3. Use symbols for lines, segments, rays, and distances; find
4. Name angles and find their measures.
distances.
1-1 Points, Lines, and Planes
SUMMARY
NOTES
– Week
2 how many different colors do you
picture,
a color television
When
you look at
see?
Actually, the picture is made up of just three colors-red, green, and
___________________________________________________________________
blue. Most color television screens are covered with more than 300,000 colored
INDUCTIVE
the enlarged diagram on the right below. Each dot glows
dots,
as shown in REASONING:
the dots are
small, and
Ifanconclusions
or Since
predictions
aresobased
on so
anclose
observed pattern…
when it is struck by
electron beam.
individual
dots.
rather
than
images
seesathree
superimposed
together, your eye“If
pattern exists it will continue”
BEWARE: Conclusions derived from observed patterns are not always true.
ooloo
ooaa30
ooaoo
ooooco
ooooo
DEDUCTIVE REASONING
If conclusions or predictions are based on a definition,rule or logical reasoning
“If A then B, if B then C, if C then D so if A then D”
______________________________________________________________________________________________
UNDEFINED TERMS: Terms use as building blocks for other terms.
POINT:
Names a location and has no size
No Dimension
Notation: Capital Letter
.A
Called:
Point A
Each small dot on a television screen suggests the simplest frgno size, it isand
a point
Although
ure
studied
in geometry-apoint.
LINE:
A straight path
that has
nohas
thickness
by adimensional
dot and named by a capital letter. Two
often representedOne
points, A arrd B, are pictured at the right.
has consist
at least
two points.
of points.
One familiar geometric
All geometricItfltgures
figure is a line, which extends in two directions without ending.
Although a picture of a line has some thickness, the line itself has no
extends forever (has infinite length)
B
thickness.
Often a line is referred to by a single lower-case letter, such as
line l. lf yo:u know that a line contains the points ,4 and B, you can
also call it line AB (denoted AB) or line BA (BA).
A
!##" !##"
ABPlanes,
Points,i.e.
Lines,
points
, BAandorAngles
a miniscule
/I
Notation: Use at least two
Line
AB, Line BA, Line l
A geometric plane is Called:
suggested by
a floor, wall, or table top. Unlike
letter
l
a table
top, a plane extends without ending and has no thickness. Although a plane
has no edges,
we usuallyApicture
a plane by
drawing
a four-sided
figure
as extends forever (has infinite length and
PLANE:
flat surface
that
has not
thickness
and
shown below. We often label a plane with a capital letter.
width, Two dimensional
Plane
M
PlaneN
Notation:
at least
non-collinear
3 Points on the plane or the Capital Letter in the
In geometry, the termsp
oint, line, Use
and plane
are accepted
as intuitive ideas
and are not defined. These
corner
of the
undefined
termsplane
are then used in the definitions of
other terms, such as those
below.
Called: Plane ABC, Plane N
Space isSPACE:
the set of all
points.
The
set of all points (a Defined Term derived from other terms)
Collinear points are points all in one line.
a
Collinear points
Coplanar points are points all
a
Noncollinear points
in one plane.
nes
and are not defined. These undefined terms are then used in the definitions of
other terms, such as those below.
\4/hen Lines and Planes
Collinear points are points all in
line.
and
-Are Parallel
OTHER
Collinear,
Coplanar, Parallel Line, Perpendicular Line, Skew Line
Space isTERMS:
the set of
all points.
one
lel lines,
skew lines.
ction of two parallel planes
Obiectioes
a
a
l. Distinguish
and
between
intersecting
lines,
parallel
lines,
skew lines.
are cut by a transversal.
Collinear points
Noncollinear points
2. lines.
State and apply the theorem about the intersection of two parallel planes
s about parallel
aareplane. all
el and a Coplanar
perpendicular
by
a to
third
points
points
in one plane.
3. Identify the angles formed when two lines are cut by a transversal.
4. State and apply the postulates and theorems about parallel lines.
5. State and apply the theorems about a parallel and a perpendicular to a
given line through a point outside the line.
or skew.
planar.
2-l
Coplanar points
Definitions
Noncoplanar points
Two lines that do not intersect are either parallel or skew.
Some expressions commonly used to describe relationships between
and are coplanar.
Parallel
hnes)
do notInintersect
points, lines,lines
and (ll
planes
follow
these expressions, intersects means "meets"
andfigures
are notis coplanar.
not intersect
"cuts."lines
or Skew
Thedo
intersectioil
of two
the set of points that are in both
figures. Dashes in the diagrams indicate parts hidden from view in figures in
space.
nd
k are skew
A
lines.
re also called parallel. For
Aisinl.
and AB ll CD.
are parallel lines.
andmthe
plane IX,and
part of
Aisonl.
/ isA.parallel to m Qll ml.
definitions.
d the following
I contains
as
I
and h intersect in O.
O is the intersection of / and h.
j and k are skew lines.
/ passes through l.
2
/
sect.
Segments and rays contained in parallel lines are also called parallel. For
Chapter I
example, in the figure at the left above, AB ll CD and AB ll CD.
E
Thinking of the top of the box pictured below as part of plane X, and the
bottom as part of plane Y, may help you understand the following definitions.
!##" !##" !##" !##" !##"
GF intersect.
|| AD || CB ; HE
Parallel planes 1ll planes)HE
do|| not
is next
parallel to plane y (X ll y).
Plane
on X
the
is given
anes
proof.
Perpendicular Lines
!##" !##"
!##"
skew with FB ; HE skew with GC ;
Lines that intersect at a 90
A line and a plane are parallel
if they do not intersect.
