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AssignmentMultiple Choice

Euclidean Geometry

Question 1 • (A. Costello)WCA 5324 Geometry 1st Nine Weeks 26-27

Which statement represents the parallel postulate in Euclidean geometry, but not elliptical or spherical geometry?

Answer
A
Through a given point not on a line, there exists no lines parallel to the given line through the given point.
B
Through a given point not on a line, there exists exactly one line parallel to the given line through the given point.
C
Through a given point not on a line, there exists more than one line parallel to the given line through the given point.
D
Through a given point not on a line, there exists exactly three lines parallel to the given line through the given point.

Explanation

In Euclidean geometry, the parallel postulate says that through a point not on a given line, exactly one line can be drawn parallel to that line. Elliptical and spherical geometries have no parallel lines, while some other non-Euclidean geometries have more than one. Therefore, the statement that there is exactly one parallel line is the Euclidean postulate.

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