Download 2.7 Angle Pair Relationships

Warm-Up
The graph shown represents a hyperbola. Draw
the solid of revolution formed by rotating the
hyperbola around the y-axis.
Warmer-Upper
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?
3.1 Identify Pairs of Lines and Angles
Objectives:
1. To differentiate between parallel,
perpendicular, and skew lines
2. To compare Euclidean and Non-Euclidean
geometries
Vocabulary
As a group, define each
of these without your
book. Draw a picture
for each word and
leave a bit of space
for additions and
revisions.
Parallel Lines Skew Lines
Perpendicular Euclidean
Lines
Geometry
Transversal
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.
Transversal
A line is a transversal
if and only if it
intersects two or
more coplanar lines.
– When a transversal
cuts two coplanar
lines, it creates 8
angles, pairs of
which have special
names
Transversal
• <1 and <5 are
corresponding angles
• <3 and <6 are alternate
interior angles
• <1 and <8 are alternate
exterior angles
• <3 and <5 are
consecutive interior
angles
Example 3
Classify the pair of numbered angles.
Example 4
List all possible answers.
1. <2 and ___ are
corresponding <s
2. <4 and ___ are
consecutive interior <s
3. <4 and ___ are
alternate interior <s
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.
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 are 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.