikipedia, the free encyclopedia
		Congruence, as opposed to equivalence or approximation, is a relation which implies a kind of equivalence, though not complete equivalence.
		In mathematics, congruence is formally represented by a triple-tilde
		In cladistics, congruence is a test of homology, or shared, derived character states, in which the distributions of supposed homologies among taxa are compared for consistency.
	en.wikipedia.org /wiki/Congruence (246 words)

	Congruence (geometry) - MSN Encarta
		All of the corresponding sides of two congruent figures are the same length, and all of the corresponding angles are equal.
		Inversely congruent figures are mirror images of each other that must be flipped over to be superimposed.
		Congruence is most often encountered when studying triangles.
	encarta.msn.com /encyclopedia_762505265/Congruence_(geometry).html (216 words)

	congruent - Search Results - MSN Encarta
		Congruence (geometry), the relationship between two- or three-dimensional figures having the exact same size and shape.
		The concept of congruence is related to similarity.
		Two triangles are congruent if they satisfy any of the three following sets of conditions: (1) two angles and a side of one triangle are equal to the...
	ca.encarta.msn.com /congruent.html (144 words)

	Math Forum - Problems Library - Geometry - Similarity, Congruence
		Prove that two congruent chords in a circle are equal distances from the center of the circle.
		Seven congruent rectangles are arranged to form a larger rectangle.
		Nine congruent rectangles form a larger rectangle whose area is 720 units^2.
	mathforum.org /library/problems/sets/geo_similarity.html (1206 words)

	PlanetMath: congruence axioms (Site not responding. Last check: 2007-10-27)
		Cross-references: properties, mean, axiom, congruent, sides, obvious, angles, triangles, line segments, boundary, closed half plane, line, intersect, lying on, point, ray, satisfies, endpoints, closed line segments, betweenness, ordered geometry, equivalence relation, transitive, imply, symmetric, Reflexive, relation on, relations
		This is version 6 of congruence axioms, born on 2005-10-08, modified 2005-10-12.
		(Geometry :: Metric geometry :: Congruence and orthogonality)
	planetmath.org /encyclopedia/CongruenceAxioms.html (277 words)

	Congruence (geometry)
		In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections.
		The third is a different size, and so is similar but not congruent to the first two; the fourth is different altogether.
		Two triangles are congruent if their corresponding sides and angles are equal in measure.
	www.ebroadcast.com.au /lookup/encyclopedia/co/Congruent.html (263 words)

	PlanetMath: angle
		This can be remedied with an additional set of axioms on the geometry: the axioms of congruence.
		In an ordered geometry satisfying the congruence axioms, we have the concept of angle congruence.
		In an ordered geometry satisfying the congruence axioms, supplementary free angles are defined if each contains a representative that is supplementary to one another.
	planetmath.org /encyclopedia/Angle.html (558 words)

	Proofs Using Congruence
		As geometry has become a cornerstone of the high school mathematics curriculum, educators have questioned this approach with some suggesting the adolescent mind as being incapable of reasoning at this level of abstraction.
		It is a restatement of Euclid's fifth postulate and is the basis of Euclidean Geometry.
		Although Euclidean Geometry is a good approximation on a local and macroscopic level, it is not the geometry of the universe we live in, either in the large-scale or perhaps in the microscopic.
	www.andrews.edu /~calkins/math/webtexts/geom05.htm (2213 words)

	SparkNotes: Congruence: Proving Congruence of Triangles
		When proving that triangles are congruent, it is not necessary to prove that all three pairs of corresponding angles and all three pairs of corresponding sides are congruent.
		For example, if two pairs of corresponding angles are congruent, then the third angle pair is also congruent, since all triangles have 180 degrees of interior angles.
		A second way to prove the congruence of triangles is to show that two sides and their included angle are congruent.
	www.sparknotes.com /math/geometry2/congruence/section2.rhtml (437 words)

	GEOMETRY (Site not responding. Last check: 2007-10-27)
		Geometry uses problem situations, physical models, and appropriate technology to investigate and justify geometric concepts and relationships.
		The concepts and topics emphasized in the course include measurement, geometric patterns, coordinate geometry, two- and three-dimensional figures, transformational geometry, congruence, similarity, inductive and deductive reasoning, logic, and proof.
		Then they transfer the design to a piece of 8" X 11" pane of plexiglass and paint the pane to create a “stained glass.” Students construct one of the regular 3-dimensional solid and compute the volume and surface area.
	www.wcs.edu /frvhs/Bull/GEOMETRY.htm (2214 words)

