Let s Practice Geometry
description
Transcript of Let s Practice Geometry
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Let's Practice...
A resource for Teachers, Students, and Parents.
By: Brent Tuller
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Ray AB, AB
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ABCABDCBD
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HMK or KMH
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AB CD
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RT=TS Bisect10=x
LP=PM Bisect5=PMso LM=10
SQ=QT Midpoint4=x
MO=ON Midpoint
9=ON9+9=18
GH=HI Midpoint x+4=2x-6-x -x 4=x-6 +6 +6 10=x
True XZ=ZY so Z is the midpoint
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AE=CE Bisect5=x
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Acute, Scalene
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a2+b2=c2
82+72=x264+49=x2
113=x2113=x2
10.6in=x
a2+b2=c2
x2+92=112x2+81=121
-81 -81 x2=40 x2=40 x=6.3in
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a2+b2=c2
82+112=142
64+121=196185=196
True 289=289
a2+b2=c2
82+152=17264+225=289289=289
False 185196
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a
2
+b
2
=c
2
a2+b2=c2 a2+b2=c2
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3-4-5
Scale factor=2
Scale Factor=2
x=2*5
x=10
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3-4-5
Scale factor=9
x=9*5x=45
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P=7+5+7+12P=31cm
P=8+6+2+18+6+24P=64cm
a2+b2=c222+42=c24+16=c2
20=c2 20=c2
4.47=c
P=4.47+5+8+9+2P=28.47cm
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heig
ht
A= bh 2
A=bh
A=bh
Area=(base)(height)
Area=(base)(height)
base
heigh
t
base
base
height
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(diagonal 1)(diagonal 2) 2
Area= (base+top)(height)
2A= (b+t)h
2
base
top
height
Area= (d1)(d2) 2A=
diagon
al1
diag
onal
2
ra
diu
s
A=r2 A=()(radius2)
A=s2 A=side2
side
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RectangleA=bhA=(15cm)(3cm)A=45cm2
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TriangleA=bh/2A=(10cm)(5.7cm)/2A=57cm2/2A=28.5cm2
a2+b2=c2
42+h2=72
16+h2=49-16 -16
h2
=33 h2=33 h=5.7
h=5.7
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A=bh 72=9h
9 9 8in=h
A= bh 2 8h 2
2*46= *2
92=8h
8 811.5=h
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B=bhB=6*5B=30
V=BhV=30*8V=240in3
V=r2h
V=(62)10V=36*10V=360V1130.97in3
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B=bhB=4*5B=20V=1/3(Bh)V=1/3(20*9)V=1/3(180)=60cm2
V=1/3r2hV=1/3(32)4V=1/39*4V=1/3(36)V=13V40.84in3
3-4-5 righttriangle scalefactor 1. 1*4=4
B=1/2bhB=1/2(3*4)B=1/2(12)B=6V=1/3(Bh)V=1/3(6*5)V=1/3(15)=5ft3
4ft
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a2+b2=c2
52+42=c2
25+16=c2
41=c2
6.4=c4.6m
5m
5m
5m 4m
4m3m
3m
4m
4.6m
3m
4cm
15cm10cm-6cm=4cm
10cm+5cm=15cm
4cm 4cm
4cm
15cm
6cm 6cm
15cm
4cm
4cm
15cm
8cm
8cm
8cm
8cm
8cm
8cm
5cm
10cm10cm
6cm
a2+b2=c2 h2+32=1002
h2+9=100 -9 -9 h2=91 h=9.54
3in
9.54in
6in 6in 6in 6in
9.54in 9.54in
6in
6in
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51+38=89m ABD=89
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x+33=90
.
x+33=90-33-33
x=57
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x+2x=180 3x=180 3 3 x=60
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x+43=180-43 -43
x=137m DEF=137
x
43
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x=138 Vertical Angle Theorem x=75 Vertical Angle Theorem
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x+45=2x+19 Vertical Angle Theorem-x -x 45=x+19 -19 -19 26=x
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m SOT=m NOEVertical Angle Theorem
138=m NOE
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m HAE=m DASVertical Angle Theorem
x+24=2x-8-x -x 24=x-8 +8 +8 32=xm DAS=2x-8m DAS=2(32)-8m DAS=64-8m DAS=56
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alternate exterior angles
vertical angles
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Alternate interior, m 1=m 2
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Alternate interior
46=x
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Alternate exterior
2x+19=x+23
-x -x x+19=23 -19-19 x=4
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x
68
Correspondingm ACB=m EFC 68=m EFC
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Linear Pairm ACH+m DCH=180(x+24)+(x-8)=180 x+24+x-8=180 2x+16=180 -16 -16
x+24x-8
2x=1642 2x=82
m ACH=x+24m ACH=82+24
m ACH=106
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x+x+x=180Triangle Sum Theorem 3x=180 3 3 x=60
2x-12=60equilateral triangle
2x-12=60 +12 +12 2x=72 2 2
x=36
x+x+30=180Triangle Sum Theorem 2x+30=180 -30 -30 2x=150 2 2 x=75
2x-27=x+53 iscosceles triangle 2x-27=x+53 -x -x x-27=53 +27+27 x=80
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Triangle Sum Theorem24+88+x=180
112+x=180-112 -112
x=68
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Triangle Sum Theorem18+94+x=180
112+x=180-112 -112 x=68 m A=68
18
94
x
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Triangle Sum Theoremx+2x+(x-16)=180x+2x+x-16=180
4x-16=180+16 +16
4x=196
4 4 x=49
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41+87+y=180
y=_________
_______+x=180
x=128
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m 1+m 2=m 3
114+38=x
152=x
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m 1+m 2=m 3 29+x=87 -29 -29 x=______
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m 1+m 2=m 3(x+3)+(x+4)=125 x+3+x+4=125 2x+7=125 -7 -7 2x=118 2 2 x=59
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m 1+m 2=m 3 m Z=x+71 (x+71)+x=6x m Z=(_____)+71
m Z=______+71 m Z=______
6x
x
x+71
48
93
x
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What could I use to find x?
B1: 53+m 1+x=180 triangle sumtheorem
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What could I use to find x?B1: x=m 7 correspondingWhat could I use to find m 7?B2: m 5=m 7 corresponding
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What could I use to find x?B1: m 2+71+x=180 triangle sumtheoremWhat could I use to find m 2?B2: m 1=m 2 vertical anglesWhat could I use to find m 1?B3: m 1+48+53=180 triangle sumtheorem
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How can I find x?B1: m 3=x alternateinterior anglesHow can I find m 3?B2: m 3+36+58=180triangle sum therorem
B2: m 3+36+58=180m 3+94=180
-94 -94 m 3=86
B1: m 3=x=18086=x
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s1+s2=s318+15>27
33>27 True!So... yes they do!
s1+s2=s387+91>456178>456 False!No triangle here!
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s1+s2=s34+7>1111>11 false!
no triangle
s1+s2=s32+5>97>9 False!no triangle
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max.=7+12=19 and the min. is 12-7=5 so 5
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max.=4+3=7min.=4-3=1
1in
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ASS
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BC EF givenC F given
AC DF given
So ABC DEF by SAS.
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Vertical Angles
Shared Line
CA BD givenCAD ADB given
AD AD shared lineSo CAD BDAby SAS.
So PQR TSR
by ASA.P T given
PR RT given PRQ SRT vertical
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False.... the triangles may not
be congruent.
False.... the triangles arecongruent, but the parts don'tcorrespond. They aren't thesame part.
True the triangles arecongruent by SSS. So,
M P because CPCTC.
TV=WY CPCTC13=x
C=m F CPCTC x+5=2x-7-x -x 5=x-7 +7 +7 12=x
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25
6x
= =
2(x)=5(6) 2x=30 2 2 x=15
57
x15= =
7(x)=5(15) 7x=75 7 7 x=10.71
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MNO~MPQ by AAAor...MNO~MPQ by AA
M M same angle MON MQP corresponding MNO MPQ corresponding
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3
4=
32
42
9
16
2
=
4
5=
43
53
64
125
3
=
16
25=
16
25
4
5=
8
27
8
27
2
3= =
3 3
3
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Corresponding Angles
x=71
Corresponding Angles
x+35=2x-21 -x -x
35=x-21+21 +21
56=x
8=2x2 24=x
x+10=2(x-6) x+10=2x-12 -x -x
10=x-12+12 +12
22=x
10=x x+9=2x+4
-x -x9=x+4
-4 -4 5=x
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If Amy goes to the store then she buys milk.
If two angles add to 180 then they are supplementary.
If x=2 then 2x=4.
p hypothesis q conclusion
p hypothesis q conclusion
p hypothesis q conclusion
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You can drive it to the store.
He has a tail.
90
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She lost her patients.
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a 45 angle
A scalene triangle Anne Hathaway(or any bruenette or red-head)
A student 30 and 40
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If an animal has a beak then it is agoose. -False, a robin has a beak.
If an animal is not a goose then itdoes not have a beak. -False, again, arobin has a beak.
If an animal does not have a beak
then it is not a goose. -True!
If a triangle has 3 equal sidesthen it is equilateral. -true
If a triangle is not equilateralthen it does not have 3 equal sides.
-true.
If a triangle does not
have 3 equal sides then it is notequilateral -true
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If a figure is a rhombus then it isa square.
If a figure is not a square then it isnot a rhombus.
If a figure is not a rhombus then it isnot a square.
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If two angles add to 90then they arecomplementary. - True
If two angles aren't complementarythen they don't add to 90. - True
If two angles don't add to 90 thenthey aren't complementary. -true
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If two lines in a plane never intersect then they are parallel.-True, the statement is bi-conditional.
If an angle is acute then it measures 28.-False, 38 is acute the statement is not bi-conditional.
Iff two lines are parallel then they never intersect.
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If a triangle has twocongruent sides then it is isoceles.
Iff a triangle is Isoceles then it has twocongruent sides.
