Download Points, Lines, Planes, and Angles

Chapter 9
Geometry
© 2008 Pearson Addison-Wesley.
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Chapter 9: Geometry
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
Points, Lines, Planes, and Angles
Curves, Polygons, and Circles
Perimeter, Area, and Circumference
The Geometry of Triangles: Congruence,
Similarity, and the Pythagorean Theorem
Space Figures, Volume, and Surface Area
Transformational Geometry
Non-Euclidean Geometry, Topology, and Networks
Chaos and Fractal Geometry
9-1-2
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Chapter 1
Section 9-1
Points, Lines, Planes, and Angles
9-1-3
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Points, Lines, Planes, and Angles
• The Geometry of Euclid
• Points, Lines, and Planes
• Angles
9-1-4
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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  ,  , or  .

l
A
D
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
Figure
Symbol
A
AB or BA
A
Half-line BA
B
A
Ray AB
A
Ray BA
B
A
BA
AB
B
A
Segment AB or
segment BA
AB
B
B
B
BA
AB or BA
9-1-8
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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
9-1-9
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Parallel and Intersecting Planes
Parallel planes never meet.
Two distinct intersecting planes meet and form
a straight line.
Parallel
Intersecting
9-1-10
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Angles
An angle is the union of two rays that have a
common endpoint. An angle can be named with the
letter marking its vertex, B , and also with three
letters: ABC - the first letter names a point on the
side; the second names the vertex; the third names a
point on the other side.
A
Vertex
B
C
9-1-11
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Angles
Angles are measured by the amount of rotation. 360°
is the amount of rotation of a ray back onto itself.
45°
90°
10°
150°
360°
9-1-12
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Angles
Angles are classified and named with reference to their
degree measure.
Measure
Between 0° and 90°
90°
Greater than 90° but
less than 180°
180°
Name
Acute Angle
Right Angle
Obtuse Angle
Straight Angle
9-1-13
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Protractor
A tool called a protractor can be used to measure
angles.
9-1-14
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Intersecting Lines
When two lines intersect to form right
angles they are called perpendicular.
9-1-15
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Vertical Angles
In the figure below the pair ABC and DBE
are called vertical angles. DBA and EBC
are also vertical angles.
A
D
B
E
C
Vertical angles have equal measures.
9-1-16
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Example: Finding Angle Measure
Find the measure of each marked angle below.
(3x + 10)°
(5x – 10)°
Solution
3x + 10 = 5x – 10
2x = 20
x = 10
Vertical angels are equal.
So each angle is 3(10) + 10 = 40°.
9-1-17
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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.
(2x + 45)°
(x – 15)°
Solution
2x + 45 + x – 15 = 180
Supplementary angles.
3x + 30 = 180
3x = 150
x = 50
Evaluating each expression we find that the
angles are 35° and 145° .
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9-1-19
Angles Formed When Parallel Lines are
Crossed by a Transversal
The 8 angles formed will be discussed on the next
few slides.
1
3
5
7
2
4
6
8
9-1-20
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Angles Formed When Parallel Lines are
Crossed by a Transversal
Name
Alternate
interior
angles
5
4
Angle
measures are
equal.
(also 3 and 6)
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
Angle
measures
add to 180°.
4
(also 3 and 5)
Corresponding
angles
2
6
(also 1 and 5, 3
and 7, 4 and 8)
Angle
measures are
equal.
© 2008 Pearson Addison-Wesley. All rights reserved
9-1-22
Example: Finding Angle Measure
Find the measure of each marked angle below.
(x + 70)°
(3x – 80)°
Solution
x + 70 = 3x – 80 Alternating interior angles.
2x = 150
x = 75
Evaluating we find that the angles are 145°.
9-1-23
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