äèPlease answer ê followïg questions about poïts, lïes,
å planes.
â
èèè The undefïed terms ï geometry are poït, lïe, å plane.
éS We are goïg ë develop geometry just as Euclid did by begïnïg
with undefïed terms, makïg defïitions from ê undefïed terms, sta-
tïg axioms usïg undefïed terms å defïitions, å fïally provïg
êorems by usïg ê same undefïed terms, defïitions, axioms, å
possibly oêr proven êorems.èThis is actually ê way that all maê-
matical systems are developed.è
è For example, you begï ê study ç algebra with ê undefïed termsèet å element.èThese undefïed terms
set å element.èThese undefïed terms are used ë make defïitions ïèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèèare used ë make defïitions ï algebra.èThe defïition ç equal sets
algebra.èThe defïition ç equal sets is "two sets A å B are equal if
å only if êy have ê same elements."èYou can see that ê first
defïition ï algebra for equal sets uses ê undefïed terms set å
element.
è The undefïed terms ï geometry are poït, lïe, å plane.èPoïts
have no area or width, lïes extend ïdefïitely far ï both directions,
å a plane extends ïdefïitely far ï all directions.
èèAn example ç a poït is ê place where a surveyor places ê sharp
end ç a stake ï ê ground ë locate a corner ç a parcel ç lå.èIf
you stretch a wire as tightly as possible from one corner stake ë an-
oêr, you can thïk ç this wire as a lïe segment that represents ê
boundary ç ê lot.èThis wire extended ïdefïitely far ï both direc-
tions is a lïe.èIf you thïk ç ê floor ç a house, a wall, or one
side ç ê roç ç a house extended ï all directions ên you have ex-
amples ç planes.
è We can now use êse three undefïed terms (poït, lïe, å plane)
ë state some defïitions.
Defïition 1.2.1èSPACE:èThe set ç all poïts.
Defïition 1.2.2èGEOMETRIC FIGURE:èAny collection ç poïts, lïes, or
planes ï space.
Defïition 1.2.3èCOLLINEAR:èA collection ç poïts is collïear if
êre is a lïe contaïïg all ç ê poïts.
Defïition 1.2.4èCOPLANAR:èA collection ç poïts is coplanar if êre
is a plane contaïïg all ç ê poïts.
Defïition 1.2.5èCOORDINATE:èIf poït A is located at a certaï real
number on a scaled number lïe, ên that real number is ê coordïate
ç ê poït A.
Defïition 1.2.6èDISTANCE:èIf two poïts, A å B, are located on a
number lïe, ên ê positive difference ç êir coordïates is ê
distance between A å B.
Defïition 1.2.7èLINE SEGMENT:èIf poïts A å B are on lïe m, ên
ê lïe segment determïed by A å B is ê endpoït A å B combïed
with all ç ê poïts between A å B.
Defïition 1.2.8èENDPOINTS:èThe endpoïts ç ê lïe segment determ-
ïed by A å B are ê poïts A å B.
Defïition 1.2.9èLENGTH:èThe length ç a lïe segment determïed by A
å B is ê distance from A å B.
Defïition 1.2.10èCONGRUENT:èTwo lïe segments are congruent if êy
have ê same length.
Defïition 1.2.11èRAY:èA ray is a half-lïe.
è You can see that we have made eleven defïitions from ê three unde-
fïed terms (poït, lïe, å plane).èEach ç êse defïitions usesèèèèè
one or more ç ê three undefïed terms.èLater, we will use êse un-
defïed terms å defïitions ë state some axioms.
è It is goïg ë be convenient for us ë use symbols ë represent
poïts, lïes, lïe segments, congruent lïe segments, å rays.èThe
notations or symbols used for êse geometric figures are given below.
èèèèèè GEOMETRIC FIGUREèèèèèèèèèèèSYMBOL
èèèè PoïtèèèèèèèèèèèèèCapital letters like A,B,C...èè
èèèè Lïeèèèèèèèèèèèèè ░╡
èèèè Lïe Segmentèèèèèèèèè ▒┤
èèèè Congruent Lïe Segmentsèèèè▒┤ ╧ ╖║è
èèèè Rayèèèèèèèèèèèèèè▒╡
1
èèèèèèèèGive ê geometric defïition ç space.
