Let s Practice Geometry

Let's Practice...

A resource for Teachers, Students, and Parents. By: Brent Tuller

Ray AB,

AB

ABC ABD CBD

HMK or

KMH

AB CD

LP=PM Bisect 5=PM so LM=10

RT=TS Bisect 10=x

MO=ON Midpoint 9=ON 9+9=18

SQ=QT Midpoint 4=x

GH=HI Midpoint x+4=2x-6 -x -x 4=x-6 +6 +6 10=x

True XZ=ZY so Z is the midpoint

AE=CE Bisect 5=x

Acute, Scalene

a2+b2=c2 82+72=x2 64+49=x2 113=x2 √113=√x2 10.6in=x a2+b2=c2 x2+92=112 x2+81=121 -81 -81 x2=40 √x2=√40 x=6.3in

a2+b2=c2

82+112=142 64+121=196 185=196

False 185≠196

a2+b2=c2 82+152=172 64+225=289 289=289 True 289=289

a2+b2=c2

a2+b2=c2

a2+b2=c2

Scale Factor=2 x=2*5 x=10

3-4-5 Scale factor=2

3-4-5 Scale factor=9 x=9*5 x=45

P=8+6+2+18+6+24 P=64cm

P=7+5+7+12 P=31cm

a2+b2=c2 22+42=c2 4+16=c2 20=c2 √20=√c2 4.47=c

P=4.47+5+8+9+2 P=28.47cm

Area=(base)(height)

height

A=bh

A=bh Area=(base)(height)

height

base

base

bh 2 height

A=

base

(b+t)h 2

Area=

(base+top)(height) 2

top height

A=

base

(d1)(d2) Area= (diagonal 1)(diagonal 2) 2 2

di ag

a on

l2

d iu ra

ag

1

A=s2 A=side2

di

al

s

on

A=πr2 A=(π)(radius2)

side

A=

Rectangle A=bh A=(15cm)(3cm) A=45cm2

a2+b2=c2 42+h2=72 16+h2=49 -16 -16 2 h =33 √h2=√33 h=5.7 Triangle A=bh/2 A=(10cm)(5.7cm)/2 A=57cm2/2 A=28.5cm2

h=5.7

bh 2 8h 2*46= *2 2 92=8h 8 8 11.5=h A=

A=bh 72=9h 9 9 8in=h

B=bh B=6*5 B=30

V=πr2h V=π(62)10 V=π36*10 V=360π V≈1130.97in3

V=Bh V=30*8 V=240in3

4ft V=1/3πr2h V=1/3π(32)4 V=1/3π9*4 V=1/3(36π) V=13π V≈40.84in3

B=bh 3-4-5 right B=4*5 triangle scale B=20 factor 1. 1*4=4 V=1/3(Bh) V=1/3(20*9) V=1/3(180)=60cm2

B=1/2bh B=1/2(3*4) B=1/2(12) B=6 V=1/3(Bh) V=1/3(6*5) V=1/3(15)=5ft3

5m

a2+b2=c2 52+42=c2 25+16=c2 √41=√c2 6.4=c

4.6m

5m

15cm

15cm 2

6in

3in

10cm

8cm

9.54in

2

6cm

8cm

8cm

5cm

a +b =c 9.54in h2+32=1002 2 h +9=100 -9 -9 6in 6in 2 √h =√91 6in h=9.54 2

4m

8cm

10cm

8cm

15cm

8cm

4cm

4cm

4cm

4cm

3m

4.6m

10cm+5cm=15cm 6cm 6cm

4cm

5m

4m 3m

10cm-6cm=4cm

15cm

4m 3m

4cm 9.54in 6in

6in

51°+38°=89° m ABD=89°

x+33°=90° . x+33°=90° -33°-33° x=57°

x+2x=180° 3x=180° 3 3 x=60°

x 43°

x+43°=180° -43° -43° x=137° m DEF=137°

x=138°

Vertical Angle Theorem

x=75°


x+45°=2x+19° -x -x 45°=x+19° -19° -19° 26°=x


m SOT=m NOE


138°=m NOE

m HAE=m DAS


x+24°=2x-8° -x -x 24°=x-8° +8° +8° 32°=x m DAS=2x-8° m DAS=2(32)-8° m DAS=64-8° m DAS=56°