	Latin-Square Geometry: Orthogonal Latin Squares as Skew Lines
		We present two results -- one old, one new -- on the geometry of Latin squares.
		A net is a point-line geometry which is a natural weakening of an affine plane."
		For further background, here is material on finite geometry from the paper Symplectic spreads (pdf), 15 Sept. 2003, by
	finitegeometry.org /sc/gen/ortho.html (2140 words)

	Northern Illinois University, MATH 302
		To increase the students' awareness and appreciation of the many diverse areas of mathematics related to geometry which are not usually encountered in high school or even undergraduate math courses.
		Since lines are such an important part of geometry, we spend a lot of time trying to understand the prototype of all lines, namely the real number line.
		With the discovery of non-Euclidean geometries we know now that the obvious and intuitive first impression is not always the truth.
	www.math.niu.edu /~richard/Math302 (1338 words)

	Mathematical Content by Strand - Geometry (Site not responding. Last check: 2007-10-27)
		Within the geometry strand are inter-related and complementary ideas: shape and transformations of shape, measurement and visualization.
		Each geometry unit may focus on just one of these, but in fact these ways of thinking about geometry are not distinct.
		This is the first unit in the geometry strand that will develop students’ ability to recognize, display, analyze, measure, and reason about the shapes and visual patterns that are important features of our world.
	connectedmath.msu.edu /parents/content/geometry.html (1246 words)

	Individual Restricted Events
		Geometry Restricted could involve solving problems requiring the application of: constructions, logic, similarity and congruence, geometry of the triangle and other polygons, geometry of the circle, area, perimeter, volume, three-dimensional geometry, transformations, and coordinate geometry.
		Geometry could involve solving problems requiring the application of constructions, logic, similarity and congruence, geometry of the triangle and other polygons, geometry of the circle, area, perimeter, volume, three-dimensional geometry, transformations, and coordinate geometry.
		Geometry Team is restricted to students enrolled in geometry this school year.
	math.missouristate.edu /pummill/EventDescription.html (1151 words)

	Mathematics - Geometry (Site not responding. Last check: 2007-10-27)
		Geometry and Measurement 1: Using Geometry; Geometry and Measurement 2: Congruent Angles, Supplementary and Complementary Angels, Two of a Kind, Congruent Triangles Part 1.
		Geometry and Measurement 1: The Pythagorean Theorem Geometry and Measurement 1; Geometry and Measurement 2: The Pythagorean Theorem 2.
		Geometry and Measurement 2: Tangents, Arcs and Chords, Inscribed Angles, Circles and Angles, Circles and Segments.
	www.tulare.k12.ca.us /profdevelopment/CR15535.HTM (2292 words)

	Euclidean Geometry at the Library of Math (Free Online Mathematics)
		The word geometry comes from the Greek geometrein (geo meaning earth, and metrein meaning to measure); geometry was originally the science of measuring the land.
		It wasn't until after the discovery of non-Euclidean geometry that mathematicians began examining the foundations of Euclidean geometry and formulating precise sets of axioms for it.
		These axioms, which do give rise to all theorems in Euclidean geometry, are not minimal in nature and are meant to move the student almost immediately to more interesting and less intuitively obvious results.
	libraryofmath.com /Euclidean_Geometry.html (2264 words)

	Mathematics Course Descriptions
		Topics include a study of two-dimensional and three-dimensional symmetric figures, similarity, congruence, basic geometrical constructions, properties and relationships of the right triangle, measurement and calculation of areas and volumes, and the use of logic and geometrical thought to solve common application problems.
		Topics include plane analytical geometry, function theory including limits and continuity, the differential and integral calculus of algebraic and trigonometric functions of one independent variable, curve sketching, maxima and minima, related rates, areas under and between curves, and volume.
		Topics include solid analytical geometry, the calculus of more than one independent variable, surfaces and curves in space, cylindrical and spherical coordinate systems, vectors and vector-valued functions, partial derivatives, multiple integrals, and applications.
	www.nscc.edu /catalog/desc/math.html (867 words)