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Statements ReasonsF1: AB DE 1. GivenF2: AC DF 2. GivenF3. BC EF 3. GivenB1. ABC DEF 4. SSS
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HO
A
H
O
A cos 63= A
H
cos 63= 17x
cos 63= 17x *x
x(cos 63)=17 cos 63 cos 63
x= 17c0s 63
x= 170.4540x37.44
H
O
A
tan 56= OA
tan 56=9x
tan 56=9x *x
x*
x(tan 56)= 9 tan 56 tan 56
x= 9tan 56
x= 91.4826
x6.07
sin 39= OH
sin 39=x27
*2727* sin 39=x27
27(sin 39)=x27(0.6293)=x 16.99x
A
H
O
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tanx=OA
89
tanx=
tanx=0.8889
x=tan-10.8889
tan-1* *tan-1
23
x=6
23
x=632
32
x=9
tanx=0.8889
tan2nd = x41.63
sinx= OH
913
sinx=
sinx=0.6923sin-1* *sin-1
x=sin-10.6923
x43.81
* *
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Tanx= 8 6
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x=6y=62
a
b=a
c=a2
a
b=a
c=a2
a
b=a
c=a2
x=4y=4
x=32
y=322y=34y=3*2y=6
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c=2a
c=2ac=2*23c=43 so..y=43
b=a3 c=2ac=2a4=2a2 22=ay=2
b=a3
a a
c=2ab=a3
a
c=2ab=a3
ab=a3b=23*3b=29 so..b=2*3b=6 so...x=6
b=a35=a33 353
=a
5 *33 *3
539
533
y= 533
x=2a
x=2533
x=1033
=a
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Rectangle
A B
CDRhombus
True
False
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n=6 and (n-2)180 so (6-2)180=4*180=720
n=6 and (n-2)180 so (6-2)180=4*180=720=120n 6 6 6
(n-2)180=2,340(n-2)180=2,340 180 180 n-2=13 +2 +2 n=15 sides
n=5(n-2)180(5-2)1803*180540
94+78+156+91+x=540 419+x=540 -419 -419
x=121
n=6(n-2)180(6-2)1804*180720
x+7+2x-31+x+24+2x+x-13+2x+11=720 9x-2=720 +2 +2
9x=722
9 9 x=80.22
n=5 (n-2)180n
(5-2)1805
108
2x+8=108 -8 -8 2x=100 2 2 x=50
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n=6(n-2)180(6-2)1804*180720
x+81+4x-10+x+35+5x-60+3x+23+x+21=720
15x+90=720 -90 -90
15x=630
15 15 x=42
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21+27+58+42+53+34+45+x=360 280+x=360 -280 -280 x=80
n=8 360 360=45
n 8
n=8360 n360
845
360n
=18360
n=18*nn* 360=18n 360=18n
18 1820=n so...20 sides
2x-15
2x-15=45 +15 +15 2x=60 2 2 x=30
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x=2cm because all radii in acircle are equal.
GI=4JK=5
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C=DC=721.99
C=2rC=29C=1856.55
mDEF+mDE=360mDEF+127=360 -127-127mDEF= 233
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m B=mAC x=35
2(m B)=mAC 2x=44 2 2 x=22
2(m B)=mAC2(75)=mAC 150=mAC
m D=mAC x=150
m D=mAB 86=mAB
x=1/2mAB x=1/2(86) x=43
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mIJK+mKLI=360 190+mKLI=360-190 -190 mKLI=170
m J=1/2mKLI x=1/2(170) x=85
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m R=1/2mQSm R=1/2(48)m R=24
m Q+m R+m S=180 x+24+65=180 x+89=180 -89 -89 x=91
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3x=6*23x=123 3 x=4
2(x+4)=1(x+10) 2x+8=x+5 -x -x x+8=10 -8 -8 x=2
AB=AC 8=x
EF=EG x+3=2x-5 -x -x 3=x-5 +5 +5 8=x
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10=x
3-4-5 right triangle
x=10
17+x=90-17 -17 x=73
134/2=mFG 67=mFG
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3-4-5 righttriangle
scale factor =2
4*2=8
x=8
m DCE+m ACD=90 35+m ACD=90 -35 -35 m ACD=55
90+48+x=180 138+x=180 -138 -138 x=42
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A (5,-6) B(-3,6) x
1y
1 x
2 y
2
x1+x
2y
1+y
2 2 2 5+(-3) -6+6 2 2
2 12 2 2 (1,6)=MP
A (5,-6) B(x2,y
2) C(4,3)
x1
y1
xm
ym
x1+x
2 2
y1+y
2 2
5+x2
2-6+y
2 2
=xm
=ym
=4 =3
5+x2
2-6+y
2 2=4 =32* *2 *22*
5+x2=8
-5 -5x
2=3
-6+y2=6
+6 +6y
2=12
(3,12)=B
A (5,7) B(-6,-5) x
1y
1 x
2 y
2
D=(x2-x
1)2+(y
2-y
1)2
D=(-6-5)2+(7-(-5))2
D=(-11)2+(12)2
D=121+144D=265D16.78
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A'
B'
C'
F'
D'
E'
H'I'
G'
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J'
L'
K'
O'
N'
M'
Q'
P'
R' Q"
P"
R"
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Let's Practice...
Answers...
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Line CD, CD
Segment EF, EFPoint G, G
Plane HAT Ray NI, NI
Ray ON, ONPoint I, I
Line ME, ME
Line WE, WE
Line KJ, KJ
Plane COW
Segment HI, HI
Point U, U Ray BO, BO
Plane OLE
Segment LM, LM
Segment XY, XY
Ray AB, AB
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ABC or CBA EFG or GFE or F
JIK or KIJ PNO or ONP
TRQ or QRT XVW or WVX
ACB or BCA
IGH, HGI or G
LKJ or JKL QPR or RPQ
UYT or TYUBGC or CGB
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NPO or OPN
RTU or UTR
VZX or XZV
ABD or DBA
EHI or IHE
KOM or MOKVYX or XYV
DCB or BCD OPJ or JPO
TSU or UST VZY or YZV
HMK or KMH
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A
B
C
D
E
F
H
I
J
G
K
L
-
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EH FG
JK LMH T
NR OR PR QR
SW UW,VW TW
AC DB,
DE EB
GL FK,
GIH FJH
QN OP,NO PQ
RSE USL
RSU ESLRS SL
RT WURTS UWV TRS WUV
AC DBDE EBAE EC
AB CD
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FG HGLM JIFGL HGJ
GT ET AT RT
AG RE
ST TQOU NPNP SQ
WR OKWO ORWKO RKO
C OA BR N
AB BC CD DE EA
GO DL,GD OL
GM FHGM LKFH LK
NO QPQ O
RT VT ST UT VRU RUV UST RST
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IK=KL Bisect7=x
OS=SQ Bisect13=x
PQ=QS Bisectx=9
FG=GH Midpoint
12=x
ST=TU Midpoint19=TU19+19=38
VW=WX Midpoint11=WX11+11=22
XY=YZ Midpoint18=YZ18+18=36
AE=CE Bisect5=x
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AB=BC Midpoint x-5=2x-12
-x -x -4=x-12 +12 +12 8=x
DH=HF Bisect x+4=2x-8
-x -x 4=x-8 +8 +8 12=x
JM=ML Bisect 2x+15=3x+4-2x -2x 15=x+4 -4 -4 11=x
NO=OP Midpoint x+4=2x-6-x -x 4=x-6 +6 +6 10=x
QR=RS Midpoint x+7=2x-10-x -x 7=x-10 +10 +10 17=x
GH=HI Midpoint 4x+11=2x+19-2x -2x 2x+11=19 -11-11 2x=8 2 2 x=4
JM=ML Bisect x+5=2x-7
-x -x 5=x-7 +7 +7 12=x
GH=HI Midpoint 3x-2=x+8 -x -x 2x-2=8 +2+2 2x=10 2 2 x=5
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Obtuse, Scalene
Acute, Scalene Acute, Isosceles
Equilateral Right, Scalene
Right, Scalene
Acute, Scalene Obtuse, Scalene
Obtuse, Isosceles
Equilateral
Acute, Scalene
Right, Isosceles
Acute, Isosceles
Acute, Scalene
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a2+b2=c272+32=x2
49+9=x2
58=x2 58=x2
7.6ft=x
a2+b2=c2102+102=x2
100+100=x2 200=x2 200=x2
14.1cm=x
a2+b2=c2x2+62=152
x2
+36=225 -36 -36 x2=189 x2=189 x=13.7mm
a2+b2=c2x2+262=382
x2+676=1444 -676 -676 x2=768 x2=768 x=27.7in
a2+b2=c2x2+902=982
x2+8100=9604 -8100 -8100 x2=1504
x2
=1504 x=38.8yd
a2+b2=c2232+x2=452529+x2=2025
-529 -529 x2=1496 x2=1496
x=38.7m
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a2+b2=c2192+442=x2
361+1936=x2 2297=x2 2297=x2
47.9m=x
a2+b2=c2542+1132=x2
2916+12769=x2 25583=x2 25685=x2
125.2in=x
a2+b2=c232+42=x29+16=x2
25=x2 25=x2
5ft=x
a2+b2=c2132+922=x2
169+8464=x2 8633=x2 8633=x2
92.9mi=x
a2+b2=c2x2+62=8.52x2+36=72.25
-36 -36 x2=36.25 x2=36.25 x=6in
a2+b2=c2x2+312=672x2+961=4489
-961 -961 x2=3528 x2=3528 x=59.4mm
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x
12in
16in
x
17in
23in
x
15m
45m x
6ft
19ft
x
55cm x
37mm
21mm79cm
a2+b2=c2162+122=x2
256+144=x2
400=x2 400=x2
20in=x
a2+b2=c2232+172=x2
529+289=x2
818=x2
818=x2
28.6in=x
a2+b2=c2792+552=x2
6241+3025=x2 9266=x2
9266=x
2
96.3cm=x
a2+b2=c2x2+152=452
x2+225=2025 -225 -225 x2=1800 x2=1800 x=42.4mm
a2+b2=c2x2+62=192
x2+36=361 -36 -36 x2=325 x2=325 x=18ft
a2+b2=c2x2+212=372