alternate exterior angles

vertical angles

Alternate interior, m 1=m 2

Alternate interior 46°=x

Alternate exterior 2x+19°=x+23° -x -x x+19°=23° -19°-19° x=4°

68° x

Corresponding m ACB=m EFC 68°=m EFC

x+24°

2x=164° 2 2 x=82° x-8°

m ACH=x+24° m ACH=82+24° m ACH=106°

Linear Pair m ACH+m DCH=180° (x+24°)+(x-8°)=180° x+24°+x-8°=180 2x+16°=180° -16° -16°

x+x+x=180° Triangle Sum Theorem 3x=180° 3 3 x=60°

x+x+30°=180° Triangle Sum Theorem 2x+30°=180° -30° -30° 2x=150° 2 2 x=75°

2x-12=60° equilateral triangle 2x-12=60° +12 +12° 2x=72° 2 2 x=36° 2x-27°=x+53° iscosceles triangle 2x-27°=x+53° -x -x x-27°=53° +27°+27° x=80°

Triangle Sum Theorem 24°+88°+x=180° 112°+x=180° -112° -112° x=68°

18° 94°

x

Triangle Sum Theorem 18°+94°+x=180° 112°+x=180° -112° -112° x=68° m A=68°

Triangle Sum Theorem x+2x+(x-16°)=180° x+2x+x-16°=180° 4x-16°=180° +16° +16° 4x=196° 4 4 x=49°

41°+87°+y=180°

y=_________

_______+x=180°

x=128°

m 1+m 2=m 3 114°+38°=x°

152°=x

m 1+m 2=m 3 29°+x=87° -29° -29° 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°

93°

x

48°

x 6x

x+71°

m 1+m 2=m 3 (x+71°)+x=6x

m m m m

Z=x+71° Z=(_____)+71° Z=______+71° Z=______

What could I use to find x? B1: 53º+m 1+x=180° triangle sum theorem

What could I use to find x? B1: x=m 7 corresponding What could I use to find m 7? B2: m 5=m 7 corresponding

What could I use to find x? B1: m 2+71°+x=180° triangle sum theorem What could I use to find m 2? B2: m 1=m 2 vertical angles What could I use to find m 1? B3: m 1+48°+53°=180° triangle sum theorem

How can I find x? B1: m 3=x alternate interior angles How can I find m 3? B2: m 3+36°+58°=180° triangle sum therorem

B2: m 3+36°+58°=180° m 3+94°=180° -94° -94° m 3=86° B1: m 3=x=180° 86°=x

s1+s2=s3 18+15>27 33>27 True! So... yes they do! s1+s2=s3 87+91>456 178>456 False! No triangle here!

s1+s2=s3 4+7>11 11>11 false! no triangle

s1+s2=s3 2+5>9 7>9 False! no triangle

max.=7+12=19 and the min. is 12-7=5 so 5
max.=4+3=7 min.=4-3=1 1in
ASS

BC EF given C F given AC DF given So ABC by SAS.

DEF

Vertical Angles Shared Line

CA BD given CAD ADB given AD AD shared line So CAD by SAS.

BDA

So PQR P T given by ASA. PR RT given PRQ SRT vertical

TSR

False.... the triangles may not be congruent.

False.... the triangles are congruent, but the parts don't correspond. They aren't the same part.

TV=WY CPCTC 13=x

True the triangles are congruent by SSS. So, M P because CPCTC.

C=m F CPCTC x+5=2x-7 -x -x 5=x-7 +7 +7 12=x

2 = 5

6 = x

2(x)=5(6) 2x=30 2 2 x=15

x 5 = 15 = 7 7(x)=5(15) 7x=75 7 7 x=10.71

M M same angle MON MQP corresponding MNO MPQ corresponding

∆MNO~∆MPQ by AAA or... ∆MNO~∆MPQ by AA

3 2 32 9 = 2= 4 16 4

16 = √16 = 4 25 √25 5 3

4 43 64 = 3= 5 5 125 3

3 8 √ 8 2 =3 = 27 √27 3

8=2x 2 2 4=x

x+10=2(x-6) x+10=2x-12 -x -x 10=x-12 +12 +12 22=x

Corresponding Angles Corresponding Angles x=71°

10=x

x+35=2x-21 -x -x 35=x-21 +21 +21 56°=x

x+9=2x+4 -x -x 9=x+4 -4 -4 5=x

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 p hypothesis

p hypothesis

q conclusion q conclusion q conclusion

If the ipod is on then it can play a song.

