Introductory Geometry

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Section 9.1

• Introductory Geometry

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School of Athens
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Euclid
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Who was Euclid?

• Born about 325 BC
• Died about 265 BC in Alexandria, Egypt
• The Elements dominated mathematical teaching for
  2000 years
• Over 1000 editions
• Best Selling book of all time?

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Euclids Quotations

• A youth who had begun to read geometry with
  Euclid, when he had learnt the first proposition,
  inquired, "What do I get by learning these
  things?" So Euclid called a slave and said "Give
  him threepence, since he must make a gain out of
  what he learns."Stobaeus, Extracts

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Euclids Quotations

• That, if a straight line falling on two straight
  lines makes the interior angles on the same side
  less than two right angles, the two straight
  lines, if produced indefinitely, meet on that
  side on which the angles are less than two right
  angles.the 5th postulate

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Euclids Quotations

• There is no royal road to geometry.
• Source
• http//www-groups.dcs.st-andrews.ac.uk/history/

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Basic Notions

• Words without Definition
• Points
• Lines
• Planes

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Linear Notions

• A line has no thickness
• A line is determined by two points
• A line goes on forever in two directions

D
B
A
C
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Linear Notions Lines
B
A
C
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Linear Notions Segments
B
A
C
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Linear Notions Rays
B
A
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Planar Notions

• Points in the same plane are coplanar.
• How many points are needed?
• Noncoplanar points cannot be placed in the same
  plane.
• How many points are needed?
• Lines in the same plane are coplanar lines.
• How many points are needed?

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Planar Notions

• Given a pair of lines what are the possibilities
  about intersections?
• What do we call a pair of lines that do not
  intersect?
• Must every pair of nonintersecting lines be
  parallel?

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Planar Notions

• Skew lines are lines that do not intersect and no
  plane contains them both.
• Intersecting lines are two coplanar lines with
  exactly one point in common.
• Concurrent lines are lines that contain the same
  point.
• Collinear lines are lines that contain the same
  segment.

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Now Try This 9-1 page 500

• How many different lines can be drawn through two
  points?
• Can Skew lines be parallel? Why?
• On a globe a line is a great circle, that is, a
  circle the same size as the equator. How many
  different lines can be drawn through two
  different points?

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Properties of Points, Lines, and Planes

• There is exactly one line that contains any two
  distinct points.
• If two points lie in a plane, then the line
  containing the points lines in the plane.
• If two points intersect, then there intersection
  is a line.
• There is exactly one plane that contains any
  three distinct noncollinear points.

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Properties of Points, Lines, and Planes

• A line and a point not on the line determine a
  plane.
• Two parallel lines determine a plane.
• Two intersecting lines determine a plane.

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Euclid assumed the first four statements and then
proved 5, 6, 7

• 7. Two intersecting lines determine a plane.
• Two intersecting planes have one point in common.
• Each line contains at least one other point not
  on the other line.
• Thus there are at least three noncollinear points
  on the two intersecting lines.
• Those points determine a plane containing the
  lines.

QED
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Now Try This 9-2 page 502

• Show that statements 5 and 6 follow logically
  from the first four statements.
• A line and a point not on the line determine a
  plane.
• Two parallel lines determine a plane.

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Problem Solving

• Given 15 points, no three of which are collinear
  , how many lines can be drawn through the 15
  points?

B
A
E
D
C
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More Terms

• Half-plane
• Angle
• Side
• Vertex

• Adjacent angles
• Protractor
• Degree
• Minutes
• Seconds

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Now Try This 9-4 page 504

• Convert 8.42o to degrees, minutes, and seconds.

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Angles Paper Folding page 506

• Right angle
• 90o
• Acute

• Right
• Obtuse
• Straight

Mnemonic
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Now Try This 9-5 page 509

• Is it possible for a line intersecting a plane to
  be perpendicular to exactly one line in the plane
  through its intersection with the plane?
• Is it possible for a line intersecting a plane to
  be perpendicular to two distinct lines in a plane
  through its point of intersection with the plane,
  and yet not be perpendicular with the plane?

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Homework 9.1 page 509

• 1, 2, 3, 4, 5, 6, 7, 12, 13