x2+441=1369 -441 -441
x
2
=928 x2=928 x=30.5mm
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False 4001225
202+212=292
400+441=841 841=841
a2+b2=c2
True 841=841
a2+b2=c2 122+162=352
144+256=1,225 400=1,225
False 145169
a2+b2=c2 82+92=132
64+81=169 145=169
152+342=482
225+1,156=2,304 1,381=2,304
a2+b2=c2
False 1,3812,304
282+452=532
784+2,025=2,809 2,809=2,809
a2+b2=c2
True 2,809=2,809
1192+1202=1692
14,161+14,400=28,561 28,561=28,561
a2+b2=c2
True 28,561=28,561
622+732=942
3,844+5,329=8,836 9,173=8,836
a2+b2=c2
False 9,1738,836
202+992=1012
400+9,801=1,0201 10,201=10,201
a2+b2=c2
True 10,201=10,201
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112+632=882
121+3,969=7,744 4,090=7,744
a2+b2=c2
False 4,0907,744
122+302=432
144+900=1,849 1,044=1,849
a2+b2=c2
False 1,0441,849
312+452=862
961+2,025=7,396 2,986=7,396
a2+b2=c2
False 2,9867,396
842+1872=2052
7,056+34,969=42,025 42,025=42,025
a2+b2=c2
True 42,025=42,025
462+652=802
2,116+4,225=6,400 6,341=6,400
a2+b2=c2
False 6,3416,400
282+322=602
784+1,024=3,600 1,808=3,600
a2+b2=c2
False 1,808,3,600
92+402=412
81+1,600=1,681 1,681=1,681
a2+b2=c2
True 1,681,=1,681
162+522=822
256+2,704=6,724 2,960=6,724
a2+b2=c2
False 2,9606,724
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3-4-5
Scale factor=4
3-4-5
Scale factor=9
5-12-13
Scale factor=2
7-24-25
Scale factor=2
8-15-17
Scale factor=3
Scale Factor=3
x=3*4x=12
3-4-5
Scale factor=2
Scale Factor=2
x=2*5
x=10
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3-4-5
Scale factor=7
3-4-5
Scale factor=3
3-4-5
Scale factor=5
3-4-5
Scale factor=6
3-4-5
Scale factor=8
5-12-13
Scale factor=11
3-4-5
Scale factor=11
8-15-17
Scale factor=2
5-12-13
Scale factor=2
5-12-13
Scale factor=3
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3-4-5
Scale factor=4x=4*4x=16
3-4-5
Scale factor=8x=8*4x=32
5-12-13
Scale factor=1x=1*13x=13
3-4-5
Scale factor=2x=2*5x=10
3-4-5
Scale factor=6x=6*3x=18
8-15-17
Scale factor=1x=1*8
x=8
5-12-13
Scale factor=2x=2*5
x=10
3-4-5
Scale factor=9
x=9*5x=45
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8-15-17
Scale factor=2x=2*8x=16
3-4-5
Scale factor=7x=7*4x=28
3-4-5
Scale factor=6x=6*4x=24
7-24-25
Scale factor=1x=1*25x=25
3-4-5
Scale factor=8x=8*3x=24
3-4-5
Scale factor=5x=5*3x=15
3-4-5
Scale factor=1x=1*4x=4
7-24-25
Scale factor=3x=3*7
x=21
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7-24-25
Scale factor=1x=1*7x=7
3-4-5
Scale factor=12x=12*5x=60
5-12-13
Scale factor=4x=4*13x=52
3-4-5
Scale factor=4x=4*5x=20
3-4-5
Scale factor=11
x=11*4x=44
3-4-5
Scale factor=6x=6*5x=30
8-15-17Scale factor=2x=2*17x=34
8-15-17
Scale factor=3x=3*8x=24
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3-4-5Scale factor=2h=2*4h=8
a2+b2=c292+h2=102
81+h2=100-81 -81 h2=19 h2=19 h=4.4
5-12-13Scale factor=1h=1*5h=5
a2+b2=c292+h2=142
81+h2=196-81 -81 h2=115 h2=115 h=10.7
3-4-5Scale factor=4h=4*3h=12
a2+b2=c2
172+h2=272289+h2=729-289 -289 h2=440 h2=440 h=21
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5-12-13Scale factor=2s=2*13s=26
8-15-17Scale factor=1h=1*15h=15
3-4-5Scale factor=5h=5*4
h=20
7-24-25Scale factor=1s=1*25
s=25
3-4-5
Scale factor=7s=7*5s=35
a2+b2=c292+82=s2
81+64=s2 145=s2 145=s2
s=12.
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11+21+11+21=64 13+19+13+19=64
38+17+39+52=14614+2+5+19+5+4+14+25=88
36+19+27=827+17+11+9=44
426+449+236=1108 7*10=70
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3-4-5 scale factor=4
a2+b2=c242+52=x2
16+25=x2 41=x2 41=x2
6.4=x
2010
10
12
paralellogram
3-4-5 scale factor=2
1937-18=19
38
(37-6)+7
6.4+9+7+13+4=39.4
6.4
19+37+7+18+12+19=112
16+20+12=4810+12+10+12=44
38+45+7+55+37+96+6+4=288
(96+4)-45
55
8-15-17 scale factor=1
8
19+23+17+8+29=96
12
19-7
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8-15-17 scale factor=2
34
34+33.1+16+14=97.1
25
7-24-25 scale factor=1
16
24-8
(12)
24/2
13
5-12-13 scale factor=1
32
32
32
square
8
10-2
(parallelogram)
a2+b2=c2a2+72=92
a2+49=81 -49 -49 a2=32 a2=32 a=5.7
4
13-9
5.7
5.7+9+4+11+10=39.7
17
8-15-17 scale factor=1
10+17+10+17=54
17
32*4=128
13
13+24+13=50
13+25+7+4+16+9+8=82
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5
3-4-5 scale factor=1
5+3+7+6+4=25
8-15-17scale factor=2
34
7-24-25 scale factor=1
25
25+27+24+20=96
a2+b2=c2a2+302=412
a2+900=1681 -900 -900 a2=781 a2=781 a=27.9
27.9
27.9+41+34+16=118.9
a2+b2=c2132+122=c2
169+144=c2 313=c2 313=c2
17.7=c
3-4-5 scale factor=3
15
16
15
15+28+15+28=86
13
17.717.7
17.7+17.7+26=61.4
a2+b2=c2
5
2
+6
2
=c
2
25+36=c2 61=c2 61=c2
7.8=c
7.8
12+25+7+19+7.8=70.8
7
5
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12
12*4=48in
14
14
11 11
11+14+11+14=50ft
27
27*4=108mm
3240
3-4-5 scale factor=8
24
32+40+24=96m
17
17
9 9
9+17+9+17=52ft
49
4+8.1+9=21.1in
a2+b2=c2a2+42=92
a2+16=81 -16 -16 a2=65 a2=65 a=8.1
8.1
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24*4=96mm
24 11
11
8.5 8.5
11+8.5+11+8.5=39in
915
3-4-5 scale factor=312
9+15+12=36ft
19
19
66
6+19+6+19=50in
10
10
8 8
8+10+8+10=36in
725
7-24-25 scale factor=1
24
7+24+25=56ft
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ParallelogramA=bhA=(5in)(4in)A=20in2
TriangleA=bh/2A=(9cm)(17cm)/2A=153cm2/2A=76.5cm2
TrapezoidA=(b+t)h/2A=(6cm+15cm)(13cm)/2A=(21cm)(13cm)/2A=273cm2/2
A=136.5cm2
ParallelogramA=bh
A=(11cm)(7cm)A=77cm2
Trapezoid
A=(b+t)h/2A=(16cm+10cm)(9cm)/2A=(26cm)(9cm)/2A=234cm2/2A=117cm2
RectangleA=bhA=(15cm)(3cm)A=45cm2
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RectangleA=bhA=(12cm)(4cm)A=48cm2
Trapezoid
A=(b+t)h/2A=(11cm+7cm)(13cm)/2A=(18cm)(13cm)/2A=234cm2/2A=117cm2
ParallelogramA=bhA=(9in)(4in)A=36in2
TriangleA=bh/2A=(13m)(24m)/2A=312m2/2A=156m2
TrapezoidA=(b+t)h/2A=(23ft+16ft)(10ft)/2A=(39ft)(10ft)/2A=390ft2/2A=195ft2
CircleA=r2
A=(8cm)2
A=64
cm
2
A=201cm2
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Circle
A=r2A=(14cm)2
A=196cm2
A=615.75cm2
Triangle
A=bh/2A=(12m)(7m)/2A=84m2/2A=42m2
SquareA=s2
A=(16ft)
2
A=256ft2
Trapezoid
A=(b+t)h/2A=(17mi+13mi)(6mi)/2A=(30mi)(6mi)/2A=180mi2/2A=90mi2
TrapezoidA=(b+t)h/2A=(31mm+27mm)(13mm)/2A=(58mm)(13mm)/2A=754mm2/2A=377mm2
RhombusA=(d
1)(d
2)/2
A=(10yd)(6yd)/2A=60yd2/2A=30yd2
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TrapezoidA=(b+t)h/2A=(21cm+13cm)(4.5cm)/2A=(34cm)(4.5cm)/2A=153cm2/2A=76.5cm2
TrapezoidA=(b+t)h/2A=(23in+15in)(7.1in)/2A=(38in)(7.1in)/2
A=269.8in
2
/2A=134.9in2
TriangleA=bh/2A=(24m)(13.4m)/2A=321.6m2/2A=160.8m2
ParallelogramA=bhA=(15in)(4in)A=60in2
h=4 3-4-5 rt. triangle
h=4
a2+b2=c2
72+h2=102
49+h
2
=100-49 -49 h2=51 h2=51 h=7.1
h=7.1
a2+b2=c2
122+h2=182 144+h2=324-144 -144 h2=180 h2=180 h=13.4
a2+b2=c2
42+h2=62
16+h2=36-16 -16 h2=20
h2=20 h=4.5
h=4.5
h=13.4
TriangleA=bh/2A=(10cm)(5.7cm)/2A=57cm2/2A=28.5cm2
a2+b2=c2
42+h2=72
16+h2=49-16 -16
h2
=33 h2=33 h=5.7
h=5.7
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TrapezoidA=(b+t)h/2A=(54mi+34mi)(24mi)/2A=(88mi)(24mi)/2
A=2112mi2
/2A=1056mi2
TrapezoidA=(b+t)h/2A=(8.9mm+4mm)(7mm)/2A=(12.9mm)(7mm)/2A=90.3mm2/2A=45.2mm2
Rhombus
A=(d1)(d2)/2A=(10cm)(6cm)/2A=60cm2/2A=30cm2
ParallelogramA=bhA=(18ft)(5.7ft)A=102.6ft2
CircleA=r2
A=(17cm)2
A=289cm2A=907.9cm2
a2+b2=c2
72+h2=92 49+h2=81-49 -49 h2=32 h2=32 h=5.7
Find theradius r.
