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Chapter
11
Introductory
Geometry
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NCTM Standard: Geometry
In grades preK–2, all students should
 recognize, name, build, draw, compare, and
sort two- and three-dimensional shapes;
 describe attributes and parts of two-and threedimensional shapes;
 investigate and predict the results of putting
together and taking apart two- and threedimensional shapes. (p. 96)
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NCTM Standard: Geometry
In grades 3–5 all students should
 sort, build, draw, model, trace, measure, and
construct, the capacity to visualize geometric
relationships
 learn to reason and to make, test, and justify
conjectures
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NCTM Standard: Geometry
In grades 6–8, all students should
 precisely describe, classify, and understand
relationships among types of two- and threedimensional objects using their defining
properties;
 use visual tools such as networks to represent
and solve problems. (p. 240)
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11-1 Basic Notions
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Linear Notions
Planar Notions
Other Planar Notions
Angles
Angle Measurement
Types of Angles
Perpendicular Lines
A Line Perpendicular to a Plane
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Undefined terms: points, lines, and
planes
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Lines
A line has no thickness and it extends forever in
two directions.
Given two points, there is one and only one line
that connects these points.
If P and Q are any two points, we can create a
number line on the line PQ such that there is oneto-one correspondence between the points on the
line and real numbers.
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Linear Notions
Collinear points
 Line ℓ contains points A, B, and C.
 Points A, B, and C belong to line ℓ.
 Points A, B, and C are collinear.
 Points A, B, and D are not collinear.
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Linear Notions
Between
 Point B is between points A and C on line ℓ.
 Point D is not between points A and C.
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Linear Notions
AB or BA
Line segment
A subset of a line that contains two points of the
line and all points between those two points.
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Linear Notions
AB
Ray
A subset of the line AB that contains the
endpoint A, the point B, all points between A
and B, and all points C on the line such that B
is between A and C.
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Planar Notions
Coplanar
 Points D, E, and G are
coplanar.
 Points D, E, F, and G
are not coplanar.
 Lines DE, DF, and FE are coplanar.
 Lines DE and EG are coplanar.
 Lines DE and EG are intersecting lines; they
intersect at point E.
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Planar Notions
Skew lines
 Lines GF and DE are
skew lines. They do not
intersect, and there is no
plane that contains them.
Concurrent lines
 Lines DE, EG, and EF are concurrent lines;
they intersect at point E.
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Planar Notions
Parallel lines
Line m is parallel to line n. They
have no points in common.
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Axioms About Points, Lines, and Planes
 There is exactly one line that contains any two
distinct points.
 If two points lie in a plane, then the line
containing the points lies in the plane.
 If two distinct planes intersect, then their
intersection is a line.
 There is exactly one plane that contains any
three distinct noncollinear points.
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Theorems About Points, Lines, and
Planes
 A line and a point not on the line determine a
plane.
 Two parallel lines determine a plane.
 Two intersecting lines determine a plane.
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Other Planar Notions
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Half plane
Line AB separates plane  into two halfplanes.
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Angles
Angle – formed by two rays
with the same endpoint.
Sides of an angle – the two
rays that form an angle.
Vertex – the common endpoint of the two rays
that form an angle.
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Angles
Adjacent angles – two angles with a common
vertex and a common side, but without
overlapping interiors.
QPR is adjacent to RPS.
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Angle Measurement
Degree :
Minute :
Second :
of a rotation about a point
of a degree
of a minute
Protractor
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Example 11-1
a. Find the measure of BAC
if m1 = 47°45′ and
m2 = 29°58′ .
mBAC = 17°47′
b. Express 47°45′36″ as a number of degrees.
47°45′36″ = 47°+ 0.75° + 0.01° = 47.76°
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Types of Angles
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Types of Angles
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Perpendicular Lines
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A Line Perpendicular to a Plane
A line perpendicular to a plane is a line that is
perpendicular to every line in the plane through
its intersection with the plane.
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Perpendicular Planes
If P is any point on AD, Q in
plane α, and S in plane β so that
PQ  AD and PS  AD, then
QPS is also a right angle.
Since QPS measures 90°, the
planes are perpendicular.
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Dihedral Angles
A dihedral angle is formed by
the union of two half-planes
and the common line defining
the half-planes.
A dihedral angle is measured
by any of the associated planar
angles such OPD, where
PO  AC and PD  AC.
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