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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
In your notebook, 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
Lines
Euclidean
Geometry
Transversal
Alternate
exterior angles
Consecutive
Interior angles
Alternate
interior angles
Consecutive
Exterior angles
Consecutive
Corresponding
Exterior angles angles
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 AB
and containing point A.
2. Line(s) skew to CD and AH
containing point A.
Example 2
3. Line(s) perpendicular to
CD and containing point
A. AD
4. Plane(s) parallel to
plane EFG and
containing point A.
ABC
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.
Corresponding
Alt. Ext
Alt. Int.
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
Answer in your notebook
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
DEFINITION: Parallel lines are infinite lines in the same
plane that do not intersect.
In the figure above, Hyperbolic Line BA and Hyperbolic
Line BC are both infinite lines in the same plane. They
intersect at point B and , therefore, they are NOT parallel
Hyperbolic lines. Hyperbolic line DE and Hyperbolic Line
BA are also both infinite lines in the same plane, and since
they do not intersect, DE is parallel to BA. Likewise,
Hyperbolic Line DE is also parallel to Hyperbolic Line
BC. Now this is an odd thing since we know that in Euclidean
geometry:
If two lines are parallel to a third line, then the two lines are
parallel to each other.
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
•Elliptic geometry is a non-Euclidean
geometry, in which, given a line L and
a point p outside L, there exists no line
parallel to L passing through p. As all
lines in elliptic geometry intersect
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
(curves inward, like a
bowl)
Positive
(curves outward, like a
ball)
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?
Infinite
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?
Each is between the other 2
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.
Each is between the other 2
Compare Triangles
Notice the difference in the sum in each picture