Download angle - Cloudfront.net

Section 9.1
Points, Lines,
Planes, and
Angles
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
Points
Lines
Planes
Angles
9.1-2
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Basic Terms
A point, line, and plane are three basic
terms in geometry that are NOT given a
formal definition, yet we recognize them
when we see them.
9.1-3
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Basic Terms
Description
Diagram
Line AB
A
Ray AB
B
B
A
Line segment AB
A
AB
AB
B
A
Ray BA
9.1-5
Symbol
B
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Plane
Two lines in the same plane that do
not intersect are called parallel lines.
9.1-7
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Plane
A line in a plane divides the plane into three
parts, the line and two half planes.
9.1-8
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
9.1-9
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Plane
Two planes that do not intersect are
said to be parallel planes.
9.1-10
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Angles
An angle is the union of two rays with a
common endpoint; denoted  .
9.1-11
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
9.1-12
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
9.1-13
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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
9.1-14
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Angles
9.1-15
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
9.1-16
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: Determining
Complementary Angles
In the figure, we see that ABC  28
ABC & CBD are complementary angles.
Determine CBD
9.1-17
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: Determining
Complementary Angles
Solution
mABC  mCBD  90
28  mCBD  90
mCBD  90  28  62
9.1-18
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: Determining
Supplementary Angles
In the figure, we see that ABC  28
ABC & CBE are supplementary angles
Determine mCBE
9.1-19
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: Determining
Supplementary Angles
Solution
mABC  mCBE  180
28  mCBE  180
mCBE  180  28  152
9.1-20
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Definitions
When two straight lines intersect, the
nonadjacent angles formed are called
Vertical angles.
Vertical angles have the same measure.
9.1-21
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Definitions
A line that
intersects two
different lines, at
two different
points is called a
transversal.
9.1-22
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Definitions
Special names
are given to the
angles formed
by a transversal
crossing two
parallel lines.
9.1-23
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Special Names
9.1-24
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
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
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.
9.1-25
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 6: Determining Angle
Measures
The figure
shows two
parallel lines
cut by a
transversal.
Determine the
measure of 1
through 7 .
9.1-26
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 6: Determining Angle
Measures
Solution
6  49
5  131
7  131
9.1-27
8 & 6 Vertical angles
8 & 5 Supplementary angles
5 & 7
Vertical angles
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 6: Determining Angle
Measures
Solution
1  131
4  49
2  49
3  131
9.1-28
1& 7 Alternate exterior
4 & 6 Alternate exterior
6 & 2 Corresponding angles
3 & 1 Vertical angles
Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Homework
P. 487 # 1 – 20all, 45 – 87 (x3)
9.1-29
Copyright 2013, 2010, 2007, Pearson, Education, Inc.