34=2r2 217=r
3-4-5 rt
triangle
4*6=24h=24mi
3-4-5 rttriangle
h=4cm
a2+b2=c2
72+h2=82
49+h2=64-49 -49 h2=17 h2=15 h=3.9
3-4-5 rttriangle
4*2=8h=8yd
RhombusA=(d
1)(d
2)/2
A=(16yd)(12yd)/2A=192yd2/2A=96yd2
h=4
h=5.7
r=17h=24
3.9
h=8
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188=(18)(d2)18 1810.4ft=d
2
(13+t)2 2
(11+12)h 2
A=bh 15=5h 5 5 3in=h
A= bh 2 6h 2
2*32= *2
64=6h 6 610.7=h
A= bh 2 2h 2
2*8= *2
16=2h 2 2 8in=h
A= (b+t)h 2
2*68= *2
136=23h23 23 5.9=h
(23)h 2
(13+t)4 2
A= (b+t)h 2
87= 2
43.5=13+t-13 -13 30.5m=t
68=
87=
A= (d1)(d2) 2 (18)(d
2)
22*94= *2
A=bh 72=9h
9 9 8in=h
A= bh 2 8h 2
2*46= *2
92=8h
8 811.5=h
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A= bh 2 4h 2
2*12= *2
A= bh 2 26b 2
2*98= *2
196=26b26 26 7.5in=b
24=4h 4 4 6mi=h
(b+9)11 2
A= (b+t)h 2
2*65= *2
130=(b+9)11 11 11
11.8=b+9-9 -9 2.8mm=b
(3+7)h
2
A= (b+t)h 2
2*19= *2
38=10h10 10 3.8in=h
(10)h 2
19=
62=(14)(d2
)14 144.4ft=d
2
A= (d1)(d2) 2 (14)(d
2)
22*31= *2
A=r2
64=r2
20.4=r2
4.5ft=r
A=r2
121=r2
38.5=r2
6.2in=r
A=s2
169=s2
169=s2
13m=s
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bh7*321cm2
A=bh2*816cm2
7
3 2
8cm
8A=
+
A=42+12=54ft2
+
7
6
4
3A= +
bh7*642ft2
A=bh3*412ft2
+
3ft
A= +
17
10
4
7
a2+b2=c2
42+b2=72
16+b2=49
-16 -16 b2=33 b=5.74
bh17*10
170cm2
A=
1/2bh1/2(4*5.74)1/2(22.96)11.48in2
+
5.7
4
4in
A=702-91=611m2
A=
39
18 14
13
14m
13m
bh
18*39702m2A= -
1/2bh
1/2(13*14)1/2(182)91m2
A=
12
10 3
4bh12*10120cm2
A= -1/2bh1/2(3*4)1/2(12)6cm2
A=120-6=114cm2
3cm
4cm
A=
10
10 6
4
+
bh10*10100in2
A=1/2bh1/2(6*4)1/2(24)12in2
A=100+12=112in2+
4in
6in
A=21+16=37cm2
A=170+11.48=181.48in2
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A= +
13
13
7
8
bh13*13169in2
A=bh8*756in2
+ A=169+56=225in2
8
19
7
154
A=
bh19*7133in2
A=bh4*1560in2
- A=133-60=73in2
bh10*10100in2
A=r222
12.57in2-
A=100-(12.57*4)A=100-50.28A=59.72in2
*4
10
10A= - - --
bh18*8144in2
A=
1/2bh
1/2(4*4)1/2(16)8in2
-
A=144-(8*4)
A=144-32A=112in2
*4
A= -
18
8
A=
32
16-
A=
1/2bh1/2(4*6)1/2(24)12ft2
-A=512-(12*2)A=512-24A=488ft2
*2bh32*16512ft2
A=
13
8-
A=
1/2bh1/2(3*4)1/2(12)6cm2
-A=6-(6*4)A=104-24A=80cm2
*4bh13*8104cm2
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A=
5
-5 14
bh5*525cm2
A=bh1*44cm2
- A=25-4=21cm2 bh
23*15345cm2
A=bh7*642cm2
- A=345-42-10A=293cm2
bh5*210cm2
-
A=
23
15
7
65
2- -
7cm
23cm
bh21*12252cm2
A=1/2bh1/2(5*4)10cm2
- A=252-10-18A=224cm2bh6*318cm2
-
A=
21
12
4
63- -5
5
4
bh21*16336in2
A= 1/2bh1/2(6*4)12in2
- bh8*1080in2
-
A=
2116
4 10
8- -6
A=336-12-80A=244in2
4in
6in
bh15*9135cm2
A=bh4*520cm2
- A=135-20-18A=97cm2
bh6*318cm2
-
A= 94
56
3- -
15
6cm
bh20*12240in2
A= 1/2bh
1/2(9*12)54in2
- A=240-54-16
A=170in2-
A=
20
12 12 8- -
9
1/2bh1/2(4*8)16in2
4
4in
8in
9in
3-4-5 rt triangle 9in
-
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B=bhB=4*6B=24
V=BhV=24*7V=168in3
B=bhB=10*9B=90V=BhV=90*15
V=1,350ft3
B=1/2bhB=1/2(11*13)B=1/2(143)B=71.5
V=BhV=71.5*21V=1,501.5m3
B=1/2bhB=1/2(5*8)B=1/2(40)B=20V=BhV=20*6V=120yd3
B=1/2bhB=1/2(19*6)
B=1/2(114)B=57V=BhV=57*17V=969yd3
V=BhV=86*17V=1,462in3
V=r2h
V=(22)9V=4*9
V=36V113.10in3
V=r2h
V=(92)18
V=81*18V=1458V4580.44cm3
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B=bhB=7*4B=28V=BhV=28*2V=56m3
4m
B=1/2bhB=1/2(7.14*7)B=1/2(49.98)B=24.99V=BhV=24.99*5V=124.95ft3
a2+b2=c2
72+b2=102
49+b2=100-49 -49 b2=51 b=7.14
B=1/2bhB=1/2(6*8)B=1/2(48)B=24V=BhV=24*7
V=168cm3
a2+b2=c2
72+b2=112
49+b2=121-49 -49 b2=72 b=8.49
V=r2h
V=(92)6V=81*6V=486V1,526.81yd3
D=2r18=2r2 2 9=r
B=S2
B=82
B=64V=BhV=64*23V=1,472mm3
B=1/2bh
B=1/2(7*8.49)B=1/2(59.43)B=29.72V=BhV=29.72*11V=326.92in3
7.14m
6cm
8cm
3-4-5 right
triangle scalefactor 2. 3*2=6
8.49cm
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V=r2h
V=(112)20V=121*20V=2420V7,602.65in3
D=2r22=2r2 211=r
B=bhB=8*16B=128V=BhV=128*9V=1,152ft3
B=1/2bhB=1/2(9*12)B=1/2(108)B=54V=BhV=54*13V=702ft3
3-4-5 righttriangle scale
factor 3. 3*3=9
11cm
9cm
30-60-90 triangle. 4, 43
B=1/2bhB=1/2(4*43)B=1/2(27.71)B=13.86V=BhV=13.86*8V=110.88cm3
B=bhB=16*31B=496V=BhV=496*84
V=41,664in3
Oh
sinx=
sin38=O66*
*6
6(sin38)=O6(.6257)=O 3.69=O
a2+b2=c2
3.692+b2=62
13.62+b2=36-13.62 -13.62
b2=22.38 b=4.73
B=1/2bhB=1/2(4.73*3.69)B=1/2(17.45)B=8.73V=BhV=8.73*10V=87.3m3
3.69m4.73m
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B=bhB=12*7B=84
V=BhV=84*11V=924m3
B=1/2bhB=1/2(8*6)B=1/2(48)
B=24V=BhV=24*9V=216ft3
V=r2h
V=(182)21V=324*21V=6804V21,375.4in3
B=bhB=23*28B=644V=BhV=644*11V=7,084cm3
B=S2
B=92B=81V=BhV=81*10V=810yd3
V=BhV=45*13
V=585mm3
B=S2
B=212B=441V=BhV=441*21V=9,261yd3
B=1/2bhB=1/2(6*8)B=1/2(48)B=24V=BhV=24*7V=168ft3
3-4-5 righttriangle scalefactor 2. 2*4=8
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B=bhB=3*6B=18V=1/3(Bh)V=1/3(18*8)V=1/3(144)=48cm2
B=1/2bhB=1/2(7*9)B=1/2(63)B=31.5V=1/3(Bh)V=1/3(31.5*8)V=1/3(252)V=84ft3
V=1/3r2hV=1/3(52)11
V=1/325*11V=1/3(275)V=91.67V287.99in3
B=s2
B=62
B=36V=1/3(Bh)V=1/3(36*10)
V=1/3(360)V=120ft2
B=1/2bhB=1/2(13*18)B=1/2(234)B=117
V=1/3(Bh)V=1/3(117*15)V=1/3(1755)V=585mm3
V=1/3r2hV=1/3(122)24V=1/3144*24V=1/3(3456)
V=1152V3,619.11cm3
B=s2
B=32
B=9V=1/3(Bh)V=1/3(9*4)V=1/3(36)V=12ft2
V=1/3(Bh)
V=1/3(35*4)V=1/3(140)V=46.67ft2
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B=1/2bhB=1/2(3*4)B=1/2(12)B=6V=1/3(Bh)
V=1/3(6*6)V=1/3(36)V=12ft3
3-4-5 righttriangle scalefactor 1. 1*3=3
3ft
5.6
7ft
a2+b2=c2
72+h2=92
49+h2=81-49 -49 h2=32 h=5.67
B=bhB=7*8B=56V=1/3(Bh)V=1/3(56*5.67)V=1/3(317.52)V=105.84ft3
3-4-5 righttriangle scalefactor 2. 2*3=6
B=1/2bhB=1/2(6*8)B=1/2(48)B=24V=1/3(Bh)V=1/3(24*8)V=1/3(192)V=64cm3
V=1/3r2hV=1/3(92)23V=1/381*23V=1/3(1,863)V=621V1,950.93in3
d=2r18=2r2 2
9=r
V=1/3r2h
V=1/3
(6
2
)12V=1/336*12V=1/3(432)V=144V452.39m3
d=2r12=2r2 2 6=r
B=s2
B=52
B=25V=1/3(Bh)V=1/3(25*4.9)V=1/3(122.5)V=40.83cm2
a2+b2=c2 52+h2=72
25+h2=49-25 -25 h2=24 h=4.9
6cm
6m
4.9cm
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a2+b2=c2
42+h2=112
16+h2=121-16 -16 h2=105
h=10.25
B=bhB=8*9B=72V=1/3(Bh)V=1/3(72*10.25)V=1/3(738)V=246in3
B=1/2bhB=1/2(6*6)B=1/2(36)B=18V=1/3(Bh)V=1/3(18*7)V=1/3(126)V=42in3
45-45-90 righttriangle b=6
10.25in
6in
a2+b2=c2
62
+h2
=132
36+h2=169-36 -36 h2=133 h=11.53
11.53cm
B=bhB=7*10B=70V=1/3(Bh)V=1/3(70*11.53)V=1/3(807.1)
V=269.3cm3
V=1/3r2hV=1/3(82)14V=1/364*14V=1/3(896)V=144V2,814.87m3
d=2r16=2r2 2 8=r
8ft
3-4-5 righttriangle scalefactor 3. 4*3=12
B=1/2bhB=1/2(9*12)B=1/2(108)B=54V=1/3(Bh)V=1/3(54*11)V=1/3(594)V=198ft3
