Transcript POINTS, LINES, & PLANES

CHAPTER 1
SECTION 1-1
A Game and Some
Geometry
SECTION 1-2
Points, Lines and
Planes
POINT– it indicates a
specific location and is
represented by a dot and a
letter, but it has no
dimensions
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LINE – is a set of points that
extends without end in two
opposite directions
R
S
«—•—————•—»
line RS
PLANE – is a set of
points that extends in all
directions along a flat
surface
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COLLINEAR POINTS –
points that lie on the
same line
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C
D
E
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COPLANAR POINTS –
are points that lie in the
same plane
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A
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•B
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D
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INTERSECTION – set of
all points common to two
geometric figures
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SECTION 1-3
Segments, Rays
and Distances
RAY – a part of a line
that begins at one point,
called the ENDPOINT
and extends without end
in one direction
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» «
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J
LINE SEGMENT - part of
a line that begins at one
endpoint and ends at
another
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G
POSTULATES - accepted as
true without proof
RULER POSTULATE
The points on any line can
be paired with the real
numbers in such a way that
any point can be paired with
0 and any other point can
be paired with 1.
The real number paired
with each point is the
coordinate of that point.
The distance between any
two points on the line is
equal to the absolute
value of the difference of
their coordinates.
THE SEGMENT
ADDITION
POSTULATE
If point B is between
points A and C, then
AB +BC = AC
Given the figure below:
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A
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B
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C
AC = 47, AB = n – 5,
and BC = n + 8, Find
AB
AC = AB + BC
47 = (n – 5) + (n + 8)
47 = 2n + 3
44 = 2n
22 = n, therefore
AB = 22-5 or 17
Congruent segments
segments that are equal
in length
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12
RS
12
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MIDPOINT– the point that
divides a segment into two
segments of equal length.
BISECTOR of a
SEGMENT– is any line,
segment, ray, or plane that
intersects the segment at its
midpoint.
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M
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SECTION 1-4
Angles
ANGLE– the union of two
rays with a common endpoint.
The rays are called sides
VERTEX – endpoint of an
angle
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A
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C
C
PROTRACTOR
POSTULATE
Let O be a point on AB such
that O is between A and B.
Then ray OA can be paired
with O° and ray OB can be
paired with 180°
P•
180º
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B
Q
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O
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A
0º
If OP is paired with x
and OQ is paired with y,
then the number paired
with measure of angle
POQ is | x – y |. This is
called the measure of
angle POQ.
ANGLE ADDITION
POSTULATE
If point B lies in the interior
of angle AOC, then:
mAOB + m BOC = m AOC
A•
O
B
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C
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CONGRUENT ANGLES
angles that have equal
measures
40°
40°
ADJACENT ANGLES
two angles in the same
plane that share a
common side and a
common vertex, but
have no interior points
in common
ADJACENT
ANGLES
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B
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C
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AOB and BOC
BISECTOR of an
ANGLE is the ray that
divides the angle into
two congruent adjacent
angles.
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B
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M AOB = M  BOC
Section 1-5
Postulates and Theorems
Relating Points, Lines, and
Planes
POSTULATE 5
• A line contains at least two
points; a plane contains at
least three points not all in
one line; space contains at
least four points not all in
one plane.
POSTULATE 6
• Through any two points
there is exactly one
line (Two points
determine a line)
POSTULATE 7
• Through any three points,
there is at least one plane,
and through any three
noncollinear points there is
exactly one plane.
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POSTULATE 8
• If two points are in a
plane, then the line
that contains the
points is in that plane.
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POSTULATE 9
• If two planes
intersect, then their
intersection is a line
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W
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U
THEOREMS – Statements
that have been proven.
THEOREM 1-1
• If two lines intersect,
then they intersect in
exactly one point.
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A
P
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B
THEOREM 1-2
• Through a line and a
point not in the line
there is exactly one
plane.
THEOREM 1-3
• If two lines intersect,
then exactly one plane
contains the lines.
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F
••
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D
THE END