aps09_0901
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Section 9.1
Points, Lines,Planes, and
Angles
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
What You Will Learn
PointsLinesPlanes
Angles
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Basic Terms
Apoint, line, andplaneare three basicterms in geometry that are NOT given a
formal definition, yet we recognize them
when we see them.
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Lines, Rays, Line SegmentsA lineis 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 rayis a half line including the endpoint.
A line segment is part of a line betweentwo points, including the endpoints.
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Basic Terms
BA
ALine segmentAB
Ray BA
RayAB
LineAB
SymbolDiagramDescription
AB
AB
BA
AB
A B
A B
B
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Plane
We can think of a planeas a two-
dimensional surface that extends infinitely inboth directions.Any three points that are not on the sameline (noncollinear points) determine a unique
plane.
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Plane
Two lines in the same plane that do
not intersect are called parallel lines.
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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.
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Plane
Any line and a point not on the line
determine a unique plane.The intersection oftwo distinct,non-parallel
planes is a line.
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Plane
Two planes that do not intersect are
said to be parallel planes.
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
AnglesAn angleis the union of two rays with a
common endpoint; denoted .
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AnglesThe vertexis the point common to both
rays.The sidesare the rays that make theangle.There are several ways to name an angle:
, ,ABC CBA B
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
AnglesThe measure of an angle is the amount of
rotation from its initial to its terminal side.Angles can be measured in degrees,radians, orgradients.
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Angles
Angles are classified by their degree
measurement.
Right Angle is 90
Acute Angle is less than 90
Obtuse Angle is greater than 90 but lessthan 180
Straight Angle is 180
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Angles
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Types of Angles
Adjacent Angles - angles that have acommon 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.
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: DeterminingComplementary Angles
In the figure, we see that
Determine mRCBD.
RABC 28.RABC& RCBDare complementary angles
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 3: DeterminingComplementary Angles
mRABC mRCBD 90
28 mRCBD 90mRCBD 90 28 62
Solution
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Example 3: DeterminingSupplementary Angles
In the figure, we see that
Determine mRCBE.
RABC 28.RABC& RCBEare supplementary angles.
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Example 3: DeterminingSupplementary Angles
mRABC mRCBE 180
28 mRCBE 180mRCBE 180 28 152
Solution
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DefinitionsWhen two straight lines intersect, the
nonadjacent angles formed are calledVertical angles.Vertical angles have the same measure.
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Definitions
A line that
intersects twodifferent lines, attwo different
points is called atransversal.
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Definitions
Special names
are given to theangles formedby a transversal
crossing twoparallel lines.
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Special Names
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Special Names
5 6
1 2
4
87
3
One interior and oneexterior angle on the
same side of thetransversalhave thesame measure
Correspondingangles
1 & 5, 2 & 6,3 & 7, 4 & 8
Exterior angles onthe opposite sides of
the transversalhave
the same measure
Alternateexterior angles
1 & 8; 2 & 7
Interior angles onthe opposite side of
the transversalhave
the same measure
Alternateinterior angles
3 & 6; 4 & 5
5 6
1 2
4
87
3
5 6
1 2
4
87
3
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Parallel Lines Cut by a TransversalWhen 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.
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E l 6 D t i i A l
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Example 6: Determining AngleMeasures
The figureshows twoparallel linescut by atransversal.Determine themeasure of
through .R7
R1
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Example 6: Determining Angle
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Example 6: Determining AngleMeasuresSolutionR6 49 R8 & R6 vertical angles
R5 131 R8 & R5 supplementary angles
R
7
131
R5 & R7 vertical angles
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Example 6: Determining Angle
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Copyright 2013, 2010, 2007, Pearson, Education, Inc.
Example 6: Determining AngleMeasuresSolutionR1 131 R1 & R7 alternate exteriorR4 49 R4 & R6 alternate interior
R2 49 R6 & R2 corresponding angles
R3 131 R3 & R1 vertical angles
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