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What is the locus of points in space that are equidistant from two parallel planes?
A plane midway between the two given planes and parallel to them.
How many planes contain a given line in space?
Given a line, there are an infinite number of different planes that it lies in.
What is a solid with congruent circular bases in parallel planes?
The given description fits that of a cylinder
How many planes will contain 2 points?
Infinitely many planes contain any two given points- it takes three (non-collinear) points to determine a plane.
How many planes can contain two given points?
If 2 points determine a line, then a line contains infinitely many planes.
Given a line and a point on that line how many different planes contain both of them?
1
Given a line and a point not on that line how many different planes contain both of them?
1
Why hilbert axiom of parallelism assert the existence of only at most one parallel line'?
There is a subtle distinction between Euclidean, Hilbert and Non-Euclidean planes. Euclidean planes are those that satisfy the 5 axioms, while Non-Euclidean planes do not satisfy the fifth postulate. This means that in Non-Euclidean planes, given a line and a point not on that line, then there are two (or more) lines that contain that point and are parallel to the original line. There are geometries where there must be exactly one line through that point and parallel to the original line and then there are also geometries where no such line contains that point and is parallel to the original line.Basically, the fifth postulate can be satisfied by multiple geometries.
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.
How many planes can contain a given line in space?
There's no limit to the number of them. A more cool and sexy way to say the same thing might be: "An infinite number."
Through a given point not on a given line there is exactly one line parallel to the given line?
The Playfair Axiom (or "Parallel Postulate")
Which conjecture justifies the construction of a line parallel to a given line through a given point?
Euclid's parallel postulate.