12ft
a2+b2=c2
92+h2=262
81+h2=676-81 -81 h2=595 h=24.39
24.39m
B=s2
B=172
B=289V=1/3(Bh)V=1/3(289*24.39)V=1/3(7048.71)V=2,349.57m3
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V=1/3r2hV=1/3(42)14V=1/316*14V=1/3(224)V=74.67V234.58m3
10
.25cm
a2+b2=c2
82+h2=132
64+h2=169-64 -64 h2=105
h=10.25
d=2r8=2r2 2 4=r
4cm
16=8 bisect2
8cm
a2+b2=c2
82+h2=92
64+h2=81-64 -64 h2=17 h=4.12
B=bhB=12*11B=132V=1/3(Bh)V=1/3(132*4.12)V=1/3(543.84)V=181.28cm3
4.12cm
45-45-90 right
triangle
8in8in
B=1/2bhB=1/2(8*8)B=1/2(64)B=32V=1/3(Bh)V=1/3(32*8)V=1/3(256)V=85.33in3
82=42 bisect2
42ft
a2+b2=c2
(42)2
+h2
=112
(16*2)+h2=121 32+h2=121-32 -32 h2=89 h=9.43
9.43ft
B=s2
B=82
B=64V=1/3(Bh)V=1/3(64*9.43)
V=1/3(603.52)V=201.17ft3
45-45-90 righttriangle
3in
a2+b2=c2
32+h2=62
9+h2=36-9 -9 h2=27 h=5.2
5.2in
B=s2
B=(32)2
B=9*2B=18V=1/3(Bh)V=1/3(18*5.2)V=1/3(93.6)V=31.2in3
45-45-90 righttriangle
42in
42in
square
a2+b2=c2
42+h2=72
16+h2=49-16 -16 h2=33 h=5.74
B=s2
B=(42)2
B=16*2B=32V=1/3(Bh)V=1/3(32*5.74)V=1/3(183.68)V=61.23in3
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45-45-90 righttriangle digonal=72
72=3.524.952
h=4.953h8.57
B=s2
B=(32)2
B=9*2B=18V=1/3(Bh)V=1/3(18*5.2)
V=1/3(93.6)V=31.2in3
45-45-90 righttriangle diagonal=9292=4.526.362
tan65=
4.95in
8.57in
h6.36
tan65= h6.36
*6.366.36*
6.36
6.36(tan65)=h6.36(2.1445)=h
13.64h
B=s2
B=92
B=81V=1/3(Bh)V=1/3(18*13.64)V=1/3(1104.84)V=368.28cm3
13.64cm
45-45-90 righttriangle diagonal=9282=425.662
5.66
tan42= h5.66
tan42= h6.36 *5.665.66*
5.66(tan42)=h5.66(.9004)=h 5.1h
5.1ft
B=s2
B=82B=64V=1/3(Bh)V=1/3(64*5.1)V=1/3(326.4)V=108.8ft3
a2+b2=c2
62+92=d2
36+81=d2
117=d2
10.82=d10.82=5.41 2
4.23m
5.41m
tan38= h5.41
tan38= h6.36
*5.45.41*
5.41(tan38)=h5.41(.7813)=h 4.23h
B=bhB=6*9B=56V=1/3(Bh)V=1/3(56*4.23)V=1/3(236.88)V=78.96m3
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B=bhB=9*8B=72
V=1/3(Bh)V=1/3(72*11)V=1/3(792)V=264ft3
V=1/3r2hV=1/3(52)14V=1/325*13
V=1/3(325
)V=108.33V340.34ft3
V=1/3r2hV=1/3(62)14V=1/336*12V=1/3(432)V=144V1,021.02ft3
d=2r12=2r2 2 6=r
B=s2
B=72
B=49V=1/3(Bh)V=1/3(49*2)V=1/3(98)V=32.67in3
B=1/2bhB=1/2(5*14)B=1/2(70)B=35V=1/3(Bh)V=1/3(35*15)V=1/3(525)V=175cm3
B=1/2bhB=1/2(5*12)B=1/2(60)B=30V=1/3(Bh)V=1/3(30*11)V=1/3(330)V=110cm3
5-12-13 righttriangle scalefactor 1 12*1=12
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d=2r8=2r2 24=r
4cm
3-4-5 righttriangle scalefactor=2
5*2=10
a2+b2=c2
a2+72=122
a2+49=144 -49 -49 a2=95 a=9.75
30-60-90right triangle
1053
11in
11in11in
11in
9in
9in
9in 9in
6in 6in
6in
6in
4cm
4cm
4cm
10m 10m
10m
10m
8m 6m
10m
10m 10m
9.75ft
6m 8m
8m6m
9.75ft
7ft
7ft
7ft
13ft 13ft
12ft
12ft
12ft13ft
9.75ft
10in10in
10in
10in
8in
8in8in
5in
5in
5in53in53in
53in
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11-4=7
4+6=10
27-13=14
19-11=8
18-8=10
12-5=7
10cm
7cm 10cm 7cm6cm
6cm
6cm
8cm
8cm
8cm 8cm
8cm
8cm
4cm
4cm
4cm
4cm
4cm 4cm
11cm
11cm 11cm
11ft
11ft 11ft
13ft
13ft
13ft
19ft
19ft
19ft 27ft
27ft
27ft14ft
14ft
14ft
8ft
8ft 8ft8ft
8ft8ft
8ft
8ft
8ft
18cm
18cm12cm
8cm
10cm
8cm
10cm
7cm
7cm
5cm
5cm
18cm
10cm
8cm
7cm
5cm
6cm
6cm
6cm
6cm
6cm6cm
4/2=2
a2+b2=c2
a2+22=82
a2+4=64 -4 -4 a2=60 a=7.75
14ft
8ft
7.75in
2in
4in 4in 4in 4in
4in
4in
7.7
5in
7.7
5in
7.7
5in
7.7
5in
3-4-5 rt.triangle. scalefactor =1h=4
45-45-90 triangle62
62in
a2+b2=c2
a2+322=52
a2+18=25 -18 -18 a2=7 a=2.65
4in
6in
4in
6in 62in2.6
5in
6in
6in
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6in
5in
6in
5in 5in
5in6in 6in
9in
9in 9in 9in
8ft
8ft
8ft
8ft
8ft 8ft 8ft
8ft
16ft
16ft 16ft 16ft
12cm 16cm 12cm 16cm
3-4-5 right
trianglescale factor=45*4=20
20cm 20cm
20cm
20cm
12cm16cm
24cm
24cm
2
4cm
5ft
5ft
5ft15ft
30in2 30in2 45in2
45in254in2 54in2
384cm
2
288cm
2
480cm
2
96cm2 96cm2
78.54ft2
52
78.54ft2
52 2(5)(15)
471.24ft2
64ft2 64ft2
128ft
2
128ft
2
128ft
2
128ft
2
30+30+45+45+54+54=258in2
96+96+384+288+480=1,344cm2
78.54+78.54+471.24=628.32ft2
64+64+128+128+128+128=640ft2
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14-5=9
16-9=7
22-13=9
11-4=7
9in
5in
9in 9in
5in
16in
9in16in 5in
7in
7in
7in7in
7in
7in
7in
9in
9in
14in16in
7in
9in
7in
13in
13in
4in 4in22in
22in
7in
6in
6in
6in6in 6in 6in
11in
13in
22in4in 9in 7in
7in
9in
10/2=5
a2+b2=c2
a2+42=102
a2+16=100 -16 -16 a2=84 a=9.17
3-4-5 righttrianglescale factor=25*2=10
a2
+b2
=c2
a2+42=92
a2+16=81 -16-16 a2=65 a=8.06
8/2=410m
a2+b2=c2
a2+32=92
a2+9=81 -9 -9 a2=72 a=8.49
6/2=3
a2+b2=c2
a2+52=92
a2+25=81 -25 -25 a2=56 a=7.48
10/2=5
8cm9.1
7cm
8cm9.1
7cm
8cm9.1
7cm
8cm9.1
7cm
8cm
8cm
8m
10m6m
6m
8m
10m8.4
9m
7.48
m
8.0
6m
81in
2
81in
2
80in
235in
2
63in
2
49in
2
98in2112in2
63
in2
64cm2
24m2
25.47m2
32.24m2
37.4m2
24in
2
54in2 42in2
132in2
78in2154in2
154in2
52in2
52in2
66in2
81+80+81+80+35+63+49+98+112+63=742in2
36.68+36.68+36.68+36.68+64=210.72cm2
24+32.24+25.47+37.4=119.11m2
154+52+154+52+66+24+54+42+78+132=808in2
36.68cm2 36.68cm236.68cm236.68cm2
80in
2
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10-2=8
7+1=8
8in 8in
7in7in
2in1in1in
8in
8in
6in
6in
6in
6in 6in6in
10in 8in2in7inin
8in
8in
4.5cm
4.5cm
4.5cm 11cm
9/2=4.5
a2+b2=c2
a2+22=92
a2+4=81 -4 -4 a2=77
a=8.77
4/2=2 a2+b2=c2
a2+2.52=92
a2+6.25=81 -6.25-6.25 a2=74.75 a=8.65
5/2=2.5
4.5cm
45-45-90triangle3
3ft3ft
3ft
3ft
3ft
3ft
3ft 3ft
6ft6ft
6ft
32ft
4ft 5ft
4ft
5ft
8.6
5ft
8.7
7ft
63.62cm2
4.52 2(4.5)(11)
311.02cm263.62cm2
4.52
17.54ft2 21.63ft2
4ft
5ft
8.6
5ft
8.7
7ft17.54ft
2
21.63ft2
20ft2
4.5ft2 4.5ft2
25.46ft2
18ft2 18ft2
42in2
48in260in2
64in2 64in2
12in2
2in
1in6in2
8in
48in2
63.62+63.62+311.02=438.26cm2
17.54+21.63+21.63+17.54+20=98.34cm2
4.5+4.5+18+18+25.46=70.46ft2
64+2+64+2+6+42+60+12+46+48=348in2
2in2 2in2
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31+43=74m ABD=74
163-44=119m CBD=119
125-36=89
m CBD=89
78-21=57m CBD=57
112-29=83m ABD=83
97-48=49
m CBD=49
42+35=77m ABD=77
51+38=89m ABD=89
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45+49=96
m ABD=94
119-67=52m CBD=52
115+65=180m ABC=180
123-67=56m CBD=56
58+37+59=154
m ABE=154
22+28+78=128m ABE=128
61+25+15=101
m ABE=101
25+41+36=102m ABE=102
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111-31-52=28m CBD=28
136-35-42=59m ABC=59
170-21-19-23=107m CBD=107
58+48+31+43=180m ABF=180
136-67=69
m DBE=69
(77+89)-136=30
m CBD=30
169-39-26-23=107
m DBE=81 165-36-21-57=51m CBD=51
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Two angles whose measures
add to 90.
