T
r
a
ns
f
orm
a
t
i
ons
,
C
on
g
ru
e
n
ce
&
Si
m
ila
r
i
ty
—
NC Ma
t
h
2
CDM
S
tu
d
y
G
u
ide
1
.
TRANSFORMATIONS
A
tr
a
ns
f
orm
a
t
i
on
mov
e
s
a
fi
g
ur
e
(
pr
ei
m
age
)
to
a
n
e
w
pos
i
t
i
on
(
i
m
age
).
P
o
i
nts
a
r
e labeled
w
i
t
h
pr
i
m
e
s
:
A
→
A
'.
T
r
a
ns
la
t
i
on
(
s
lide
)
E
v
e
ry
po
i
nt
mov
e
s
t
he
s
a
m
e di
st
a
n
ce i
n
t
he
s
a
m
e di
r
ec
t
i
on
.
N
o
rot
a
t
i
on
,
no
fl
i
p
,
no
r
e
s
i
z
e
.
R
u
le f
orm
:
(
x
,
y
)
→
(
x
+
a
,
y
+
b
)
a
=
h
or
i
zont
al
s
hif
t
(
r
igh
t
if
pos
i
t
i
v
e
,
lef
t
if
n
ega
t
i
v
e
)
b
=
v
e
rt
ical
s
hif
t
(
up
if
pos
i
t
i
v
e
,
d
own
if
n
ega
t
i
v
e
)
E
x
a
mp
le
:
T
r
a
ns
la
t
e
(
3
,
2
)
us
i
n
g
(
x
,
y
)
→
(
x
-
4
,
y
+
5
)
→
(-
1
,
7
)
T
r
a
ns
la
t
i
ons
a
r
e
r
igid
(
pr
e
s
e
rv
e
s
i
z
e a
n
d
s
ha
p
e
)
—
t
he i
m
age i
s
c
on
g
ru
e
nt
to
t
he
pr
ei
m
age
.
Re
fl
ec
t
i
on
(
fl
i
p
)
Fli
ps
a
fi
g
ur
e
ov
e
r
a li
n
e
(
t
he li
n
e
o
f
r
e
fl
ec
t
i
on
),
c
r
ea
t
i
n
g a
m
i
rror
i
m
age
.
C
ommon
ru
le
s
:
Re
fl
ec
t
ov
e
r
R
u
le
x
-
a
x
i
s
(
x
,
y
)
→
(
x
,
-
y
)
y
-
a
x
i
s
(
x
,
y
)
→
(-
x
,
y
)
li
n
e
y
=
x
(
x
,
y
)
→
(
y
,
x
)
li
n
e
y
=
-
x
(
x
,
y
)
→
(-
y
,
-
x
)
Re
fl
ec
t
i
ons
a
r
e
r
igid b
ut
r
e
v
e
rs
e
or
ie
nt
a
t
i
on
(
cl
o
ck
w
i
s
e bec
om
e
s
c
ount
e
r
cl
o
ck
w
i
s
e
).
R
ot
a
t
i
on
(
turn
)
T
urns
a
fi
g
ur
e a
roun
d a
fix
ed
po
i
nt
(
ce
nt
e
r
o
f
rot
a
t
i
on
)
b
y
a gi
v
e
n
a
n
gle
,
usu
all
y
c
ount
e
r
cl
o
ck
w
i
s
e
un
le
ss
st
a
t
ed
ot
he
rw
i
s
e
.
R
u
le
s
ab
out
t
he
or
igi
n
:
R
ot
a
t
i
on
R
u
le
90
°
CCW
(
x
,
y
)
→
(-
y
,
x
)
180
° (
x
,
y
)
→
(-
x
,
-
y
)
270
°
CCW
(=
90
°
CW
) (
x
,
y
)
→
(
y
,
-
x
)
R
ot
a
t
i
ons
a
r
e
r
igid a
n
d
pr
e
s
e
rv
e
or
ie
nt
a
t
i
on
.
C
ompos
i
t
i
on
o
f T
r
a
ns
f
orm
a
t
i
ons
T
wo
or
mor
e
tr
a
ns
f
orm
a
t
i
ons
a
pp
lied i
n
s
e
qu
e
n
ce
.
O
r
de
r
m
a
tt
e
rs
—
d
o
t
he
tr
a
ns
f
orm
a
t
i
ons
i
n
t
he
or
de
r
li
st
ed
,
a
pp
l
y
i
n
g
t
he
r
e
su
l
t
o
f
t
he
first
to
t
he
s
ec
on
d
.
E
x
a
mp
le
:
Re
fl
ec
t
ov
e
r
t
he
x
-
a
x
i
s
,
t
he
n
tr
a
ns
la
t
e
(
x
,
y
)
→
(
x
+
2
,
y
)
1
.
(
3
,
4
)
→
r
e
fl
ec
t
ov
e
r
x
-
a
x
i
s
→
(
3
,
-
4
)
2
.
(
3
,
-
4
)
→
tr
a
ns
la
t
e
→
(
5
,
-
4
)
A c
ompos
i
t
i
on
o
f
two
r
e
fl
ec
t
i
ons
ov
e
r
p
a
r
allel li
n
e
s
=
a
tr
a
ns
la
t
i
on
.
A c
ompos
i
t
i
on
o
f
two
r
e
fl
ec
t
i
ons
ov
e
r
i
nt
e
rs
ec
t
i
n
g li
n
e
s
=
a
rot
a
t
i
on
.
2
.
CONGRUENCE
(
T
r
ia
n
gle C
on
g
ru
e
n
ce Sh
ort
c
uts
)
T
wo
tr
ia
n
gle
s
a
r
e c
on
g
ru
e
nt
if
t
hei
r
c
orr
e
spon
di
n
g
s
ide
s
a
n
d a
n
gle
s
a
r
e all e
qu
al
.
