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Q: Do Through a point not on a line one and only one line always can be drawn parallel to the given line?

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Yes. That's always possible, but there's only one of them.

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In Geometry

Any line that is not parallel to the given lines. The transversal that contains the shortest distance between the two parallel lines, is perpendicular to them.

Yes, this statement is true. However, it is controversial between Euclid and Lobachevsky. In Euclid, this is alwaystrue. In Lobachevsky, however, this could be both true and untrue. Did this help?

Related questions

Assume there are no lines through a given point that is parallel to a given line or assume that there are many lines through a given point that are parallel to a given line. There exist a line l and a point P not on l such that either there is no line m parallel to l through P or there are two distinct lines m and n parallel to l through P.

False.

The Playfair Axiom (or "Parallel Postulate")

Euclid's parallel postulate.

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.

Yes. That's always possible, but there's only one of them.

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... given line. This is one version of Euclid's fifth postulate, also known as the Parallel Postulate. It is quite possible to construct consistent systems of geometry where this postulate is negated - either many parallel lines or none.

In Geometry

Probably the best known equivalent of Euclid's parallel postulate, contingent on his other postulates, is Playfair's axiom, named after the Scottish mathematician John Playfair, which states:In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.

Write an equation in slope-intercept form for the line that passes through the given point and is parallel to the given line (-7,3); x=4

"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."

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