Two angles who are supplementary
and adjacent.
Two angles whose measures
add to 180.
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y+134=180
-134-134
y=46
z+67=180-67 -67 z=113
z+67=180-67 -67 z=113 x+64=90
-64-64
x=26
y+44=180-44 -44 y=136
z+90=180-90 -90 z=90
x+12=90-12 -12 x=78
x+64=180-64 -64 x=116
x+33=90
.
x+33=90-33-33
x=57
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x+(3x+18)=90 x+3x+18=90 4x+18=90 -18-18 4x=72
4 4 x=18
(x+21)+(x+58)=180 x+21+x+58=180 2x+79=180 -79 -79 2x=101 2 2 x=50.5
(2x+8)+(x-24)=180 2x+8+x-24=180 3x-16=180 +16 +16 3x=196 3 3 x=65.3
(x+15)+(x-12)=180 x+15+x-12=180
2x+3=180 -3 -3 2x=177 2 2 x=88.5
131+y=180-131 -131 y=49
49+x=180 -49 -49
x=131
The value of x is the same asthe measure of the angle onthe other side of theintersecting lines, 131.
x+2x=180 3x=180 3 3 x=60
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143+x=180 linear pair
-143 -143x=37
m TRS=37
x143
59+x=90-59 -59
x=31
m H=31x
59
x
78 78+x=180-78 -78
x=102
m O=102
x+27
x (x+27)+x=180 linear pair x+27+x=180 2x+27=180
-27 -272x=153 2 2 x=76.5
m ROK=76.5
2x+11
(2x+11)+x=180 linear pair 2x+11+x=180
3x+11=180 -11 -113x=169
3 3 x=56.3
check your work
x
x+43=180-43 -43
x=137m DEF=137
x
43
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x=121 Vertical Angle Theorem x=37 Vertical Angle Theorem
x=28 Vertical Angle Theorem x=165 Vertical Angle Theorem
x=48 Vertical Angle Theorem
x=90 Vertical Angle Theorem
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x+13=2x-23 Vertical Angle Theorem-x -x 13=x-23 +23 +23 36=x
x+117=3x+9 Vertical Angle Theorem-x -x 117=2x-9 +9 +9 126=2x 2 2
63=x
2x-17=x+12 Vertical Angle Theorem-x -x x-17=12 +17 +17 x=29
x+91=3x-37 Vertical Angle Theorem-x -x 91=2x-37
+37 +37 128=2x 2 2 64=x
x+33=2x-23 Vertical Angle Theorem-x -x 33=x-23 +23 +23 56=x
x+45=2x+19 Vertical Angle Theorem-x -x 45=x+19 -19 -19 26=x
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m EAM=m IALVertical Angle Theorem
63=m IAL
m TKE=m AKSVertical Angle Theorem
87=m AKS
m GER=m AETVertical Angle Theorem147=m AET
m IKS=m HKEVertical Angle Theorem
51=m HKE
m DCE=m ACBVertical Angle Theorem
153=m ACB
87
138
63
147
51 153
m SOT=m NOEVertical Angle Theorem
138=m NOE
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m ABD=m CBEVertical Angle Theorem
2x-37=x+65-x -x x-37=+65 +37 +37 x=102
m CBE=x+65m CBE=102+65m CBE=102+65m CBE=167
m GET=m AENVertical Angle Theorem 3x-51=2x-17-2x -2x x-51=-17 +51 +51 x=34m GET=3x-51m GET=3(34)-51m GET=102-51
m GET=51
m BOE=m NOLVertical Angle Theorem
5x+36=3x+42-3x -3x 2x+36=42 -36-36 2x=6
2 2 x=3m NOL=3x+42m NOL=3(3)+42m NOL=9+42m NOL=51
m LEN=m BEDVertical Angle Theorem
X+89=2x+89-x -x 89=x+89 -89 -89
0 =x
m BED=2X+89m BED=2(0)+89m BED=0+89m BED=89
m LIO=3xm LIO=3(51)m LIO=153
m PIT=m LIOVertical Angle Theorem
4X-51=3X-3x -3x x-51=0 +51 +51 x=51
x+24
2x-8
2x-37 x+65
3x-512
x-17 x+89
2x+89
5x+363x+42
4x-51
3x
m HAE=m DASVertical Angle Theorem
x+24=2x-8
-x -x 24=x-8 +8 +8 32=xm DAS=2x-8m DAS=2(32)-8m DAS=64-8m DAS=56
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1,2; 2,3; 3,4; 4,5; 5,1
1,3;
1,2; 2,3
1,2; 2,3; 3,4; 4,5; 5,6; 6,1
1,4; 2,5; 3,6
1,2;2,3;3,4;4,5;5,6;6,7;7,8;8,1
1,5; 2,6; 3,7; 4,8
1,2; 2,3; 3,4; 4,5; 5,6; 6,7; 7,8 8,1
1,2; 3,4; 5,6; 7,8
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linear pair
vertical angles
adjacent angles
supplementary
complementary
linear pair
vertical angles
complementary
linear pair
supplementary
vertical angles
vertical angles
complementary
vertical angles
supplementary
two angles that are supplementary and adjacent
two angles that share a common ray and vertex
two angles that are on opposite sides of two intersecting lines
complementary
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1, 5; 3, 7; 2, 6; 4, 8
3, 6; 4, 5;
1, 8; 2, 7;
3, 5; 4, 6;
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consecutive angles
corresponding angles corresponding angles
corresponding angles alternate interior angles
alternate exterior angles alternate interior angles
consecutive angles corresponding angles
a linear pair, or supplementary
vertical angles
corresponding angles vertical angles
alternate interior angles
a linear pair, or supplementary a linear pair, or supplementary
alternate exterior angles
vertical angles
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G
D
I
E
C
B
F
A
H
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Corresponding, m 1=m 2
Corresponding, m 1=m 2
Alternate interior, m 1=m 2
Alternate exterior, m 1=m 2
Alternate exterior, m 1=m 2
Consecutive, m 1+m 2=180
Corresponding, m 1=m 2
Consecutive, m 1+m 2=180
Alternate interior, m 1=m 2
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Corresponding
111=x
Alternate exterior
x=37
Corresponding
175=x
Alternate interior
x=58
Corresponding
x=26
Consecutive
x+68=180-68 -68 x=112
Alternate interior
46=x
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Corresponding
x=31
Alternate interior
x=83
Vertical
x=126
Alternate interior
x=132
Corresponding
x=119
Alternate exterior
x=96
Linear Pair
x+113=180-113 -113 x=67
Consecutive
x+21=180-21 -21 x=159
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Corresponding
2x-61=x+3 -x -x x-61=3 +61+61 x=64
Alternate interior
2x-12=x+34 -x -x x-12=34 +12+12 x=46
Vertical
3x+21=x+57 -x -x 2x+21=57 -21-21 2x=36 2 2
x=18
Corresponding
3x-35=2x+52-2x -2x x-35=52 +35 +35 x=87
Consecutive
(2x+42)+(x-23)=180 2x+42+x-24=180 3x+18=180 -18 -18 3x=162 3 3 x= 54
Linear pair
(x+23)+(x+61)=180 x+23+x+61=180
2x+84=180 -84 -84 2x=96 2 2 x= 48
Alternate exterior
2x+19=x+23
-x -x x+19=23 -19-19 x=4
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Vertical
2x-17=x+27 -x -x x-17=27 +17+17
x=44
Corresponding
2x-37=x+79 -x -x x-37=79 +37+37 x=116
Alternate interior
2x+5=x+59 -x -x x+5=59 -5 -5 x=54
Corresponding
2x+17=x+77 -x -x x+17=77 -17-17 x=60
Alternate exterior
3x-26=x+64 -x -x 2x-26=64 +26+26 2x=90
2 2 x=45
Vertical
2x+32=x+97 -x -x x+32=97 -32-32 x=65
Corresponding
3x+71=5x+37-3x -3x 71=2x+37 -37 -37 34=2x
2 217=x
Consecutive
(x+23)+(x+54)=180 x+23+x+54=180 2x+77=180 -77 -77 2x=10 3 2 2 x=5 1.5
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x
x
x
x
x
x
85
91103
47
63
Alternate exterior
m HFG=m BCD 103=m BCD
Vertical
m GEH=m BEF 47=m BCD Linear Pair
m DCH+m ACH=180 103+x=180-103 -103 x=77 m ACH=77
Alternate interior
m DCH=m BFE 85=m BFE
Consecutivem EFB+m DCH=180
91+x=180-91 -91 x=89 m ACH=89
x
68
Correspondingm ACB=m EFC 68=m EFC
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Alternate Exterior
m ACB=m HFG2x-45=x+23-x -x
x-45=23 +45+45 x=68m HFG=x+23m HFG=68+23m HFG=91
2x-45
x+23
Corresponding
m EFH=m ACH2x-142=x+16-x -x x-142=x-16 +142 +142 x=126m ACH=x-16m ACH=126-16
m ACH=110
Corresponding
m BCD=m BFG x+26=2x-24-x -x 26=x-24 +24 +24
50=xm BFG=2x-24m BFG=2(50)-24m BFG=100-24m BFG=76
Alternate Interior
m ADF=m HEC2x+4=4x-14-2x -2x 4=2x-14 +14 +14 18=2x
2 2 9=xm HEC=4x-14m HEC=4(9)-14m HEC=36-14m HEC=22
Consecutive
m GF+m DCH=180(x+32)+(x+24)=180 x+32+x+24=180
2x=164
2x+56=180 -56 -56 2x=124 2 2 x=62
m DCH=x+24m DCH=62+24
m DCH=86
2x-124x+16
x+32
x+24
x+26
2x-242x+4
4
x-14
Linear Pairm ACH+m DCH=180(x+24)+(x-8)=180 x+24+x-8=180 2x+16=180 -16 -16
x+24x-8
2x=1642 2x=82
m ACH=x+24m ACH=82+24
m ACH=106
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AB CD 1 and 2 arecorresponding so m 1=m 2
AB CD 1 and 2 are Alternateexterior so m 1=m 2
AB CD 1 and 2 arecorresponding so m 1=m 2
AB CD 1 and 2 are Alternateinterior so m 1=m 2
AB is not to CD 1 m 1=m 2 isfalse.