Y
ou
d
on
'
t
n
eed
to
check all
6
—
t
he
s
e
s
h
ort
c
uts
prov
e c
on
g
ru
e
n
ce
w
i
t
h le
ss
i
n
f
o
.
Sh
ort
c
ut
Mea
ns
N
ot
e
s
SSS
All
3
s
ide
s
m
a
t
ch N
o
a
n
gle
s
n
eeded
SAS
2
s
ide
s
+
t
he i
n
cl
u
ded a
n
gle
(
a
n
gle be
tw
ee
n
t
he
m
)
m
a
t
ch
A
n
gle MUST be be
tw
ee
n
t
he
two
s
ide
s
ASA
2
a
n
gle
s
+
t
he i
n
cl
u
ded
s
ide
(
s
ide be
tw
ee
n
t
he
m
)
m
a
t
ch
Side MUST be be
tw
ee
n
t
he
two
a
n
gle
s
AAS
2
a
n
gle
s
+
a
non
-
i
n
cl
u
ded
s
ide
m
a
t
ch Side i
s
NOT be
tw
ee
n
t
he a
n
gle
s
HL
H
ypot
e
nus
e
+
on
e Leg
m
a
t
ch
Righ
t
tr
ia
n
gle
s
on
l
y
NOT
v
alid
:
SSA
("
a
n
gle
-
s
ide
-
s
ide
")
a
n
d AAA d
o
NOT
prov
e c
on
g
ru
e
n
ce
.
AAA
on
l
y
prov
e
s
s
i
m
ila
r
i
ty
.
Q
u
ick check
pro
ce
ss
:
m
a
r
k
w
ha
t
'
s
gi
v
e
n
/
s
h
own
i
n
a diag
r
a
m
(
t
ick
m
a
r
k
s
f
or
s
ide
s
,
a
r
c
s
f
or
a
n
gle
s
),
ide
nt
if
y
w
hich
s
ide
s
/
a
n
gle
s
a
r
e i
n
cl
u
ded
vs
.
not
,
m
a
t
ch
to
a
s
h
ort
c
ut
.
3
.
PROOFS
CPCTC
—
C
orr
e
spon
di
n
g Pa
rts
o
f C
on
g
ru
e
nt
T
r
ia
n
gle
s
a
r
e C
on
g
ru
e
nt
O
n
ce
you
'
v
e
prov
e
n
two
tr
ia
n
gle
s
c
on
g
ru
e
nt
(
v
ia SSS
,
SAS
,
ASA
,
AAS
,
or
HL
),
CPCTC le
ts
you
c
on
cl
u
de
t
ha
t
ANY
ot
he
r
c
orr
e
spon
di
n
g
p
a
rts
(
a
s
ide
or
a
n
gle
not
y
e
t
us
ed
)
a
r
e al
so
c
on
g
ru
e
nt
.
T
yp
ical
stru
c
tur
e
:
1
.
S
t
a
t
e gi
v
e
n
i
n
f
o
2
.
P
rov
e
tr
ia
n
gle
s
c
on
g
ru
e
nt
(
n
a
m
e
t
he
s
h
ort
c
ut
)
3
.
S
t
a
t
e c
on
cl
us
i
on
us
i
n
g CPCTC
(
e
.
g
.,
"
∠
B
≅
∠
E b
y
CPCTC
")
C
orr
e
spon
di
n
g Pa
rts
Whe
n
two
tr
ia
n
gle
s
a
r
e c
on
g
ru
e
nt
,
t
he
y
'
r
e
n
a
m
ed
so
t
ha
t
c
orr
e
spon
di
n
g
v
e
rt
ice
s
li
n
e
up
i
n
or
de
r
.
△
ABC
≅
△
DEF
m
ea
ns
:
A
↔
D
,
B
↔
E
,
C
↔
F
AB
≅
DE
,
BC
≅
EF
,
AC
≅
DF
∠
A
≅
∠
D
,
∠
B
≅
∠
E
,
∠
C
≅
∠
F
O
r
de
r
i
n
t
he c
on
g
ru
e
n
ce
st
a
t
e
m
e
nt
t
ell
s
you
t
he c
orr
e
spon
de
n
ce
—
d
on
'
t
a
ssum
e
,
r
ead
t
he
le
tt
e
rs
.
C
oor
di
n
a
t
e P
roo
f
s
U
s
e c
oor
di
n
a
t
e
s
a
n
d f
ormu
la
s
(
not
protr
ac
tors
/
ru
le
rs
)
to
prov
e c
on
g
ru
e
n
ce
,
s
i
m
ila
r
i
ty
,
or
prop
e
rt
ie
s
o
f
s
ha
p
e
s
.
T
oo
l
s
you
'
ll
n
eed
:
Di
st
a
n
ce f
ormu
la
:
d
=
√
[(
x₂
-
x₁
)
²
+
(
y₂
-
y₁
)
²
]
—
to
c
omp
a
r
e
s
ide le
n
g
t
h
s
Sl
op
e f
ormu
la
:
m
=
(
y₂
-
y₁
)/(
x₂
-
x₁
)
—
to
check
p
a
r
allel
(
e
qu
al
s
l
op
e
s
)
or
p
e
rp
e
n
dic
u
la
r
(
oppos
i
t
e
r
eci
pro
cal
s
l
op
e
s
,
e
.
g
.,
r
igh
t
a
n
gle
s
)
Mid
po
i
nt
f
ormu
la
:
((
x₁
+
x₂
)/
2
,
(
y₁
+
y₂
)/
2
)
—
to
check bi
s
ec
t
ed
s
eg
m
e
nts
End of Document
327
