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Copyright 2013, 2010, 2007, Pearson, Education, Inc.

Section 9.1

Points, Lines,Planes, and

Angles

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What You Will Learn

PointsLinesPlanes

Angles

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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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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.

9.1-4

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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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Plane

A line in a plane divides the plane into three

parts, the line and two half planes.

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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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Plane

Two planes that do not intersect are

said to be parallel planes.

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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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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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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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Example 3: DeterminingComplementary Angles

In the figure, we see that

Determine mRCBD.

RABC 28.RABC& RCBDare complementary angles

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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.

9.1-21

D fi iti

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Definitions

A line that

intersects twodifferent lines, attwo different

points is called atransversal.

9.1-22

D fi iti

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Definitions

Special names

are given to theangles formedby a transversal

crossing twoparallel lines.

9.1-23

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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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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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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