He has a tail.

90°
You can drive it to the store.

She lost her patients.

a 45° angle

A scalene triangle

A student

Anne Hathaway (or any bruenette or redhead)

30° and 40°

If an animal has a beak then it is a goose. -False, a robin has a beak.

If a triangle has 3 equal sides then it is equilateral. -true

If an animal is not a goose then it does not have a beak. -False, again, a robin has a beak.

If a triangle is not equilateral then it does not have 3 equal sides. -true.

If an animal does not have a beak then it is not a goose. -True!

If a triangle does not have 3 equal sides then it is not equilateral -true

If a figure is a rhombus then it is

a square.

If a figure is not a square then it is not a rhombus. If a figure is not a rhombus then it is not a square.

If two angles add to 90°then they are complementary. - True If two angles aren't complementary then they don't add to 90°. - True If two angles don't add to 90° then they aren't complementary. -true

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.

If a triangle has two congruent sides then it is isoceles. Iff a triangle is Isoceles then it has two congruent sides.

Statements F1: AB DE F2: AC DF F3. BC EF B1. ABC DEF

Reasons 1. Given 2. Given 3. Given 4. SSS

H

O A

H

A

cos 63= A H

O O

tan 56°= O A

H

17 x 17 cos 63°= x *x cos 63°=

9 x 9 x* tan 56°= x *x tan 56°=

A

A O H

O sin 39°= H

x sin 39°= 27

x 27* sin 39°= 27 *27

x(cos 63°)=17 cos 63° cos 63° 17 x= c0s 63° 17 x= 0.4540 x≈37.44 x(tan 56°)= 9 tan 56° tan 56° 9 x= tan 56° 9 x= 1.4826 x≈6.07

27(sin 39°)=x 27(0.6293)=x 16.99≈x

tanx°= O A

tanx°=

8 9

tanx°=0.8889

2 x=6 3 2 3 3 * 3 x=6* 2 2

x=9

tan-1* tanx°=0.8889 *tan-1 x=tan-10.8889

2nd

tan

sinx°=

=

O H

sinx°=

x≈41.63°

9 13

-1 sin-1* sinx°=0.6923 *sin

x=sin-10.6923

x≈43.81°

Tanx=

8 6

x=6 y=6√2 c=a√2

a

a

b=a

b=a

x=3√2 c=a√2 a

b=a

c=a√2

y=3√2√2 y=3√4 y=3*2 y=6

x=4 y=4

b=a√3

c=2a

b=a√3

c=2a

a

c=2a 4=2a 2 2 2=a y=2

a

c=2a

b=a√3

a

c=2a c=2*2√3 c=4√3 so.. y=4√3 b=a√3 b=2√3*√3 b=2√9 so.. b=2*3 b=6 so... x=6

c=2a

b=a√3

a x=2a

x=25√3 3 x= 10√3 3

b=a√3 5=a√3 √3 √3 5 =a √3 5 *√3 √3 *√3

5√3 √9

5√3 =a 3 y=

5√3 3

Rectangle

A

B

D

C

Rhombus

True False

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°=120° 6 6 n 6 (n-2)180=2,340° (n-2)180=2,340° 180 180 n-2=13 +2 +2 n=15 sides

n=5

94°+78°+156°+91°+x=540° 419°+x=540° -419° -419° x=121°

n=6

(n-2)180° (5-2)180 3*180 540°

(n-2)180° (6-2)180 4*180 720°

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)180° n (5-2)180 5 108°

2x+8=108 -8 -8 2x=100 2 2 x=50

n=6 (n-2)180° (6-2)180 4*180 720° x+81+4x-10+x+35+5x-60+3x+23+x+21=720°

15x+90=720° -90 -90° 15x=630° 15 15 x=42°

n=8 360° n

360°=45° 8

360° =18° n

n*

360° =18° *n 360°=18°n 360°=18°n 20=n so... n 18° 18° 20 sides

2x-15

21°+27°+58°+42°+53°+34°+45°+x=360° 280°+x=360° -280° -280° x=80°

2x-15°=45° +15° +15° 2x=60° 2 2 x=30°

n=8 360° n 360° 8 45°

x=2cm because all radii in a circle are equal.