èèèèèA)èA collection ç poïts all ï ê same plane.
èèèèèB)èAny collection ç poïts, lïes, or planes.
èèèèèC)èThe set ç all poïts.
èèèèèD)èA collection ç poïts all ï ê same lïe.
ü
èèèèèèèèèSpace is ê set ç all poïts.
Ç C
2
èèèèèèèèGive ê defïition ç a geometric figure.
èèèèèA)èA collection ç poïts all ï ê same plane.
èèèèèB)èAny collection ç poïts, lïes, or planes.
èèèèèC)èThe set ç all poïts.
èèèèèD)èA collection ç poïts all ï ê same lïe.
ü
è A geometric figure is any collection ç poïts, lïes, or planes.
Ç B
3
èèèèèèèèGive ê defïition ç collïear poïts.
èèèèèèA)èA collection ç poïts all ï ê same plane.
èèèèèèB)èAny collection ç poïts, lïes, or planes.
èèèèèèC)èThe set ç all poïts.
èèèèèèD)èA collection ç poïts all ï ê same lïe.
ü
èèèèèèè A collection ç poïts all ï ê same lïe.
Ç D
4
èèèèèèèèèèGive ê defïition ç coplanar poïts.
èèèèèèA)èA collection ç poïts all ï ê same plane.
èèèèèèB)èAny collection ç poïts, lïes, or planes.
èèèèèèC)èThe set ç all poïts.
èèèèèèD)èA collection ç poïts all ï ê same lïe.
ü
èèèèèèA collection ç poïts all ï ê same plane.
Ç A
5èèèèè Give ê coordïate ç poït C.
èèèèèèèèèè
èèèèèèèèèèèèCèèè Dèèèèè E
èèèèèèèèè╓╓£╓╓╓£╓╓╓£╓╓╓£╓╓╓£╓╓╓£╓╓╓£╓╓╓£╓╓¥
èèèèèèèèè -3è-2è-1è 0è 1è 2è 3è 4è
èèèèèèèèèèèèèè
èèèèè A)è3èèèè B)è-2èèèè C)è0èèèè D)èNoneèèèè
ü
èèèèèèèèèèThe coordïate ç poït C is -2.
Ç B
6èèèèèè Fïd ê distance from D ë E.èè
èèèèèèèèèè
èèèèèèèèèèèè DèèèèèèèèèèE
èèèèèèèèèèè ╓╓£╓╓£╓╓£╓╓£╓╓£╓╓£╓╓£╓╓£╓¥èèèèè
èèèèèèèèèèèè-3 -2 -1è0è1è2è3è4è
èèèèèè
èèèèè A)è7èèèè B)è1èèèèèC)è3èèèè D)èNoneèèèè
ü
èèèèThe positive difference ç ê coordïates ç E å D is
èèèèèèèèèèèèèèèè 4-(-3)
èèèèèèèèèèèèèèè = 4 + 3èèèè
èèèèèèèèèèèèèèè = 7èè
Ç A
7èèèè Fïd ê length ç ê lïe segment ▒┤.èè
èèèèèèèèèè
èèèèèèèèèèèèèèAèèèèèèèBèèèèèèèèèèèèèèèè
èèèèèèèèèèèè╓£╓╓ÿ¢¢Ü¢¢Ü¢¢Ü¢¢Ü¢¢ÿ╓╓£╓╓¥èèèèè
èèèèèèèèèèèè-3 -2 -1è0è1è2è3è4è
èèèèèè
èèèèè A)è3èèèè B)è1èèèèèC)è5èèèè D)èNoneèèèè
ü
èè The length ç ê lïe segment ▒┤ is ê distance from A ë B.
èèèèèèèèèèèèè
èèèèèèèèèèèèèèèè 3-(-2)
èèèèèèèèèèèèèèè = 3 + 2èèèè
èèèèèèèèèèèèèèè = 5èè
Ç C
8èèGive a correct statement about lïe segments ╞└ å ╠├.èè
