Parallel And Perpendicular Lines Worksheet Page 42
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50A Follow-Up of Lesson 3-6
Non-Euclidean Geometry
So far in this text, we have studied
plane Euclidean geometry
, which is based on a
system of points, lines, and planes. In
spherical geometry
, we study a system of
points, great circles (lines), and spheres (planes). Spherical geometry is one type of
non-Euclidean geometry
.
Plane Euclidean Geometry
Spherical Geometry
Longitude lines
and the equator
A
model great
m
A great circle
circles on Earth.
divides a sphere
P
P
into equal halves.
P
E
Plane
contains line
and point A not on .
Sphere
E
contains great
Polar points are endpoints of a
circle m and point P not
diameter of a great circle.
E
on m. m is a line on sphere
.
The table below compares and contrasts lines in the system of plane Euclidean
geometry and lines (great circles) in spherical geometry.
Plane Euclidean Geometry
Spherical Geometry
Lines on the Plane
Great Circles (Lines) on the Sphere
1. A line segment is the shortest path
1. An arc of a great circle is the shortest
between two points.
path between two points.
2. There is a unique line passing through
2. There is a unique great circle passing
any two points.
through any pair of nonpolar points.
3. A line goes on infinitely in two directions.
3. A great circle is finite and returns to its
original starting point.
4. If three points are collinear, exactly one
4. If three points are collinear, any one of
is between the other two.
the three points is between the other two.
A is between B and C.
C
A
B
C
B is between A and C.
C is between A and B.
B is between A and C.
A
B
In spherical geometry, Euclid’s first four postulates and their related theorems
hold true. However, theorems that depend on the parallel postulate (Postulate 5)
may not be true.
In Euclidean geometry parallel lines lie in the same plane and never
intersect. In spherical geometry, the sphere is the plane, and a great circle
A
represents a line. Every great circle containing A intersects . Thus, there
exists no line through point A that is parallel to .
(continued on the next page)
Investigating Slope-Intercept Form 165
Geometry Activity Non-Euclidean Geometry 165
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