Warm-Up

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Warm-Up

• The graph shown represents a hyperbola. Draw the
  solid of revolution formed by rotating the
  hyperbola around the y-axis.

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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?

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3.1 Identify Pairs of Lines and Angles

• Objectives
• To differentiate between parallel, perpendicular,
  and skew lines
• To compare Euclidean and Non-Euclidean geometries

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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 interior angles
Alternate exterior angles Alternate interior angles
Consecutive Interior angles Consecutive Exterior angles
Consecutive Exterior angles Corresponding angles
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Example 1

• Use the diagram to answer the following.
• Name a pair of lines that intersect.
• Would JM and NR ever intersect?
• Would JM and LQ ever intersect?

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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.
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Parallel Lines

• Two lines are parallel lines if and only if they
  are coplanar and never intersect.

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Skew Lines

• Two lines are skew lines if and only if they are
  not coplanar and never intersect.

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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?
• Line(s) parallel to CD and containing point A.
• Line(s) skew to CD and containing point A.

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Example 2

1. Line(s) perpendicular to CD and containing point
   A.
2. Plane(s) parallel to plane EFG and containing
   point A.

AD
ABC
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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

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Transversal

• lt1 and lt5 are corresponding angles
• lt3 and lt6 are alternate interior angles
• lt1 and lt8 are alternate exterior angles
• lt3 and lt5 are consecutive interior angles

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Example 3

• Classify the pair of numbered angles.

Corresponding
Alt. Ext
Alt. Int.
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Example 4

• List all possible answers.
• lt2 and ___ are corresponding lts
• lt4 and ___ are consecutive interior lts
• lt4 and ___ are alternate interior lts

Answer in your notebook
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Example 5a

• Draw line l and point P. How many lines can you
  draw through point P that are perpendicular to
  line l?

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Example 5b

• Draw line l and point P. How many lines can you
  draw through point P that are parallel to line l?

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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.

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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 Euclids Fifth Postulate

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Euclids 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.

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Example 6

• If the Parallel Postulate is false, then what
  must be true?
• 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.

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Example 6

• If the Parallel Postulate is false, then what
  must be true?
• 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.

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Example 6

• If the Parallel Postulate is false, then what
  must be true?
• 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.
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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.
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Example 6

• If the Parallel Postulate is false, then what
  must be true?
• Through a given point not on a given line, you
  can draw no line parallel to the given line.
  This makes Elliptic Geometry.

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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.
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Comparing Geometries
Parabolic Hyperbolic Elliptic
Also Known As Also Known As Also Known As
Euclidean Geometry Lobachevskian Geometry Riemannian Geometry
Geometric Model (Where Stuff Happens) Geometric Model (Where Stuff Happens) Geometric Model (Where Stuff Happens)
Flat Plane Poincare Disk Riemannian Sphere
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Comparing Geometries
Parabolic Hyperbolic Elliptic
Parallel Postulate Point P is not on line l Parallel Postulate Point P is not on line l 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) Geometric Model (Where Stuff Happens) Geometric Model (Where Stuff Happens)
Flat Plane Poincare Disk Riemannian Sphere
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Comparing Geometries
Parabolic Hyperbolic Elliptic
Curvature Curvature Curvature
None Negative (curves inward, like a bowl) Positive (curves outward, like a ball)
Geometric Model (Where Stuff Happens) Geometric Model (Where Stuff Happens) Geometric Model (Where Stuff Happens)
Flat Plane Poincare Disk Riemannian Sphere
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Comparing Geometries
Parabolic Hyperbolic Elliptic
Applications Applications Applications
Architecture, building stuff (including pyramids, great or otherwise) Minkowski Spacetime Einsteins General Relativity (Curved space) Global navigation (pilots and such)
Geometric Model (Where Stuff Happens) Geometric Model (Where Stuff Happens) Geometric Model (Where Stuff Happens)
Flat Plane Poincare Disk Riemannian Sphere
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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.

Great circles divide the sphere into two equal
halves.
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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?

Infinite
Each is between the other 2
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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
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Compare Triangles
Notice the difference in the sum in each picture