GI=4

JK=5

C=∏D C=7∏≈21.99

C=2∏r C=2∏9 C=18∏≈56.55

mDEF+mDE=360° mDEF+127°=360° -127°-127° mDEF= 233°

m B=mAC x=35°

2(m B)=mAC 2(75°)=mAC 150°=mAC m D=mAC x=150°

2(m B)=mAC 2x=44° 2 2 x=22°

m D=mAB 86°=mAB x=1/2mAB x=1/2(86°) x=43°

mIJK+mKLI=360° 190°+mKLI=360° -190° -190° mKLI=170° m J=1/2mKLI x=1/2(170°) x=85°

m R=1/2mQS m 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°

3x=6*2 3x=12 3 3 x=4

AB=AC 8=x

2(x+4)=1(x+10) 2x+8=x+5 -x -x x+8=10 -8 -8 x=2

EF=EG x+3=2x-5 -x -x 3=x-5 +5 +5 8=x

10=x

17°+x=90° -17° -17° x=73°

3-4-5 right triangle x=10

134°/2=mFG 67°=mFG

3-4-5 right triangle scale factor =2 4*2=8 x=8

90°+48°+x=180° 138°+x=180° -138° -138° x=42°

m DCE+m ACD=90° 35°+m ACD=90° -35° -35° m ACD=55°

A (5,-6) B(-3,6) x1 y 1 x2 y 2 x1+x2 y1+y2 2 2 5+(-3) -6+6 2 2 2 12 2 2 (1,6)=MP

A (5,-6) B(x2,y2) C(4,3) x1 y 1 xmy m x1+x2 2 =xm 5+x2 2 =4

y1+y2 =ym 2

-6+y2 =3 2

-6+y2 5+x2 *2 2* =4 =3 *2 2 2 -6+y2=6 5+x2=8 +6 +6 -5 -5 y =12 x2=3 2 (3,12)=B

2*

A (5,7) B(-6,-5) x1 y 1 x2 y 2 D=√(x2-x1)2+(y2-y1)2 D=√(-6-5)2+(7-(-5))2

D=√(-11)2+(12)2 D=√121+144 D=√265 D≈16.78

A' B'

C'

F'

E' D'

I'

H' G'

L' K' J'

N' M' O'

P"

P' R'

Q" Q'

R"

Let's Practice... Answers...

Ray AB,

Segment EF,

AB

Line CD,

Point G,

EF

CD

G

Ray NI,

Plane HAT

Ray ON,

ON

Line ME,

I

Plane COW

ME

Line WE,

Point U,

Point I,

WE

Line KJ,

Segment HI,

Ray BO,

U

KJ

NI

HI

BO

Plane OLE

Segment XY, Segment LM,

LM

XY

ABC or

JIK or

TRQ or

ACB or

CBA

KIJ

QRT

BCA

LKJ or

JKL

UYT or

TYU

EFG or

GFE or

PNO or

XVW or

IGH,

F

ONP

WVX

HGI or

G

QPR or

RPQ

BGC or

CGB

NPO or HMK or

OPN

KMH

RTU or

UTR VZX or

EHI or ABD or

IHE

DBA

KOM or

DCB or

XZV

MOK

BCD

TSU or

UST

VYX or

XYV

OPJ or

JPO

VZY or

YZV

C

F

K

E

J

G

L

B

A

D

I

H

AB CD

JK LM

NR OR PR QR

AC DB, DE EB

RSE USL RSU ESL RS SL

RT WU RTS UWV TRS WUV

EH FG

H

T

SW UW, VW TW

GL FK, GIH FJH

QN OP, NO PQ

AC DB DE EB AE EC

FG HG LM JI FGL HGJ GT ET AT RT AG RE

WR OK WO OR WKO RKO

C A R

GO DL, GD OL

NO QP Q O

O B N

ST TQ OU NP NP SQ

AB BC CD DE EA

GM FH GM LK FH LK

RT VT ST UT VRU RUV UST RST

AE=CE Bisect 5=x

FG=GH Midpoint 12=x

IK=KL Bisect 7=x

OS=SQ Bisect 13=x

VW=WX Midpoint 11=WX 11+11=22

ST=TU Midpoint 19=TU 19+19=38

PQ=QS Bisect x=9

XY=YZ Midpoint 18=YZ 18+18=36

DH=HF Bisect x+4=2x-8 -x -x 4=x-8 +8 +8 12=x

AB=BC Midpoint x-5=2x-12 -x -x -4=x-12 +12 +12 8=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

