1.1 Intro to Geometry

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1.1 Intro to Geometry
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Geometry
The word "geometry " comes from two Greek words
geo and metron meaning "earth measuring."
Geometry was extremely important to ancient
societies and was used for surveying, astronomy,
navigation, and building. Geometry, as we know it
is actually known as Euclidean geometry which was
written well over 2000 years ago in Ancient
Greece by Euclid, Pythagoras, Thales, Plato and
Aristotle just to mention a few. The most
fascinating and accurate geometry text was
written by Euclid, and was called Elements.
Euclid's text has been used for over 2000 years!
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Millau Bridge Sir Norman Foster
Millenium Park Frank Lloyd Wright
Fallingwaters Frank Lloyd Wright
Point, Lines, Planes, Angles
Components of Geometry Part 1
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GEOMETRYS UNDEFINED TERMS

• POINT
• LINE
• PLANE

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POINT

• A point is a_______, or a_____.

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POINT

• A point is not a physical object you can not
  touch it, feel it, or even see it.

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POINT

• A point has no size.

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POINT

• A point can be represented by a dot. The size
  that you make the dot is not important because a
  point has no size. Make it just large enough to
  see.

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POINT

• A point is named by using an upper-case block
  letter.

EXAMPLE
P
This figure would be called point P.
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POINT
EXAMPLE 2
X
This figure would be called point X.
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LINE

• A line is a set of points that extends
  infinitely in 2 opposite directions.

What does infinitely mean ?
NEVER-ENDING (Goes on forever.)
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LINE
Why does it have arrows on both ends ?
EXAMPLE
TO SHOW THAT IT EXTENDS INFINITELY IN BOTH
(OPPOSITE) DIRECTIONS
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NAMING A LINE
There are 2 ways
1. Use the names of (any) two points on the line.
What name(s) can we give to this line?
Example
X
P
Line ___or line____
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NAMING A LINE
There are 2 ways
2. Use a lower-case cursive letter (located
near one of the arrows).
Example
What name can we give to this line?
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NAMING A LINE
There are 2 ways
1. Use the names of (any) two points on
the line.
PX
EXAMPLE line PX (OR USE A SYMBOL FOR LINE
AND WRITE IT ABOVE THE LETTERS.)
2. Use a lower-case cursive letter (located
near one of the arrows).
EXAMPLE line ____
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How many names does this line have?
K
G
R
There are 7 different names that you could use to
name this line.
They are
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Lets try again.How many names does this line
have?
t
X
N
F
There are 7 different names that you could use to
name this line.
They are
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ONE MORE TIME.How many names does this line have?
N
Y
C
T
There are 12 different names that you could use
to name this line.
They are
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PLANE

• A plane is a flat surface a set of points
  that extends infinitely in 2 dimensions.

What does dimension mean ?
________,_______,__________.
A plane is infinitely long and infinitely wide,
but it has no height (or depth).
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PLANE
A plane can be drawn (represented) by a four
sided figure known as a parallelogram.
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PLANE
A plane can be named by a single upper-case
block letter (written near a corner of the
plane).
What is the name of this plane?
Q
Plane Q or ?Q
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What is the name of this plane?
R
N
ANSWER
T
_______
F
WHY CANT IT BE CALLED PLANE R or T or N ?

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• Descriptions of Defined Terms
• Space
• General Terms
• Congruent
• Similar
• Equal
• Union
• Intersection

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• Space

Refers to the set of all points. Space goes on
forever in every direction, and therefore has
length, width, and depth. Space has no special
notations. It is simply referred to as space.
Space contains at least 4 points that are not all
on the same plane.
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• General Terms

• Congruent
• Similar
• Equal
• Union
• Intersection

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• Congruent

_at_ or congruent shapes are the same shape and
size therefore, after some movement of the
shapes they can be made to fit exactly on top of
one another.
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• Congruent

_at_ or congruent shapes are the same shape and
size therefore, after some movement of the
shapes they can be made to fit exactly on top of
one another.
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• Similar

or similar shapes are the same shape, but can
be different sizes thus congruent shapes are
also similar shapes, but similar shapes are not
necessarily congruent shapes.
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• Equal

or equal can apply to sets of points being
exactly the same set or to numerical measurements
being exactly the same number values.
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• Union

È or union refers to putting all of the points
together and describing the result.
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• Intersection

Ç or intersection refers to describing only those
points that are common to all sets involved in
the intersection or to describing the points
where indicated shapes touch.
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Ray

• Symbolized by

Lets look at a ray
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Symbol alert!

• Not all symbols are created equal!

is the same as
BUT
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Symbol alert!!
The ray is different!
Notice that the initial point is listed first in
the symbol. Also note that the symbolic ray
always has the arrowhead on the right regardless
of the direction of the ray.
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Opposite Rays

• If C is between A and B,

then and are opposite rays.
C is the common initial point for the rays!
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Angles

• Rays are important because they help us define
  something very important in geometryAngles!
• An angle consists of two different rays that have
  the same initial point. The rays are sides of
  the angles. The initial point is called the
  vertex.

Notation We denote an angle with three points
and symbol. The middle point is always the
vertex. We can also name the angle with just the
vertex point. This angle can be denoted as
B
vertex
sides
A
C
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Classifying Angles

• Angles are classified as acute, right, obtuse,
  and straight, according to their measures.
  Angles have measures greater than 0 and less or
  equal to 180.

Straight angle m A 180
Obtuse angle 90lt m A lt 180
Acute angle 0lt m A lt 90
Right angle m A 90
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Intersections of lines and planes

• Two or more geometric figures intersect if they
  have one or more points in common.
• The intersection of the figures is the set of
  points the figure has in common

How do 2 line intersect? How do 2 planes
intersect? What about a line and a plane?
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Modeling Intersections

• To think about the questions on the last slide
  lets look at the following

Point E is the intersection of plane H and line EC
E
Two lines intersect at a point, like here at
point A.
H
C
G
Line BF is the intersection of the planes G and H.