Download chapter 9 - El Camino College

What You Will Learn (Review)
•
Points
•
Lines
•
Planes
•
Angles
9.1-1
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Basic Terms
Three basic items in geometry:
• Point
• Line
9.1-2
•
plane are three basic terms in geometry
•
These three items are NOT given a formal definition
•
Yet we recognize them when we see them.
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Lines, Rays, Line Segments
9.1-3
•
A line is a set of points.
•
Any two distinct points determine a unique line.
•
Any point on a line separates the line into three parts: the point
and two half lines.
•
A ray is a half line including the endpoint.
•
A line segment is part of a line between two points, including
the endpoints.
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Basic Terms
Description
Diagram
Line AB
A
Ray AB
B
B
A
Line segment AB
A
AB
AB
B
A
Ray BA
9.1-4
Symbol
B
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BA
AB
Plane
•
We can think of a plane as a two-dimensional
surface that extends infinitely in both directions.
•
Any three points that are not on the same line
(noncollinear points) determine a unique plane.
9.1-5
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Plane
Two lines in the same plane that do
not intersect are called parallel lines.
9.1-6
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Plane
A line in a plane divides the plane into three
parts, the line and two half planes.
9.1-7
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Plane
•
Any line and a point not on the line
determine a unique plane.
•
The intersection of
two distinct,
non-parallel
planes is a line.
9.1-8
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Angles
An angle is the union of two rays with a
common endpoint; denoted by .
9.1-9
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Angles
9.1-10
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Types of Angles
Adjacent Angles - angles that have a
common vertex and a common side but no
common interior points.
Complementary Angles - two angles whose
sum of their measures is 90 degrees.
Supplementary Angles - two angles whose
sum of their measures is 180 degrees.
9.1-11
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Problem solving
3x – 20
2
9.1-12
1
x
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Definitions
When two straight lines intersect, the nonadjacent
angles formed are called Vertical angles.
Vertical angles have the same measure.
9.1-13
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Definitions
A line that
intersects two
different lines, at
two different
points is called a
transversal.
9.1-14
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Special Names (l1 and l2 are parallel)
9.1-15
Alternate
interior angles
3 & 6; 4 & 5
Interior angles on
the opposite side of
the transversal–have
the same measure
Alternate
exterior angles
1 & 8; 2 & 7
Exterior angles on
the opposite sides of
the transversal–have
the same measure
Corresponding
angles
1 & 5, 2 & 6,
3 & 7, 4 & 8
One interior and one
exterior angle on the
same side of the
transversal–have the
same measure
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1
2
3
4
5 6
7 8
1
3
2
4
5 6
7 8
1
3
5 6
7 8
2
4
Parallel Lines Cut by a Transversal
When two parallel lines are cut by a transversal,
1. alternate interior angles have the same measure.
2. alternate exterior angles have the same measure.
3. corresponding angles have the same measure.
9.1-16
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Example 6: Determining Angle
Measures
The figure shows
two parallel lines cut
by a transversal.
Determine the
measure of
through
9.1-17
.
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What You Will Learn
•
Rigid Motion or Transformation
•
Reflections
•
Translations
•
Rotations
•
Glide Reflections
•
Tessellations
9.5-18
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Definitions
The act of moving a geometric figure
from some starting position to some
ending position without altering
its shape or size is
called a rigid motion
(or transformation).
9.5-19
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Reflection
A reflection is a rigid motion that
moves a geometric figure to a new
position such that the figure in the
new position is a mirror image of the
figure in the starting position. In two
dimensions, the figure and its mirror
image are equidistant from a line
called the reflection line or the axis
of reflection.
9.5-20
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Example 1: Reflection of a Triangle
Construct the reflection of triangle
ABC about reflection line l.
B’
A’
C’
9.5-21
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Translation
A translation (or glide) is a rigid
motion that moves a geometric figure
by sliding it along a straight line
segment in the plane. The direction
and length of the line segment
completely determine the translation.
9.5-22
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Translation
A concise way to indicate the direction
and the distance that a figure is moved
during a translation is with a
translation vector.
9.5-23
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Example 2: A Translated Square
Given square ABCD and translation
vector v, construct the translated
square A´B´C´D´.
9.5-24
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Example 2: A Translated Square
9.5-25
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Rotation
A rotation is a rigid motion performed
by rotating a geometric figure in the
plane about a specific point, called the
rotation point or the center of
rotation. The angle through which the
object is rotated is called the angle of
rotation.
9.5-26
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Example 5: A Rotation Point
Inside a Polygon
Given polygon ABCDEFGH and
rotation point P, construct polygons
that result from
rotations through
a) 90º
9.5-27
b) 180º.