GH=HI Midpoint 4x+11=2x+19 -2x -2x 2x+11=19 -11-11 2x=8 2 2 x=4

QR=RS Midpoint x+7=2x-10 -x -x 7=x-10 +10 +10 17=x

GH=HI Midpoint 3x-2=x+8 -x -x 2x-2=8 +2+2 2x=10 2 2 x=5

JM=ML Bisect x+5=2x-7 -x -x 5=x-7 +7 +7 12=x

Acute, Scalene

Acute, Scalene

Equilateral

Obtuse, Scalene

Acute, Isosceles

Right, Scalene

Right, Scalene Acute, Isosceles

Acute, Scalene

Obtuse, Isosceles

Right, Isosceles

Obtuse, Scalene

Acute, Scalene

Equilateral

a2+b2=c2 72+32=x2 49+9=x2 58=x2 √58=√x2 7.6ft=x

a2+b2=c2 102+102=x2 100+100=x2 200=x2 √200=√x2 14.1cm=x

a +b =c x +902=982 x2+8100=9604 -8100 -8100 x2=1504 √x2=√1504 x=38.8yd 2

2

2

2

a2+b2=c2 x2+62=152 x2+36=225 -36 -36 x2=189 √x2=√189 x=13.7mm

a2+b2=c2 x2+262=382 x2+676=1444 -676 -676 x2=768 √x2=√768 x=27.7in

a2+b2=c2 232+x2=452 529+x2=2025 -529 -529 2 x =1496 √x2=√1496 x=38.7m

a2+b2=c2 192+442=x2 361+1936=x2 2297=x2 √2297=√x2 47.9m=x

a2+b2=c2 x2+62=8.52 x2+36=72.25 -36 -36 x2=36.25 √x2=√36.25 x=6in a2+b2=c2 132+922=x2 169+8464=x2 8633=x2 √8633=√x2 92.9mi=x

a2+b2=c2 542+1132=x2 2916+12769=x2 25583=x2 √25685=√x2 125.2in=x a2+b2=c2 32+42=x2 9+16=x2 25=x2 √25=√x2 5ft=x