Lines ,and
Parallel
/ 55Fd!##
For example
frllPlanes
v ana
ll"y. !##" !##" !##"
HE ⊥ EA HE ⊥ EF
Arso, iB llx and frlt x.
degree
E angle.
Our first theorem about parallel lines and planes is given on the next
page. Notice the importance of delinitions in the proof.
l-2 Segments, Rays, and Distance
I
In the diagram, point B is between points and C.
SEGMENT
The term is undeflrned, but note lhat B lies on ,i?.
l-2 Segments, Rays, and Distance
AC,
l-2all points
pointsDistance
Segments,
and
In
B lt Rays, of
I A
that
I
Segment
consists
the diagram,d,enoted,
point is between
points
and C. and. C
and
are between
and C. Points A and C
The term is undeflrned, but note lhat B lies on ,i?.
In
the
diagram,
point
B
is
between
points I and C.
are called the endpoints of AC.
Segment
The termAC,
d,enoted, ltbutconsists
is undeflrned,
of Bpoints
note lhat
A ,i?.
and. C
lies on
Rlay
AC,
denoted
7Z,between
consists Iofand
ACC.and,all
and all points that are
Pointsother
A andpoints
C
Segment AC, d,enoted, lt
RAY
consists of points A and. C
P
such
that
Cis
betweenl
and
P.
The
endpoint
is
are
the that
endpoints
of AC. I and
and called
all points
are between
A
A
AC BC
H
BC
AC
AC
H
AC AC
H
otrt
C. Points A and C
A, the point named first.
BC
A
CP
AC AC
called
the endpoints
of AC.
AC, denoted
7Z, consists
of AC and,all other points
P
such that
betweenl
andofP.AC
The endpoint
otrt
is
Rlay
7Z, consists
points
SR AC,
and denoted
are called
SZCis
opposite
raysand,all
if
S isother
A,
the point
named first. and P. The endpoint otrt is
P
such
that
between
R Cis
andbetweenl
Z
A, the point named first.
SR and SZ are called opposite rays if S is
between
R and
SR and SZ
are Z
called opposite rays if S is
between
R
and
Z clock shown suggest opThe hands of the
posite rays.
are
Rlay
AC
CP
CP
AC
The hands of the clock shown suggest opThe hands
posite
rays. of the clock shown suggest opposite rays.
On a number line every point is paired with a number and every number is
paired with a point. Below, point .I is paired with
the coordinate of J.
-3,
On a number
line every
is paired
with proof
a number
and every
number isobvious.
POSTULATE:
A rule
this ispoint
accepted
without
because
it’ s something
On
a
number
line
every
point
is
paired
with
a
number
and
every
number
paired with a point. Below, point .I is paired
with
the coordinate
of J.is
K
M
paired with a point. Below, point .I is paired with -3, the coordinate of J.
-3,proven to be true with the facts already
THEOREM: A rule that is proven to be true (or can be
0
I
2
3
-4-3-2-t
K These rules are
M 4strong as foundations.
established in our system
of knowledge).
K
M
00 the
I distance
22 33 between
4
The length of i,-4-3-2-t
denotedby JL,is
point -Iand point
4
-4-3-2-t
PARALLEL LINE POSTULATE
: Through
aIpoint
not a given
line, there
is a line parallel to the
Z. You can find the length of a segment on a number line by subtracting
the
givenThe
linelength of i, denotedby JL,is the distance between point -Iand point
coordinates
of of
itsi,endpoints:
The length
denotedby JL,is the distance between point -Iand point
Z.
can find the length of
on a number line by subtracting the
Z. You
You can find the length of aa segment
segment on a number line by subtracting the
coordinates of its endpoints: MJ =4-(-3):7
coordinates of its endpoints:
r I I I I I I I I I'
rr I I I I I I I I I'
I I I I I I I I I'
ALWAYS,
SOMETIMES, NEVER
Notice that
since a length
MJ
must be a positive number, you subtract the
MJ =4-(-3):7
=4-(-3):7
lesser coordinate from the greater one. Actually, the distance between two
Notice
that
since
be
aa positive
number,
you
the
Notice
A
scenario
is
ALWAYS
true ifmust
it only
examples
and
no
counter
that
since aa length
length
must
beproduces
positive
number,
yousubtract
subtract
theexamples.
points
is
the
absolute
value
of
the
difference
of their
coordinates.
when
you
lesser
coordinate
from
the
greater
one.
Actually,
the
between
two
lesser coordinate from the greater one. Actually, thedistance
distance
between
two
use absolute
value, the
order
in which
you subtract
coordinates doesn't
matter.
points
is
value
of
difference
of
points
is the
theisabsolute
absolute
of the
theonly
difference
of their
their coordinates.
coordinates.
when
you
A scenario
NEVER value
true if
it
produces
counter
examples when
and
noyou
examples.
use
the
order
in
which
you
subtract
coordinates
doesn't
use absolute
absolute value,
value, the order in which you subtract coordinates doesn't matter.
matter.
J
K
L
M
PA
I |
JI I I Kt0 IllL IM
Mt4 PA
PA
'l-4-3-2-1
3
xy
2
I I I t ll II tt II ||
A scenario is SOMETIMES true if it produces both examples and counter
examples.
>
J
K
L
'l-4-3-2-1
-4-3-2-1
00 II 22 33
JL =l-3 - 2l = l-51 =
JL =l-3
JL
=l-3 -- 2l2lor== l-51
l-51 == 55
JL=12
or
or
44
5
=l5l =5
-(-3)l=l5l
JL=12 -(-3)l
JL=12
=5
-(-3)l =l5l =5
>>
xy
xy
PQ=lx-yl
PQ=lx-yl
PQ=lx-yl
or
or
PQor=ly
PQ =ly
PQ
=ly
xl
xlxl