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1.2 Points, Lines and1.2 Points, Lines and
PlanesPlanesGeometryGeometry
Mrs. SpitzMrs. SpitzFall 2005Fall 2005
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Objecties!"ssi#nment$Objecties!"ssi#nment$
%nderstand and &se t'e basic%nderstand and &se t'e basic
&nde(ned terms and de(ned terms&nde(ned terms and de(ned terms
o) #eometry.o) #eometry.
S*etc' t'e intersections o) lines andS*etc' t'e intersections o) lines and
planes.planes.
"ssi#nment$ pp. 1+15 -12 all"ssi#nment$ pp. 1+15 -12 all
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%sin# %nde(ned terms and%sin# %nde(ned terms and
de(nitionde(nition" defnition&ses *no/n /ords to
describe a ne/ /ord. n #eometry,some /ords s&c' as point, line andplane are undefnedterms or not)ormally de(ned.
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ample /'at yo&r note pa#eample /'at yo&r note pa#e
s'o&ld loo* li*e no/s'o&ld loo* li*e no/
Name____________________________
Date___________________Period_____
Class_____________________________
1.2 Points,
Lines, &
Planes
Objectives:
Assignment:sing
!nde"ined
terms and
de"initions
1.1. nderstand and !se t#e basic !nde"inednderstand and !se t#e basic !nde"ined
terms and de"ined terms o" geometr$.terms and de"ined terms o" geometr$.
2.2. %etc# t#e intersections o" lines and%etc# t#e intersections o" lines and
'lanes.'lanes.
Assignment: ''. 1()1* +1)2 allAssignment: ''. 1()1* +1)2 all
A de"inition !ses no-n -ords to describe a
ne- -ord. n geometr$, some -ords s!c#
as 'oint, line and 'lane are !nde"ined
terms or not "ormall$ de"ined.
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%sin# %nde(ned terms and%sin# %nde(ned terms and
de(nitionde(nition
3 "point'as nodimension. t is&s&allyrepresented by asmall dot.
A
Point "
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%sin# %nde(ned terms and%sin# %nde(ned terms and
de(nitionde(nition3 " lineetends in one
dimension. t is&s&ally representedby a strai#'t line /it't/o arro/'eads toindicate t'at t'e lineetends /it'o&t endin t/o directions. n
t'is boo*, lines areal/ays strai#'t lines.
A
B
l
Line lor "4
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%sin# %nde(ned terms and%sin# %nde(ned terms and
de(nitionde(nition"planeetends in t/o
dimensions. t is&s&ally represented bya s'ape t'at loo*s li*e
a tabletop or /all. o&m&st ima#ine t'at t'eplane etends /it'o&tend een t'o' t'e
dra/in# o) a planeappears to 'aeed#es.
A
B
C
M
Plane M or plane"46
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" )e/ basic concepts . . ." )e/ basic concepts . . .
M&st be commonly &nderstoodM&st be commonly &nderstood
/it'o&t bein# de(ned. One s&c'/it'o&t bein# de(ned. One s&c'
concept is t'e idea t'at a point liesconcept is t'e idea t'at a point lies
on a line or a plane.on a line or a plane.
Collinear pointsCollinearpointsare points t'at lie onare points t'at lie on
t'e same line.t'e same line.
Coplanar pointsCoplanar pointsare points t'at lie onare points t'at lie on
t'e same plane.t'e same plane.
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. 1$ 7amin# 6ollinear and. 1$ 7amin# 6ollinear and
6oplanar Points6oplanar Pointsa.a. 7ame t'ree7ame t'ree
points t'at arepoints t'at are
collinearcollinear
Sol&tion$Sol&tion$
8, and F lie on t'e8, and F lie on t'e
same line, sosame line, sot'ey are collinear.t'ey are collinear.
G
D E F
H
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. 1$ 7amin# 6ollinear and. 1$ 7amin# 6ollinear and
6oplanar Points6oplanar Pointsb.b. 7ame )o&r points7ame )o&r points
t'at are coplanar.t'at are coplanar.
Sol&tion$Sol&tion$8, , F, and G lie on t'e8, , F, and G lie on t'e
same plane, so t'eysame plane, so t'ey
are coplanar. "lso 8,are coplanar. "lso 8,
, F, and 9 are, F, and 9 are
coplanar: alt'o',coplanar: alt'o',
t'e plane containin#t'e plane containin#
t'em is not dra/n.t'em is not dra/n.
G
D E F
H
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. 1$ 7amin# 6ollinear and. 1$ 7amin# 6ollinear and
6oplanar Points6oplanar Pointsc.c. 7ame t'ree points7ame t'ree points
t'at are nott'at are notcollinear.collinear.
Sol&tion$Sol&tion$
;'ere are many;'ere are manycorrect ans/ers.correct ans/ers.
For instance,For instance,points 9, , and Gpoints 9, , and Gdo not lie on t'edo not lie on t'esame line.same line.
G
D E F
H
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More . . .More . . .
"not'er &nde(ned"not'er &nde(ned
concept in #eometry isconcept in #eometry is
t'e idea t'at a pointt'e idea t'at a point
on a line is bet/eenon a line is bet/een
t/o ot'er points ont/o ot'er points ont'e line. o& can &set'e line. o& can &se
t'is idea to de(net'is idea to de(ne
ot'er important termsot'er important terms
in #eometry.in #eometry. 6onsider t'e line "46onsider t'e line "4
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More . . .More . . .
;'e ray "4;'e ray "4
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More . . .More . . .
