The Parallel Postulate
Assignment:
Objectives:
• Supplemental
1. To differentiate
Problems (Online)
between parallel,
perpendicular, and
skew lines
2. To compare
Euclidean and NonEuclidean
geometries
Example 1
Use the diagram to answer
the following.
1. Name a pair of lines that
intersect.
2. Would JM and NR ever
intersect?
3. Would JM and LQ ever
intersect?
Parallel Lines
Two lines are parallel lines if and only if they
are coplanar and never intersect.
The red arrows
indicate that the
lines are parallel.
Parallel Lines
Two lines are parallel lines if and only if they
are coplanar and never intersect.
Skew Lines
Two lines are skew lines if and only if they are
not coplanar and never intersect.
Example 2
Think of each segment in
the figure as part of a
line. Which line or plane
in the figure appear to fit
the description?
1. Line(s) parallel to CD
and containing point A.
2. Line(s) skew to CD and
containing point A.
Example 2
3. Line(s) perpendicular to
CD and containing point
A.
4. Plane(s) parallel plane
EFG and containing
point A.
Example 3
The graph shown represents a hyperbola. Draw
the solid of revolution formed by rotating the
hyperbola around the y-axis.
Example 4
The image shown is a
print by M.C. Escher
called Circle Limit III.
Pretend you are one of
the golden fish toward
the center of the image.
What you think it means
about the surface you
are on if the other
golden fish on the same
white curve are the
same size as you?
You will be able to compare Euclidean and
Non-Euclidean geometries
Objective 2
Example 5a
Draw line l and
point P. How
many lines can
you draw through
point P that are
perpendicular to
line l?
Example 5b
Draw line l and
point P. How
many lines can
you draw through
point P that are
parallel to line l?
Perpendicular Postulate
If there is a line and a point not on the line, then
there is exactly one line through the point
perpendicular to the given line.
Parallel Postulate
If there is a line and a point not on the line, then
there is exactly one line through the point
parallel to the given line.
Also referred to as Euclid’s Fifth Postulate
Euclid’s Fifth Postulate
Some mathematicians believed
that the fifth postulate was
not a postulate at all, that it
was provable. So they
assumed it was false and
tried to find something that
contradicted a basic
geometric truth.
Example 6
If the Parallel Postulate
is false, then what
must be true?
1. Through a given point
not on a given line,
you can draw more
than one line parallel to
the given line. This
makes Hyperbolic
Geometry.
Example 6
If the Parallel Postulate
is false, then what
must be true?
1. Through a given point
not on a given line,
you can draw more
than one line parallel to
the given line. This
makes Hyperbolic
Geometry.
Example 6
If the Parallel Postulate
is false, then what
must be true?
1. Through a given point
not on a given line,
you can draw more
than one line parallel to
the given line. This
makes Hyperbolic
Geometry.
This is called a
Poincare Disk, and it
is a 2D projection of
a hyperboloid.
Example 6
If the Parallel Postulate
is false, then what
must be true?
1. Through a given point
not on a given line,
you can draw more
than one line parallel to
the given line. This
makes Hyperbolic
Geometry.
Click the
hyperboloid.
Example 6
If the Parallel Postulate
is false, then what
must be true?
2. Through a given
point not on a given
line, you can draw
no line parallel to the
given line. This
makes Elliptic
Geometry.
Example 6
If the Parallel Postulate
is false, then what
must be true?
2. Through a given
point not on a given
line, you can draw
no line parallel to the
given line. This
makes Elliptic
Geometry.
This is a
Riemannian Sphere.
Click the thing.
Comparing Geometries
Parabolic
Hyperbolic
Elliptic
Also Known As
Euclidean Geometry
Lobachevskian
Geometry
Riemannian Geometry
Geometric Model (Where Stuff Happens)
Flat Plane
Poincare Disk*
Riemannian Sphere
Comparing Geometries
Parabolic
Hyperbolic
Elliptic
Parallel Postulate: Point P is not on line l
There is one line through
P that is parallel to line l.
There are many lines
through P that are
parallel to line l.
There no lines
through P that are
parallel to line l.
Geometric Model (Where Stuff Happens)
Flat Plane
Poincare Disk*
Riemannian Sphere
Comparing Geometries
Parabolic
Hyperbolic
Elliptic
Curvature
None
Negative
Positive
Geometric Model (Where Stuff Happens)
Flat Plane
Poincare Disk*
Riemannian Sphere
Comparing Geometries
Parabolic
Hyperbolic
Elliptic
Applications
Architecture, building
Minkowski Spacetime
stuff (including pyramids,
Einstein’s General
great or otherwise)
Relativity (Curved space)
Global navigation (pilots
and such)
Geometric Model (Where Stuff Happens)
Flat Plane
Poincare Disk*
Riemannian Sphere
Great Circles
Great Circle: The
intersection of the
sphere and a plane that
cuts through its center.
•
•
Think of the equator or
the Prime Meridian
The lines in Euclidean
geometry are
considered great circles
in elliptic geometry.
l
Great circles divide
the sphere into two
equal halves.
Example 7
1. In Elliptic geometry, how many great
circles can be drawn through any two
points?
2. Suppose points A, B, and C are collinear
in Elliptic geometry; that is, they lie on the
same great circle. If the points appear in
that order, which point is between the
other two?
Example 8
For the property below from Euclidean
geometry, write a corresponding statement
for Elliptic geometry.
For three collinear points, exactly one of them is
between the other two.
The Parallel Postulate
Assignment:
Objectives:
• Supplemental
1. To differentiate
Problems (Online)
between parallel,
perpendicular, and
skew lines
2. To compare
Euclidean and NonEuclidean
geometries