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What else can I help you with?
What does it mean for a system to be consistent or inconsistent?
does it stay the same or not? Actually, a system is inconsistent if you can derive two (or more) statements within the system which are contradictory. Otherwise it is consistent. For example, Eucliadean geometry requires that given a line and a point not on that line, you can have one and only one line through the point which is parallel to the original line. However, you can have a consistent system of geometry if you assume that there is no such parallel line. This is known as the projective plane. You can assume that there will be an infinite number of parallel lines through a point not on the line. And again you can have a consistent system. Consistency or inconsistency has nothing whatsoever to do with time.
What is needed to determine a line?
Two points determine a line. Also there is one and only line perpendicular to given line through a given point on the line,. and There is one and only line parallel to given line through a given point not on the line.
A given line is always parallel to itself?
False.
To find a segment parallel to another segment and through a given point fold a piece of paper so that the fold goes through the point and the pieces of the segment on either side of the fold match?
The answer is FALSE i just did it on
Write an equation of the line containing the given point and parallel to the the given line6 and 7 3x plus y equals 8?
Given point: (6, 7) Equation: 3x+y = 8 Parallel equation: 3x+y = 25
Is hyperbolic parallel postulate a postulate of Euclid?
No, the hyperbolic parallel postulate is not one of Euclid's original five postulates. Euclid's fifth postulate, known as the parallel postulate, states that given a line and a point not on that line, there is exactly one line parallel to the original line that passes through the point. Hyperbolic geometry arises from modifying this postulate, allowing for multiple parallel lines through the given point, leading to a different set of geometric principles.
What are the characteristics of hyperbolic geometry?
A.When represented on a Poincaré Disk, a line is an arc that has endpoints.B.There is an infinite number of lines parallel to a given line through a given point.C.It can be represented by a Poincaré Disk.Triangles have less than 180 degrees.
What isn't a Euclidean postulate geometry?
Non-Euclidean geometries are those that reject or modify Euclid's fifth postulate, the parallel postulate, which states that through a point not on a line, there is exactly one line parallel to the given line. Examples include hyperbolic and elliptic geometry, where multiple parallel lines can exist through a point or no parallels exist at all, respectively. These geometries explore curved spaces and differ fundamentally from classic Euclidean geometry, which is based on flat planes.
What lines on a hyperbolic plane are considered to be?
In hyperbolic geometry, lines are typically represented by arcs of circles that intersect the boundary of the hyperbolic plane orthogonally or by straight lines that extend infinitely in both directions. Unlike Euclidean geometry, where two parallel lines never intersect, hyperbolic planes can contain multiple lines that do not intersect a given line, leading to unique properties of parallelism. This results in a richer structure where the concepts of distance and angle differ significantly from those in Euclidean space.
In Euclidean geometry if there is a line and a point not on the line then there is exactly one line through the point and the parallel to the given line. True or false?
True. In Euclidean geometry, if there is a line and a point not on that line, there exists exactly one line that can be drawn through the point that is parallel to the given line. This is known as the Parallel Postulate, which states that for a given line and a point not on it, there is one and only one line parallel to the given line that passes through the point.
Why don't parallel lines exist in elliptical geometry?
Elliptical geometry is a non-Euclidean geometry. The parallel postulate of Euclidean geometry was replaced by the statement that through any point in the plane, there exist no lines parallel to a given line. A consistent geometry - of a space with positive curvature - was developed on that basis.It is, therefore, by definition that parallel lines do not exist in elliptical geometry.
What is eullidean geometry?
"Euclidean" geometry is the familiar "standard" geometry. Until the 19th century, it was simply "geometry". It features infinitely divisible space, up to three dimensions, and, most notably, the "parallel postulate": "Given a line, and a point not on the line, there is exactly one line that can be drawn through the point and parallel to the given line."
how to negate the hyperbolic parallel postulate?
The hyperbolic parallel postulate states that given a line L and a point P, not on the line, there are at least two distinct lines through P that do not intersect L.The negation is that given a line L and a point P, not on the line, there is at most one line through P that does not intersect L.The negation includes the case where there is exactly one such line - which is the Euclidean space.
What postulate is not of euclidean geometry?
Euclidean Geometry is based on the premise that through any point there is only one line that can be drawn parallel to another line. It is based on the geometry of the Plane. There are basically two answers to your question: (i) Through any point there are NO lines that can be drawn parallel to a given line (e.g. the geometry on the Earth's surface, where a line is defined as a great circle. (Elliptic Geometry) (ii) Through any point, there is an INFINITE number of lines that can be drawn parallel of a given line. (I think this is referred to as Riemannian Geometry, but someone else needs to advise us on this) Both of these are fascinating topics to study.
What Compare and contrast the major characteristics of euclidean and non euclidean geometry?
Euclidean geometry is based on the principles outlined by Euclid, emphasizing flat spaces and relying on postulates such as the parallel postulate, which states that through a point not on a given line, exactly one parallel line can be drawn. In contrast, non-Euclidean geometry arises when this parallel postulate is altered, leading to geometries such as hyperbolic and elliptic geometry, where multiple parallels can exist or none at all. While Euclidean geometry deals with shapes and figures in two-dimensional flat planes, non-Euclidean geometry explores curved surfaces and spaces, resulting in different properties and relationships among points, lines, and angles. Overall, the key distinction lies in the treatment of parallel lines and the nature of space itself.
What is elliptical geometry and examples?
Elliptical geometry is like Euclidean geometry except that the "fifth postulate" is denied. Elliptical geometry postulates that no two lines are parallel.One example: define a point as any line through the origin. Define a line as any plane through the origin. In this system, the first four postulates of Euclidean geometry hold; through two points, there is exactly one line that contains them (i.e.: given two lines through the origin, there is one plane that contains them) and so on. However, it is nottrue that given a line and a point not on the line that there is a parallel line through the point (that is, given a plane through the origin, and a line through the origin, not on the plane, there is no other plane through the origin that is parallel to the given plane).
Which postulates led to the discovery of non-Euclidean geometry?
Adding to what Anand Mehta said, the negation of that statement has two interpretations. (i) there are zero lines through that point that are parallel to the given line (this is called Elliptic or Reimannian Geometry) (ii) there is an infinite number of lines that pass through the point and parallel to a given line (this is called Hyperbolic or Lobachevskian Geometry) I might add that the study of non-Euclidean Geometries are absolutely fascinating.