a2+b2=c2 x2+312=672 x2+961=4489 -961 -961 x2=3528 √x2=√3528 x=59.4mm

x

16in

a2+b2=c2 162+122=x2 256+144=x2 400=x2 √400=√x2 20in=x

45m

17in

a2+b2=c2 x2+152=452 x2+225=2025 -225 -225 x2=1800 √x2=√1800 x=42.4mm

15m

79cm

x

23in

12in

x

19ft

x

6ft

x

55cm

a2+b2=c2 232+172=x2 529+289=x2 818=x2 √818=√x2 28.6in=x

a2+b2=c2 792+552=x2 6241+3025=x2 9266=x2 √9266=√x2 96.3cm=x

a2+b2=c2 x2+62=192 x2+36=361 -36 -36 x2=325 √x2=√325 x=18ft

a2+b2=c2 x2+212=372 x2+441=1369 -441 37mm -441 2 x =928 √x2=√928 x=30.5mm

21mm

x

a2+b2=c2 202+212=292 400+441=841 841=841

a2+b2=c2 122+162=352 144+256=1,225 400=1,225 False 400≠1225

True 841=841

a2+b2=c2 82+92=132 64+81=169 145=169 False 145≠169

a2+b2=c2 152+342=482 225+1,156=2,304 1,381=2,304 False 1,381≠2,304

a2+b2=c2 282+452=532 784+2,025=2,809 2,809=2,809

a2+b2=c2 1192+1202=1692 14,161+14,400=28,561 28,561=28,561

True 2,809=2,809

True 28,561=28,561

a2+b2=c2 622+732=942 3,844+5,329=8,836 9,173=8,836 False 9,173≠8,836

a2+b2=c2 202+992=1012 400+9,801=1,0201 10,201=10,201 True 10,201=10,201

a2+b2=c2 112+632=882 121+3,969=7,744 4,090=7,744 False 4,090≠7,744

a2+b2=c2 312+452=862 961+2,025=7,396 2,986=7,396 False 2,986≠7,396

a2+b2=c2 462+652=802 2,116+4,225=6,400 6,341=6,400 False 6,341≠6,400

a2+b2=c2 92+402=412 81+1,600=1,681 1,681=1,681 True

1,681,=1,681

a2+b2=c2 122+302=432 144+900=1,849 1,044=1,849 False 1,044≠1,849

a2+b2=c2 842+1872=2052 7,056+34,969=42,025 42,025=42,025 True 42,025=42,025

a2+b2=c2 282+322=602 784+1,024=3,600 1,808=3,600 False 1,808,≠3,600

a2+b2=c2 162+522=822 256+2,704=6,724 2,960=6,724 False 2,960≠6,724


3-4-5 Scale factor=2 5-12-13 Scale factor=2




8-15-17

7-24-25

Scale factor=3

Scale factor=2

3-4-5

5-12-13

Scale factor=11

Scale factor=11

3-4-5

3-4-5

Scale factor=7

Scale factor=3

8-15-17

3-4-5

Scale factor=2

Scale factor=5



5-12-13

3-4-5

Scale factor=3

Scale factor=8

3-4-5 Scale factor=9 x=9*5 x=45 3-4-5 Scale factor=8 x=8*4 x=32


5-12-13 Scale factor=1 x=1*13 x=13

3-4-5

3-4-5

Scale factor=2 x=2*5 x=10


8-15-17

5-12-13



3-4-5 8-15-17 Scale factor=2 x=2*8 x=16


3-4-5

7-24-25






7-24-25 Scale factor=3 x=3*7 x=21

3-4-5 7-24-25 Scale factor=1 x=1*7 x=7

5-12-13 Scale factor=4 x=4*13 x=52

3-4-5 Scale factor=11 x=11*4 x=44




8-15-17 8-15-17 Scale factor=2 x=2*17 x=34


a2+b2=c2 92+h2=102 81+h2=100 -81 -81 2 h =19 √h2=√19 h=4.4

3-4-5 Scale factor=2 h=2*4 h=8

a2+b2=c2 92+h2=142 81+h2=196 -81 -81 2 h =115 √h2=√115 h=10.7



a2+b2=c2 172+h2=272 289+h2=729 -289 -289 2 h =440 √h2=√440 h=21

5-12-13 Scale factor=2 s=2*13 s=26


3-4-5 Scale factor=7 s=7*5 s=35

8-15-17 Scale factor=1 h=1*15 h=15

7-24-25 Scale factor=1 s=1*25 s=25

a2+b2=c2 92+82=s2 81+64=s2 145=s2 √145=√s2 s=12.

11+21+11+21=64

38+17+39+52=146

36+19+27=82

426+449+236=1108

13+19+13+19=64

14+2+5+19+5+4+14+25=88

7+17+11+9=44

7*10=70

6.4 12

3-4-5 scale factor=4 20

19 37-18=19

19+37+7+18+12+19=112

12 10

10

3-4-5 scale factor=2

paralellogram

a2+b2=c2 42+52=x2 16+25=x2 41=x2 √41=√x2 6.4=x 6.4+9+7+13+4=39.4

19-7

10+12+10+12=44

16+20+12=48

(96+4)-45 38

55

8 8-15-17 scale factor=1 19+23+17+8+29=96

(37-6)+7 38+45+7+55+37+96+6+4=288

8-15-17 scale factor=2 34 5.7


17

25

4 16

24-8

13-9

13+25+7+4+16+9+8=82

17 8-15-17 scale factor=1 10+17+10+17=54

24/2

32

(12) 13

13

8

32

32

5-12-13 scale factor=1 13+24+13=50

square

32*4=128

10-2 (parallelogram)

34+33.1+16+14=97.1

a2+b2=c2 a2+72=92 a2+49=81 -49 -49 a2=32 √a2=√32 a=5.7 5.7+9+4+11+10=39.7

16 15

5


3-4-5 scale factor=3 15+28+15+28=86

5+3+7+6+4=25


13 17.7

34

a2+b2=c2 a2+302=412 a2+900=1681 -900 -900 a2=781 √a2=√781 a=27.9 27.9+41+34+16=118.9