7ote t'at "4 is t'e7ote t'at "4 is t'e
same as 4" and "4same as 4" and "4
is t'e same as 4".is t'e same as 4".
9o/eer, "4 and9o/eer, "4 and4" are not t'e4" are not t'e
same. ;'ey 'aesame. ;'ey 'ae
di?erent initialdi?erent initial
points and etendpoints and etendin di?erentin di?erent
directions.directions.
l
Line lor "4
A
A
B
B
>ay 4"
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More . . .More . . .
) 6 is bet/een " and) 6 is bet/een " and4, t'en 6" and 64 are4, t'en 6" and 64 are
opposite rays.opposite rays.
Li*e points, se#mentsLi*e points, se#ments
and rays are collinearand rays are collinear
i) t'ey lie on t'e samei) t'ey lie on t'e same
line. So, any t/oline. So, any t/o
opposite rays areopposite rays arecollinear. Se#ments,collinear. Se#ments,
rays and lines arerays and lines are
coplanar i) t'ey lie oncoplanar i) t'ey lie on
t'e same plane.t'e same plane.
l
Line lor "4
A
C
B
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. 2$ 8ra/in# lines, se#ments. 2$ 8ra/in# lines, se#ments
and raysand rays 8ra/ t'ree noncollinear points @, A,8ra/ t'ree noncollinear points @, A,
and L. ;'en dra/ @A, AL and [email protected] L. ;'en dra/ @A, AL and L@.
J
K
L
8ra/ @, A and L
;'en dra/ @A
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. 2$ 8ra/in# lines, se#ments. 2$ 8ra/in# lines, se#ments
and raysand rays 8ra/ t'ree noncollinear points @, A,8ra/ t'ree noncollinear points @, A,
and L. ;'en dra/ @A, AL and [email protected] L. ;'en dra/ @A, AL and L@.
J
K
L
8ra/ AL
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. 2$ 8ra/in# lines, se#ments. 2$ 8ra/in# lines, se#ments
and raysand rays 8ra/ t'ree noncollinear points @, A,8ra/ t'ree noncollinear points @, A,
and L. ;'en dra/ @A, AL and [email protected] L. ;'en dra/ @A, AL and L@.
J
K
L
8ra/ L@
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. +$ 8ra/in# Opposite. +$ 8ra/in# Opposite
>ays>ays 8ra/ t/o lines. Label8ra/ t/o lines. Labelpoints on t'e lines andpoints on t'e lines and
name t/o pairs o)name t/o pairs o)
opposite rays.opposite rays.
Sol&tion$ Points M, 7,Sol&tion$ Points M, 7,
and B are collinearand B are collinear
and B is bet/een Mand B is bet/een M
and 7. So BM and B7and 7. So BM and B7
are opposite rays.are opposite rays.
P
MQ
N
X
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. +$ 8ra/in# Opposite. +$ 8ra/in# Opposite
>ays>ays 8ra/ t/o lines. Label8ra/ t/o lines. Labelpoints on t'e lines andpoints on t'e lines and
name t/o pairs o)name t/o pairs o)
opposite rays.opposite rays.
Sol&tion$ Points P, C,Sol&tion$ Points P, C,
and B are collinearand B are collinear
and B is bet/een Pand B is bet/een P
and C. So BP and BCand C. So BP and BC
are opposite rays.are opposite rays.
P
MQ
N
X
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Goal 2$ S*etc'in#Goal 2$ S*etc'in#
ntersections o) Lines andntersections o) Lines and
PlanesPlanes;/o or more #eometric intersect i) t'ey;/o or more #eometric intersect i) t'ey'ae one or more points in common.'ae one or more points in common.
;'e intersection o) t'e (#&res is t'e set;'e intersection o) t'e (#&res is t'e set
o) points t'e (#&res 'ae in common.o) points t'e (#&res 'ae in common. "ctiity$ p. 12 D Modelin# intersections."ctiity$ p. 12 D Modelin# intersections.
D %se t/o inde cards. Label t'em as s'o/n%se t/o inde cards. Label t'em as s'o/n
and c&t slots alon# eac' card.and c&t slots alon# eac' card. 6omplete t'e eercise and place t'e completed6omplete t'e eercise and place t'e completed
E&estions in yo&r lab section labelin# t'is LabE&estions in yo&r lab section labelin# t'is Lab
1.2.1.2.
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. $ S*etc'in#. $ S*etc'in#
intersectionsintersections S*etc' t'e (#&reS*etc' t'e (#&redescribed.described.
a.a. " line t'at intersects" line t'at intersects
a plane in one pointa plane in one point 8ra/ a plane and a8ra/ a plane and aline.line.
mp'asize t'e pointmp'asize t'e point
/'ere t'ey meet./'ere t'ey meet. 8as'es indicate8as'es indicate
/'ere t'e line is/'ere t'e line is'idden by t'e plane'idden by t'e plane
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. $ S*etc'in#. $ S*etc'in#
intersectionsintersections S*etc' t'e (#&reS*etc' t'e (#&redescribed.described.
b.b. ;/o planes t'at;/o planes t'at
intersect in a lineintersect in a line
8ra/ t/o planes.8ra/ t/o planes.
mp'asize t'e linemp'asize t'e line
/'ere t'ey meet./'ere t'ey meet.
8as'es indicate8as'es indicate/'ere one plane is/'ere one plane is
'idden by t'e ot'er'idden by t'e ot'er
plane.plane.
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6omin# &p soon . . .6omin# &p soon . . .
C&iz a)ter 1.+C&iz a)ter 1.+
Practice E&iz on pa#e 25Practice E&iz on pa#e 25