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Example 5: A Rotation Point
Inside a Polygon (90: (x,y)  (-y,x))
a. 90 Solution:
B
H’(-1,2)
C’(1, 4)
D’(1, -2)
E’(3, -2)
F’(3, -4)
C
A(2,3)
A’(-3,2)
B’(-3,4)
4
A
H
G(-4,1)
-4
2
H(2,1)
-2
P
D(-2,-1)
-2
F(-4,-3) E(-2,-3)
4
2
D
-4
G
G’(-1,-4)
9.5-28
B(4, 3)
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E
F
6
C(4, -1)
Example 5: A Rotation Point
Inside a Polygon (180: (x,y)  (-x,-y))
a. 180 Solution:
4
H’(-2,-1)
A(2,3)
E
A’(-2,-3)
B’(-4,-3)
C’(-4,1)
D’(2,1)
E’(2,3)
2
G(-4,1)
C
-4
H(2,1)
D
-2
P
H
D(-2,-1)
-2
B
A
F(-4,-3) E(-2,-3)
2
4
6
G C(4, -1)
-4
F’(4,3)
G’(4,-1)
9.5-29
B(4, 3)
F
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Example 5: A Rotation Point
Inside a Polygon
9.5-30
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Glide Reflection
A glide reflection is a rigid motion
formed by performing a translation
(or glide) followed by a reflection.
9.5-31
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Tessellations
A tessellation (or tiling) is a pattern
consisting of the repeated use of the
same geometric figures to entirely
cover a plane, leaving no gaps. The
geometric figures used are called the
tessellating shapes of the
tessellation.
9.5-32
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For Example
The simplest tessellations use one
single regular polygon.
9.5-33
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For Example
9.5-34
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Creating a Tessellation with a Square
Begin with a 2” square, draw a line.
Cut and rotate.
9.5-35
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What You Will Learn
Non-Euclidean Geometry
• Elliptical geometry
• Hyperbolic geometry
9.7-36
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Euclid’s Fifth Postulate
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
the two right angles.
9.7-37
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Euclid’s Fifth Postulate
The sum of angles A and B is less
than the sum of two right angles
(180º). Therefore, the two lines will
meet if extended.
9.7-38
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Playfair’s Postulate or
Euclidean Parallel Postulate
Given a line and a point not on the
line, one and only one line can be
drawn through the given point
parallel to the given line.
9.7-39
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Non-Euclidean Geometry
•
Euclidean geometry is geometry in a plane.
•
Many attempts were made to prove the fifth
postulate.
•
These attempts led to the study of geometry on
the surface of a curved object:
- Hyperbolic geometry
- Spherical, elliptical or Riemannian geometry
9.7-40
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Non-Euclidean Geometry
A model may be considered a physical
interpretation of the undefined terms
that satisfies the axioms. A model may
be a picture or an actual physical
object.
9.7-41
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Fifth Axiom of Three Geometries
Euclidean
Given a line
and a point
not on the
line, one and
only one line
can be drawn
through the
given point
parallel to the
given line.
9.7-42
Elliptical
Given a line
and a point
not on the
line, no line
can be drawn
through the
given point
parallel to the
given line.
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Hyperbolic
Given a line
and a point
not on the
line, two or
more lines
can be drawn
through the
given point
parallel to the
given line.
A Model for the Three Geometries
The term line is undefined.
It can be interpreted differently in
different geometries.
Euclidean
Elliptical
Plane
Sphere
Hyperbolic
Pseudosphere
9.7-43
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Elliptical Geometry
A circle on the surface of a sphere is
called a great circle if it divides the
sphere into two equal parts.
We interpret a line to be a great circle.
This shows the fifth axiom
of elliptical geometry to be
true. Two great circles on a
sphere must intersect;
hence, there can be no
parallel lines.
9.7-44
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Elliptical Geometry
If we were to construct a
triangle on a sphere, the
sum of its angles would be
greater than 180º.
The sum of the measures of the angles
varies with the area of the triangle and
gets closer to 180º as the area
decreases.
9.7-45
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Hyperbolic Geometry
Lines in hyperbolic
geometry are
represented by geodesics
on the surface of a
pseudosphere.
A geodesic is the shortest
and least-curved arc
between two points on
the surface.
9.7-46
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Hyperbolic Geometry
This illustrates one
example of the
fifth axiom:
through the given
point, two lines
are drawn parallel
to the given line.
9.7-47
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Hyperbolic Geometry
If we were to construct a
triangle on a pseudosphere,
the sum of the measures of
the angles of the triangle
would be less than 180º.
9.7-48
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