Download Section 9.1 Points, Lines, Planes, and Angles

Section 9.1
Points, Lines,
Planes, and
Angles
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
INB Table of Contents
2.3-2
Date
Topic
September 9, 2013
Test #1
12
September 9, 2013
Test #1 Corrections
13
September 9, 2013
Sections 9.1, 9.2 Foldable & Examples
14
September 9, 2013
Sections 9.1, 9.2 Notes
15
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Page #
What You Will Learn
Points
Lines
Planes
Angles
9.1-3
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Lines, Rays, Line Segments
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.
9.1-4
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Basic Terms
Description
Diagram
Line AB
A
Ray AB
B
B
A
Ray BA
B
A
Line segment AB
9.1-5
Symbol
A
B
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
AB

AB

BA
AB
Plane
We can think of a plane as a twodimensional surface that extends infinitely in
both directions.
Any three points that are not on the same
line (noncollinear points) determine a unique
plane.
9.1-6
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Plane
Two lines in the same plane that do
not intersect are called parallel lines.
9.1-7
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Angles
An angle is the union of two rays with a
common endpoint; denoted ∠.
9.1-8
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Angles
The vertex is the point common to both
rays.
The sides are the rays that make the
angle.
There are several ways to name an angle:
∠ABC, ∠CBA, ∠B
9.1-9
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Angles
The measure of an angle is the amount of
rotation from its initial to its terminal side.
Angles can be measured in degrees,
radians, or gradients.
9.1-10
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Angles
Angles are classified by their degree
measurement.
• Right Angle is 90º
• Acute Angle is less than 90º
• Obtuse Angle is greater than 90º but less
than 180º
• Straight Angle is 180º
9.1-11
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Angles
9.1-12
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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.
9.1-13
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Example 3: Determining
Complementary Angles
In the figure, we see that ∠ABC = 28°
∠ABC & ∠CBD are complementary
angles. Determine m∠CBD.
9.1-14
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Example 3: Determining
Supplementary Angles
In the figure, we see that ∠ABC = 28°.
∠ABC & ∠CBE are supplementary
angles. Determine m∠CBE.
9.1-16
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Definitions
When two straight lines intersect, the
nonadjacent angles formed are called
Vertical angles.
Vertical angles have the same measure.
9.1-18
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Definitions
A line that
intersects two
different lines, at
two different
points is called a
transversal.
9.1-19
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Definitions
Special names
are given to the
angles formed
by a transversal
crossing two
parallel lines.
9.1-20
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Special Names
9.1-21
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.
9.1-22
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Example 6: Determining Angle
Measures
The figure
shows two
parallel lines
cut by a
transversal.
Determine the
measure of 
R1
through 
R7 .
9.1-23
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Section 9.2
Polygons
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What You Will Learn
Polygons
Similar Figures
Congruent Figures
9.2-27
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Polygons
A polygon is a closed figure in a plane
determined by three or more straight
line segments.
9.2-28
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Polygons
Polygons are named according to
their number of sides.
9.2-29
Number
of Sides
Name
Number of
Sides
Name
3
Triangle
8
Octagon
4
Quadrilateral
9
Nonagon
5
Pentagon
10
Decagon
6
Hexagon
12
Dodecagon
7
Heptagon
20
Icosagon
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Polygons
Sides Triangles Sum of the Measures of
the Interior Angles
3
1
1(180º) = 180º
4
2
2(180º) = 360º
5
3
3(180º) = 540º
6
4
4(180º) = 720º
The sum of the measures of the
interior angles of an n-sided polygon
is (n – 2)180º.
9.2-30
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Types of Triangles
Acute Triangle
All angles are acute.
9.2-31
Obtuse Triangle
One angle is obtuse.
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Types of Triangles (continued)
Right Triangle
One angle is a right
angle.
9.2-32
Isosceles Triangle
Two equal sides.
Two equal angles.
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Types of Triangles (continued)
Equilateral Triangle
Three equal sides.
Three equal angles,
60º each.
9.2-33
Scalene Triangle
No two sides are
equal in length.
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Similar Figures
Two figures are similar if their
corresponding angles have the same
measure and the lengths of their
corresponding sides are in proportion.
9
6
4
4
3
9.2-34
6
6
4.5
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Example 3: Using Similar Triangles
to Find the Height of a Tree
Monique Currie plans to remove a tree
from her backyard. She needs to know
the height of the tree. Monique is 6 ft
tall and determines that when her
shadow is 9 ft long, the shadow of the
tree is 45 ft long (see Figure). How tall
is the tree?
9.2-35
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Example 3: Using Similar Triangles
to Find the Height of a Tree
9.2-36
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Example 3: Using Similar Triangles
to Find the Height of a Tree
9.2-37
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Congruent Figures
If corresponding sides of two similar figures
are the same length, the figures are
congruent.
Corresponding angles of congruent figures
have the same measure.
9.2-39
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Quadrilaterals
Quadrilaterals are four-sided polygons, the
sum of whose interior angles is 360º.
Quadrilaterals may be classified according to
their characteristics.
9.2-40
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Quadrilaterals
Trapezoid
Parallelogram
Two sides are parallel.
9.2-41
Both pairs of opposite
sides are parallel. Both
pairs of opposite sides
are equal in length.
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Quadrilaterals
Rhombus
Rectangle
Both pairs of opposite
sides are parallel. The
four sides are equal in
length.
9.2-42
Both pairs of opposite
sides are parallel. Both
pairs of opposite sides
are equal in length. The
angles are right angles.
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Quadrilaterals
Square
Both pairs of opposite
sides are parallel. The
four sides are equal in
length. The angles are
right angles.
9.2-43
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Example 5: Angles of a Trapezoid
Trapezoid ABCD is shown.
a) Determine the measure of the
interior angle, x.
b) Determine the measure of the
exterior angle, y.
9.2-44
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Example 5: Angles of a Trapezoid
9.2-45
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