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Chapter 9: Geometry
9.1 Points, Lines, Planes, and Angles
9.2 Curves, Polygons, and Circles
9.3 Perimeter, Area, and Circumference
9.4 The Geometry of Triangles: Congruence,
Similarity, and the Pythagorean Theorem
9.5
Space Figures, Volume, and Surface Area
9.6
Transformational Geometry
9.7 Non-Euclidean Geometry, Topology, and Networks
9.8 Chaos and Fractal Geometry
9-1-2
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Points, Lines, Planes, and Angles
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9-1-3

Points, Lines, Planes, and Angles
• The Geometry of Euclid
• Points, Lines, and Planes
• Angles
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9-1-4

The Geometry of Euclid
A point has no magnitude and no size.
A line has no thickness and no width and it extends indefinitely in two directions.
A plane is a flat surface that extends infinitely.
9-1-5
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Points, Lines, and Planes
A capital letter usually represents a point. A line may named by two capital letters representing points that lie on the line or by a single letter such as l
. A plane may be named by three capital letters representing points that lie in the plane or by a letter of the Greek alphabet such as
  

A
D l
E
9-1-6
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Half-Line, Ray, and Line Segment
A point divides a line into two half-lines , one on each side of the point.
A ray is a half-line including an initial point.
A line segment includes two endpoints.
9-1-7
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Half-Line, Ray, and Line Segment
Name
Line AB or BA
Half-line AB
Half-line BA
Ray AB
Ray BA
Segment AB or segment BA
Figure
A B
A B
A B
A B
A B
A B
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Symbol
AB or BA
AB
BA
AB
BA
AB or BA
9-1-8

Parallel and Intersecting Lines
Parallel lines lie in the same plane and never meet.
Two distinct intersecting lines meet at a point.
Skew lines do not lie in the same plane and do not meet.
Parallel
Intersecting Skew
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9-1-9

Parallel and Intersecting Planes
Parallel planes never meet.
Two distinct intersecting planes meet and form a straight line.
Parallel Intersecting
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9-1-10

Angles
An angle is the union of two rays that have a common endpoint. An angle can be named with the side; the second names the vertex; the third names a point on the other side.
A
Vertex B
C
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9-1-11

Angles
Angles are measured by the amount of rotation. 360° is the amount of rotation of a ray back onto itself.
90°
45°
150°
360°
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10°
9-1-12

Angles
Angles are classified and named with reference to their degree measure.
Measure Name
Between 0° and 90° Acute Angle
90° Right Angle
Greater than 90° but less than 180°
180°
Obtuse Angle
Straight Angle
9-1-13
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Protractor
A tool called a protractor can be used to measure angles.
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9-1-14

Intersecting Lines
When two lines intersect to form right angles they are called perpendicular .
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9-1-15

Vertical Angles
In the figure below the pair are called vertical angles.
are also vertical angles.
D
A
ABC and
DBA and
DBE
EBC
B
E C
Vertical angles have equal measures.
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9-1-16

Example: Finding Angle Measure
Find the measure of each marked angle below.
(3 x + 10)°
Solution
3 x + 10 = 5 x – 10
2 x = 20 x = 10
Vertical angels are equal.
So each angle is 3(10) + 10 = 40°.
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(5 x – 10)°
9-1-17

Complementary and Supplementary
Angles
If the sum of the measures of two acute angles is
90°, the angles are said to be complementary , and each is called the complement of the other. For example, 50° and 40° are complementary angles
If the sum of the measures of two angles is 180°, the angles are said to be supplementary , and each is called the supplement of the other. For example,
50° and 130° are supplementary angles
9-1-18
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Example: Finding Angle Measure
Find the measure of each marked angle below.
(2 x + 45)°
( x – 15)°
Solution
2 x + 45 + x – 15 = 180
3 x + 30 = 180
3 x = 150
Supplementary angles.
x = 50
Evaluating each expression we find that the angles are 35° and 145° .
9-1-19
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Angles Formed When Parallel Lines are
Crossed by a Transversal
The 8 angles formed will be discussed on the next few slides.
1 2
3 4
5 6
7 8
9-1-20
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Angles Formed When Parallel Lines are
Crossed by a Transversal
Name
Alternate interior angles
5 4
(also 3 and 6)
Angle measures are equal.
1
Alternate exterior angles
Angle measures are equal.
8 (also 2 and 7)
9-1-21
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Angles Formed When Parallel Lines are
Crossed by a Transversal
Name
Interior angles on same side of transversal
6
4
(also 3 and 5)
Angle measures add to 180°.
Corresponding angles 6
2
(also 1 and 5, 3 and 7, 4 and 8)
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Angle measures are equal.
9-1-22

Example: Finding Angle Measure
Find the measure of each marked angle below.
(3 x – 80)°
( x + 70)°
Solution x + 70 = 3 x – 80
2 x = 150 x = 75
Alternating interior angles.
Evaluating we find that the angles are 145°.
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9-1-23