AB is not to CD 1 som 1=m 2 is false.
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AB is not to CD 1 and 2 are
m 1=m 2 is false.
AB is not to CD som 1+m 2=180 is false.
This is true because vertical
angles are always equal.
AB CD, 1 and 2 areconsecutive so m 1+m 2=180 istrue.
1 and 2 are Corresponding ifcorresponding angles are equal
then AB CD.
1 and 2 are Alternateexterior if Alternate exterior
angles are equal then AB CD.
1 and 2 are Alternate interior
if Alternate interior angles areequal then AB CD
1 and 2 are Vertical angles.Vertical angles don't requireparallel lines so we don't know ifAB CD so probably false.
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Triangle Sum Theorem23+68+x=180
91+x=180-91 -91
x=89
Triangle Sum Theorem42+87+x=180
129+x=180-129 -129
x=51
Triangle Sum Theorem
31+53+x=18084+x=180-84 -84
x=96
Triangle Sum Theorem60+90+x=180
150+x=180-150 -150
x=30
Triangle Sum Theorem25+26+x=180
51+x=180-51 -51
x=129
Triangle Sum Theorem18+54+x=180
72+x=180-72 -72
x=108
Triangle Sum Theorem106+59+x=180
165+x=180-165 -165
x=15
Triangle Sum Theorem24+88+x=180
112+x=180-112 -112
x=68
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Triangle Sum Theorem53+68+x=180
121+x=180-121 -121
x=59
Triangle SumTheorem24+x+x=18024+2x=180-24 -24 2x=156
2 2 x=78
Equilateral Triangle
x=60
Triangle Sum Theorem46+71+x=180117+x=180
-117 -117 x=63
Triangle Sum Theorem25+37+x=180
62+x=180-62 -62
x=118
Triangle Sum Theorem121+39+x=180
160+x=180-160 -160
x=20
Triangle Sum Theorem32+x+x=180
32+2x=180-32 -32 2x=148 2 2 x=74
Triangle Sum Theorem30+x+x=180
30+2x=180-30 -30 2x=150 2 2 x=75
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Triangle Sum Theorem30+90+x=180
120+x=180-120 -120
x=60
m G=60
Triangle Sum Theorem138+17+x=180
155+x=180-155 -155
x=25 m M=25
Triangle Sum Theorem47+43+x=180
90+x=180-90 -90 x=90 m H=90
Equilateral Trianglex=60m O=60m T=60m P=60
Triangle Sum Theorem13+24+x=180
37+x=180-37 -37
x=143 m M=143
Triangle Sum Theorem118+26+x=180
144+x=180-144 -144
x=36 m H=36
Triangle Sum Theorem 48+x+x=180
48+2x=180-48 -48
2x=132 2 2 x=66
m T=66
30
x
17
138x
43
47
x
48x x
x
24
13
x
26118
x
x
x
Triangle Sum Theorem18+94+x=180
112+x=180-112 -112 x=68 m A=68
18
94
x
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Triangle Sum Theoremx+3x+(x+15)=180x+3x+x+15=180
5x+15=180-15 -15
5x=165 5 5 x=33
Triangle Sum Theoremx+(x+43)+(x-31)=180x+x+43+x-31=180
3x+12=180-12 -12
3x=168 3 3 x=56
Triangle Sum Theoremx+x+(x-27)=180x+x+x-27=180
3x-27=180+27 +27 3x=207 3 3 x=69
Triangle Sum Theoremx+(x+55)+(x+38)=180x+x+55+x+38=180
3x+93=180-93 -93
3x=87 3 3 x=29
Triangle Sum Theorem(x+21)+(x+23)+(x+61)=180x+21+x+23+x+61=180
3x+105=180-105-105
3x=75 3 3 x=25
Triangle Sum Theoremx+2x+(x-16)=180x+2x+x-16=180
4x-16=180+16 +16
4x=196
4 4 x=49
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Triangle Sum Theorem2x+(x-23)+(x-17)=1802x+x-23+x-17=180
4x-40=180+40 +40
4x=220 4 4 x=55 m S=2x m S=2(55)
m S=110
Triangle Sum Theorem2x+x+(2x+30)=1802x+x+2x+30=180
5x+30=180-30 -30
5x=150 5 5 x=30 m B=2x m B=2(30) m B=60
2x+30
2x
x
2x
x-23
x-17
x+5
x-7
x
x
x
3x+19
x-22
x-17
Triangle Sum Theorem90+(x-7)+(x+5)=18090+x-7+x+5=1802x+88=180
-88 -88 2x=92m Z=46-7 2 2 m Z=39 x=46 m Z=x-7
Equilateral Trianglex=60m R=60m Q=60m S=60
Triangle Sum Theorem(x-22)+(3x+19)+(x-17)=180
x-22+3x+19+x-17=1805x-20=180+20 +20
5x=200 5 5 x=40
m X=3x+19m X=3(40)+19m X=120+19m X=139
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m 1+m 2=m 3
54+63=x
117=x
m 1+m 2=m 3
114+38=x
152=x
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m 1+m 2=m 3
18+42=x
60=x
m 1+m 2=m 3
48+68=x
116=x
m 1+m 2=m 343+106=x 149=x
m 1+m 2=m 3
21+138=x 159=x
m 1+m 2=m 3 60+90=x 150=x
m 1+m 2=m 3 35+90=x 125=x
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m 1+m 2=m 3 78+x=131 -78 -78
x=53
m 1+m 2=m 3 142+x=167 -142 -142
x=25
m 1+m 2=m 3 39+x=106 -39 -39
x=67
m 1+m 2=m 3 42+x=77 -42 -42
x=35
m 1+m 2=m 3 x+x=148 2x=148
2 2 x=74
58
m 1+m 2=m 3 29+x=87 -29 -29 x=______
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m 1+m 2=m 3 (x-6)+x=148 x-6+x=148
2x-6=148 +6 +6 2x=154 2 2 x=77
m 1+m 2=m 3(2x-37)+90=3x 2x-37+90=3x 2x+53=3x -2x -2x
53=x
m 1+m 2=m 3(x+1)+(2x-17)=x+89 x+1+2x-17=x+89 3x-16=x+89 -x -x 2x-16=89 +16+16 2x=105 2 2 x=52.5
m 1+m 2=m 3(x+13)+(x+13)=135
2(x+13)=135 2x+26=135 -26 -26 2x=109 2 2 x=54.5
m 1+m 2=m 3(x+34)+(x+34)=3x 2(x+34)=3x 2x+68=3x -2x -2x 68=x
m 1+m 2=m 3(x+3)+(x+4)=125 x+3+x+4=125 2x+7=125 -7 -7 2x=118 2 2 x=59
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m 1+m 2=m 3(x-16)+(x+3)=x+42 x-16+x+3=x+42 2x-13=x+42 -x -x x-13=42 +13+13 x=55
m 1+m 2=m 3(x-19)+(x-26)=x
x-19+x-26=x 2x-45=x -2x -2x -45=-x -1 -1 45=x
35
61
x
m 1+m 2=m 3 93+48=x
141=x
m 1+m 2=m 3 35+61=x 96=x
x+71+x=6x 2x+71=6x -2x -2x 71=4x 4 4 17.75=x
17.75
17.7578.75
m 1+m 2=m 3 m Z=x+71 (x+71)+x=6x m Z=(_____)+71
m Z=______+71 m Z=______
6x
x
x+71
48
93
x
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x
422x+19
m 1+m 2=m 3 x+42=2x+19 -x -x 42=x+19 -19 -19 23=x
m 1+m 2=m 3 60+60=x 120=x
m ATS=x m ATS=120
m CAS=2x+19m CAS=2(23)+19m CAS=46+19m CAS=65
m 1+m 2=m 3 x+x=148 2x=148 2 2 x=74
m D=x
m D=74
x x
x60
60
60
1482x+51
4x+9
x
m 1+m 2=m 3(2x+51)+x=4x+9 2x+51+x=4x+9 3x+51=4x+9
-3x -3x 51=x+9 -9 -9 42=x
m HEP=4x+9m HEP=4(42)+9m HEP=168+9m HEP=177
75 3x+9
x
m GRE+m ERA=180 75+m ERA=180-75 -75 m ERA=105
105
m 1+m 2=m 3105+x=3x+9 -x -x 105=2x+9 -9 -9 96=2x 2 2 48=x
m HEP=3x+9m HEP=3(48)+m HEP=144+9m HEP=153
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What could I use to find x?