27.9

15

17.7

a2+b2=c2 132+122=c2 169+144=c2 313=c2 √313=√c2 17.7=c 17.7+17.7+26=61.4

7 25 7-24-25 scale factor=1 25+27+24+20=96

5

7.8

a2+b2=c2 52+62=c2 25+36=c2 61=c2 √61=√c2 7.8=c

12+25+7+19+7.8=70.8

14 11

11

12*4=48in

14 11+14+11+14=50ft

12

40

32

27*4=108mm

32+40+24=96m

24

27


17 9

9 17 9+17+9+17=52ft

4

9 8.1

4+8.1+9=21.1in

a2+b2=c2 a2+42=92 a2+16=81 -16 -16 a2=65 √a2=√65 a=8.1

11

24*4=96mm

8.5

8.5 11

24

11+8.5+11+8.5=39in

19

25

7

6

6

24

19


6+19+6+19=50in

7+24+25=56ft

10

9 8

8 10 8+10+8+10=36in

15 12 3-4-5 scale factor=3 9+15+12=36ft


Triangle A=bh/2 A=(9cm)(17cm)/2 A=153cm2/2 A=76.5cm2

Parallelogram A=bh A=(11cm)(7cm) A=77cm2

Parallelogram A=bh A=(5in)(4in) A=20in2

Trapezoid A=(b+t)h/2 A=(6cm+15cm)(13cm)/2 A=(21cm)(13cm)/2 A=273cm2/2 A=136.5cm2

Trapezoid A=(b+t)h/2 A=(16cm+10cm)(9cm)/2 A=(26cm)(9cm)/2 A=234cm2/2 A=117cm2



Triangle A=bh/2 A=(13m)(24m)/2 A=312m2/2 A=156m2 Circle A=πr2 A=π(8cm)2 A=64πcm2 A=201cm2

Trapezoid A=(b+t)h/2 A=(23ft+16ft)(10ft)/2 A=(39ft)(10ft)/2 A=390ft2/2 A=195ft2

Trapezoid A=(b+t)h/2 A=(11cm+7cm)(13cm)/2 A=(18cm)(13cm)/2 A=234cm2/2 A=117cm2

Triangle A=bh/2 A=(12m)(7m)/2 A=84m2/2 A=42m2

Rhombus A=(d1)(d2)/2 A=(10yd)(6yd)/2 A=60yd2/2 A=30yd2

Trapezoid A=(b+t)h/2 A=(31mm+27mm)(13mm)/2 A=(58mm)(13mm)/2 A=754mm2/2 A=377mm2

Circle A=πr2 A=π(14cm)2 A=196πcm2 A=615.75cm2

Square A=s2 A=(16ft)2 A=256ft2

Trapezoid A=(b+t)h/2 A=(17mi+13mi)(6mi)/2 A=(30mi)(6mi)/2 A=180mi2/2 A=90mi2

h=4 a2+b2=c2 42+h2=72 16+h2=49 -16 -16 2 h =33 √h2=√33 h=5.7

h=4 h=5.7

Triangle A=bh/2 A=(10cm)(5.7cm)/2 A=57cm2/2 A=28.5cm2

3-4-5 rt. triangle


h=7.1

h=4.5

a2+b2=c2 42+h2=62 16+h2=36 -16 -16 2 h =20 √h2=√20 h=4.5 Trapezoid A=(b+t)h/2 A=(21cm+13cm)(4.5cm)/2 A=(34cm)(4.5cm)/2 A=153cm2/2 A=76.5cm2

h=13.4

a2+b2=c2 72+h2=102 49+h2=100 -49 -49 2 h =51 √h2=√51 h=7.1

Trapezoid A=(b+t)h/2 A=(23in+15in)(7.1in)/2 A=(38in)(7.1in)/2 A=269.8in2/2 A=134.9in2

a2+b2=c2 122+h2=182 144+h2=324 -144 -144 2 h =180 √h2=√180 h=13.4

Triangle A=bh/2 A=(24m)(13.4m)/2 A=321.6m2/2 A=160.8m2

h=5.7

3-4-5 rt triangle h=4

h=4cm

Rhombus a2+b2=c2 A=(d1)(d2)/2 72+h2=92 A=(10cm)(6cm)/2 49+h2=81 2 A=60cm /2 -49 -49 2 2 A=30cm h =32 √h2=√32 h=5.7

Parallelogram A=bh A=(18ft)(5.7ft) A=102.6ft2

3-4-5 rt triangle h=24

r=17

Find the radius r. 34=2r 2 2 17=r

3.9

Circle A=πr2 A=π(17cm)2 A=289πcm2 A=907.9cm2 a2+b2=c2 72+h2=82 49+h2=64 -49 -49 2 h =17 √h2=√15 h=3.9