	Re: Klein's Definition of Geometry (Site not responding. Last check: 2007-10-27)
		Congruence geometry has a notion of "distance";, and inherits the\nnotions of angle and area from the less rigid similarity and areal\ngeometry.
		In euclidean geometry we have of course an obvious two\npoint invariant, and given m, the ring of m point invariants is generated\nby the pairwise squared distances.
		We can\nsay that -euclidean distance geometry is "more rigid than" euclidean\nsimilarity geometry-, because -it admits more invariants-.\n\nThis was one of Klein\'s greatest insights: we have a whole partially\nordered hierarchy of geometries, with the increasing "rigidity"\ncorresponding to inclusion and quotient relations among the releveant\ninvariant rings of the symmetry groups.
	www.physicsforums.com /showthread.php?t=87833 (8488 words)

	COURSES (Site not responding. Last check: 2007-10-27)
		Topics in geometry to include polygons, congruence and similarity, measurement, geometric transformations, coordinate geometry, and connections between numbers and geometry with selected applications.
		Study of geometry in Euclidean space by means of calculus, including theory of curves and surfaces, curvature, theory of surfaces, and intrinsic geometry on a surface.
		Study of geometry of curves and surfaces in Euclidean space; including an introduction to Riemannian geometry and theory of manifolds.
	www.csufresno.edu /math/department/courses.bottom.html (1861 words)

	Read This: Non-Euclidean Geometry
		An example is found in chapter 8 where descriptive geometry (described as "high school geometry with congruence and parallelism left out") is compared to projective geometry.
		Though he introduces the elliptic metric "by means of absolute polarity" and could have introduced the hyperbolic metric in a similar fashion, he chose to "reverse the process" in order to "follow the historical development more closely".
		Robert Stolz (rstolz@webmail.uvi.edu) is associate professor of mathematics at the University of the Virgin Islands.
	www.maa.org /reviews/coxeterneg.html (427 words)

	TN:ED:GEOMETRY
		Course Description: Geometry is a course that uses problem situations, physical models, and appropriate technology to investigate geometric concepts, relationships, and systems.
		The concepts/topics emphasized in the course include measurement, geometric patterns, coordinate geometry, two- and three-dimensional figures, transformational geometry, congruence, and similarity.
		construct bisectors of angles and line segments, perpendicular lines, congruent line segments and angles, and perpendicular bisectors using a variety of methods (e.g., patty paper, technology).
	www.state.tn.us /education/ci/cigateendofcourse/geometry2.htm (2026 words)

	Geometry
		identify corresponding parts of similar and congruent geometric figures given a diagram.
		determine congruence or similarity relations between triangles or quadrilaterals given a diagram;
		Throw randomly and count the throws that hit the board to determine which board yields the highest probability of a dart’s landing in a circle.
	jc-schools.net /curr/SPI/math-geometry.htm (1146 words)

	Chapter 13 Geometry
		Geometry is used in astronomy, navigation, surveying and many practical occupations.
		Navigation by air and sea depends on accurate geometry.
		If you experience difficulties when using this Website, phone 1300 799976, use the feedback form or email enquiries@mathsteacher.com.au.
	www.mathsteacher.com.au /year9/ch13_geometry/chapter13.htm (80 words)

	E-Example 4.2.1: Investigating the Concepts of Triangle and Properties of Polygons (Site not responding. Last check: 2007-10-27)
		Students at this level can check congruence in two dimensions by moving one shape to exactly cover another.
		Geoboard shapes can be described with a simple system of coordinate geometry; thus two shapes on a geoboard are also congruent if their constructions can be described in the same way.
		If designs are made on two different geoboards, one geoboard can be moved so that eventually the constructions can be viewed in the same way (perhaps by a flip, top to bottom, or a rotation of 90 degrees).
	standards.nctm.org /document/eexamples/chap4/4.2/index.htm (632 words)

	Amazon.ca: Euclidean and Non-Euclidean Geometry: Books: Patrick J. Ryan (Site not responding. Last check: 2007-10-27)
		The primary purpose is to acquaint the reader with the classical results of plane Euclidean and nonEuclidean geometry, congruence theorems, concurrence theorems, classification of isometries, angle addition and trigonometrical formulae.
		The aim is to link classical and modern geometry to prepare students for further study and research in group theory, Lie groups, differential geometry, topology, and mathematical physics.
		The book is intended primarily for undergraduate mathematics students who have acquired the ability to formulate mathematical propositions precisely and to construct and understand mathematical arguments.
	amazon.ca /Euclidean-Non-Euclidean-Geometry-Patrick-Ryan/dp/0521276357 (328 words)

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