B1: m 7=x corresponding angles
What could I use to find x?
B1: m 1=x corresponding angles What could I use to find x?
B1: m 2+m 3+x=180 triangle sumtheorem
What could I use to find x?
B1: m 2+m 1+x=180 triangle sumtheorem
What could I use to find x?
B1: m 9=x corresponding angles
What could I use to find x?
B1: m 1+x=180 linear pair
What could I use to find x?
B1: 35+m 7+x=180 triangle sumtheorem
*Note: Thereare manypossible waysto solve someof theseproblems.
What could I use to find x?
B1: 53+m 1+x=180 triangle sumtheorem
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What could I use to find x?B1: m 1+m 3=x remote exterior
angle theoremWhat could I use to find m 3?B2: m 3+132=180 linear pair
What could I use to find x?B1: m 2+61+x=180 triangle sumtheoremWhat could I use to find m 2?B2: m 1=m 2 vertical angles
*Note: There are many possibleways to solve some of theseproblems.
What could I use to find x?
B1: m 7+68=x remote exteriorangle theoremWhat could I use to find m 7?B2: m 7+35=90 complementary
What could I use to find x?B1: m 2+m 3+x=180 triangle sumtheoremWhat could I use to find m 3?B2: m 3=53 corresponding angles
What could I use to find x?B1: m 2+m 5+x=180 angle additionpostulate, straight angleWhat could I use to find m 5?B2: m 5+76+18=180 triangle sumtheorem.
What could I use to find x?B1: x=m 7 correspondingWhat could I use to find m 7?B2: m 5=m 7 corresponding
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What could I use to find x?B1: m 1+93+x=180 triangle sumtheoremWhat could I use to find m 1?B2: m 1+m 4+83=180 angleaddition postulateWhat could I use to find m 4?B3: m 4+132+24=180 trianglesum theorem
What could I use to find x?
B1: m 9=x corresponding anglesWhat could I use to find m 9?B2: m 9=68 alternate interior angles
What could I use to find x?B1: m 10+m 13+x=180 trianglesum theoremWhat could I use to find m 10?B2: m 10+=42 vertical anglesWhat could I use to find m 13?B3: m 13=78 corresponding angles
What could I use to find x?B1: m 1+31=x=180 remote exteriorangle theoremWhat could I use to find m 1?B2: m 1+142=180 vertical angles
What could I use to find x?B1: m 1+82=x remote exterior angletheorem
What could I use to find m 1?B2: m 1+m 2+82=180 trianglesum theoremWhat could I use to find m 2?B3: m 2=32 alternate interior angles
What could I use to find x?B1: m 2+71+x=180 triangle sumtheoremWhat could I use to find m 2?B2: m 1=m 2 vertical anglesWhat could I use to find m 1?B3: m 1+48+53=180 triangle sumtheorem
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How can I find x?B1: m 7+43=x remoteexterior angle theoremHow can I find m 7?B2: m 7+68=90complementary anglestriangle sum therorem
B2: m 7+68=90-68-68
m 7=22
B1: m 7+43=x22+43=x
65=x
How can I find x?B1: m 3=x vertical anglesHow can I find m 8?B2: m 3+39=180 linear
pair
B2: m 3+39=180 -39 -39 m 3=141
B1: m 3=x141=x
How can I find x?B1: m 1+m 2=x remoteexterior angle theoremHow can I find m 2?B2: m 2+142=180linear pairHow can I find m 1?B3: m 1+31+87=180angle addition postulate,straight angle
B3: m 1+31+87=180m 1+118=180
-118-118 m 1=62
B2: m 2+142=180-142 -142
m 2=38B1: m 1+m 2=x
38+62=x 100=x
How can I find x?B1: m 3=x alternateinterior anglesHow can I find m 3?B2: m 3+36+58=180triangle sum therorem
B2: m 3+36+58=180m 3+94=180
-94 -94 m 3=86
B1: m 3=x=18086=x
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How can I find x?B1: m 13=x alternateexterior anglesHow can I find m 13?B2: m 13=163corresponding angles
B2: m 13=163
B1: m 13=x163=x
How can I find x?B1: m 1=x vertical anglesHow can I find m 1?B2: m 1+41+72=180
triangle sum theorem
B2: m 1+41+72=180m 1+113=180
-113-113 m 1=67
B1: m 1=x67=x
How can I find x?B1: m 1+m 4=x remoteexterior angle theoremHow can I find m 1?B2: m 1+72+68=180angle addition postulate,straight angleHow can I find m 4?B3: m 4=52 verticalangles
B3: m 4=52B2: m 1+72+68=180
m 1+140=180 -140-140 m 1=40B1: m 1+m 2=x
52+40=x 92=x
How can I find x?B1: (m 1+31)+(52+24)+x=180 triangle sumtheoremHow can I find m 1?B2: m 1+63+89=180triangle sum theorem
B2: m 1+63+89=180m 1+152=180
-152-152 m 1=28
B1:(m 1+31)+(52+24)=18028+31+52+24+x=180
135+x=180 -135 -135 x=45
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How can I find x?B1: m 6+m 7=x remoteexterior angle theoremHow can I find m 6?B2: m 6+48=90complementary anglesHow can I find m 7?B3: m 7=53corresponding angles.
B3: m 7=53B2: m 6+48=90
-48-48 m 6=42B1: m 6+m 7=x
42+53=x 95=x
How can I find x?B1: m 1+55=x remoteexterior angle theoremHow can I find m 1?B2: m 1+46=90complementary angles
B2: m 1+46=90-46-46
m 6=44B1: m 1+55=x
44+55=x 99=x
How can I find x?B1: m 11+x=180 linearpairHow can I find m 11?B2: m 11+m10+20=180
triangle sum theoremHow can I find m 10?B3: m 10=78corresponding angles.
B3: m 10=78B2: m 11+m 10+20=180 m 11+78+20=180
m 11+98=180 -98-98
m 11=82B1: m 11+x=18082+x=180
-82 -82 98=x
How can I find x?B1: m 1+(38+26)+x=180triangle sum theorem
How can I find m 1?B2: m 1+139=180 linearpair
B2: m 1+139=180-139 -139
m 1=41B1: m 1+(38+26)+x=180 41+64+x=180
105+x=180 -105 -105 x=75
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How can I find x?B1: m 5+41+x=180triangle sum theoremHow can I find m 5?B2: m 5=m 3How can I find the m 3?B3: 47+21+m 3=180
B2: m 3=m 5 112=m 5B1: m 5+38+x=180
112+38+x=180 150+x=180 -150 -150 x=30
How can I find x?B1: m 1+m 2+x=180triangle sum theoremHow can I find m 1?B2: m 1=72corresponding anglesHow can I find the m 2?B3: m 2=91 vertical
angles
B3: m 2=91
B2: m 1=72
B1: m 1+m 2+x=18091+72+x=180
163+x=180 -163 -163 x=17
How can I find x?B1: m 2+m3+x=180triangle sum theorem
How can I find m 2?B2: m 2=126corresponding anglesHow can I find the m 3?B3: m 3=44 alternateexterior angles
B3: m 3=44
B2: m 2=126
B1: m 2+m 3+x=180126+44+x=180
170+x=180 -170 -170 x=10
How can I find x?B1: m 2+96+x=180triangle sum theoremHow can I find m 2?B2:(m1+m 2)+69+72=180triangle sum theoremHow can I find the m 1?B3:69+90+m 1=180triangle sum theorem
B3: 69+90+m 1=180159+m 1=180
-159 -159 m 1=21
B2:(m1+m 2)+69+72=180
21+m 2+69+72=180 162+m 2=180 -162 -162 m 2=18
B1: m 2+96+x=18018+96+x=180
114+x=180 -114 -114 x=66
B1: 47+21+m 3=18047+21+m 3=180
68+m 3=180 -68 -68 m 3=112
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How can I find x?B1: m 8=xcorresponding anglesHow can I find the m 8?B2: m 8=119 alternateinterior angles
B2: m 8=119
B1: m 8=x 119=x
How can I find x?B1: m 1+90=x remoteexterior angle theoremHow can I find m 1?B2: m 1+150=180
B2: m 1+150=180 -150 -150 m 1=30B1: m 1+90=x
30+90=x 120=x
How can I find x?B1: m 3+47=x remoteexterior angle theoremHow can I find m 1?B2: m 3+21=90
complementary angles
B2: m 3+21=90-21-21
m 3=69B1: m 3+47=x
69+47=x 116=x
How can I find x?B1: m 4=xalternate interior anglesHow can I find the m 4?B2: m 4+125=180consecutive angles
B2: m 4+125=180-125-125
m 4=55
B1: m 4=x 55=x
(Sometimes a very hard lookingproblem can have an easy solution.)
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s1+s2=s37+7>1314>13 truetriangle
s1+s2=s34+3>67>6 truetriangle
s1+s2=s36+10>1216>12 truetriangle
s1+s2=s3
21+18>3439>34 truetriangle
s1+s2=s38+9>1717>17 false
no triangle
s1+s2=s345+46>9191>91 falseno triangle
s1+s2=s348+52>80100>80 truetriangle
s1+s2=s336+25>6161>61 falseno triangle
s1+s2=s386+76>128162>128 truetriangle
s1+s2=s325+48>7073>70 truetriangle
s1+s2=s34+7>1111>11 false!no triangle
s1