4*6=24 h=24mi

Trapezoid A=(b+t)h/2 A=(54mi+34mi)(24mi)/2 A=(88mi)(24mi)/2 A=2112mi2/2 A=1056mi2

h=8

Trapezoid A=(b+t)h/2 A=(8.9mm+4mm)(7mm)/2 A=(12.9mm)(7mm)/2 A=90.3mm2/2 A=45.2mm2

3-4-5 rt triangle 4*2=8 h=8yd Rhombus A=(d1)(d2)/2 A=(16yd)(12yd)/2 A=192yd2/2 A=96yd2

bh 2 8h 2*46= *2 2 92=8h 8 8 11.5=h A=

A=bh 72=9h 9 9 8in=h

A=bh 15=5h 5 5 3in=h

(d1)(d2) 2 (18)(d ) 2*94= 2 2 *2 188=(18)(d2) 18 18 10.4ft=d2 A=

(b+t)h 2 (13+t)4 87= 2 A=

87= (13+t)2 2 2 43.5=13+t -13 -13 30.5m=t

bh 2 6h 2*32= *2 2 64=6h 6 6 10.7=h A=

(b+t)h 2 (11+12)h 68= 2 (23)h 2*68= *2 2 A=

bh 2 2h 2*8= 2 *2 16=2h 2 2 8in=h A=

136=23h 23 23 5.9=h

A=πr2 64=πr2 π π √20.4=√r2 4.5ft=r

bh 2 4h 2*12= *2 2 24=4h 4 4 6mi=h A=

(d1)(d2) 2 (14)(d ) 2*31= 2 2 *2 62=(14)(d2) 14 14 4.4ft=d2 A=

(b+t)h 2 (3+7)h 19= 2 (10)h 2*19= *2 2 A=

38=10h 10 10 3.8in=h

(b+t)h 11.8=b+9 2 -9 -9 (b+9)11 2.8mm=b 2*65= *2 2 130=(b+9)11 11 11 A=

bh 2 26b 2*98= *2 2 196=26b 26 26 7.5in=b A=

A=πr2 121=πr2 π π √38.5=√r2 6.2in=r

A=s2 169=s2 √169=√s2 13m=s

bh 7*6 42ft2

A=

bh 3*4 12ft2

+

8cm A=42+12=54ft2

3 +

A=

3ft

2

8

7 A=

bh 7*3 21cm2

+

6 + 3

A=

bh A=21+16=37cm2 2*8 16cm2

4

7

13m

bh A= 17*10 + 170cm2 1/2bh 1/2(4*5.74) 1/2(22.96) 11.48in2

4in

A=170+11.48=181.48in2

7

10 +

5.74

17

13

4

2

2

14

A= 18

a +b =c 42+b2=72 16+b2=49 bh A= 18*39 -16 -16 2 702m2 √b =√33 b=5.74 2

A=

14m

39

1/2bh 1/2(13*14) 1/2(182) 91m2

10 + 6

A= 10

6in 3cm

10 12

1/2bh bh A= 12*10 - 1/2(3*4) 120cm2 1/2(12) 6cm2

4

4in

4cm A=

A=702-91=611m2

3 4 A=120-6=114cm2

bh A= 10*10 100in2

1/2bh

+ 1/2(6*4) 1/2(24) 12in2

A=100+12=112in2

13 +

A=

8

13

A= 8

A=

bh 13*13 169in2

+

bh 8*7 56in2

10

bh A= 10*10 100in2

-

-

-

32 -

19

4

2

A=

15 bh 19*7 133in2

A=133-60=73in2

60in2

818

bh A=100-(12.57*4) A= 18*8 πr π22 *4 A=100-50.28 144in2 2 2 A=59.72in 12.57in

1/2bh 1/2(4*6) 1/2(24) *2 12ft2

bh

- 4*15

A=

2

-

1/2bh 1/2(4*4) 1/2(16) *4 8in2

A=144-(8*4) A=144-32 A=112in2

8-

A=

16 -

A=

bh A= 32*16 512ft2

A=169+56=225in

-

10-

A=

7

7

13 A=512-(12*2) A=512-24 A=488ft2

bh A= 13*8 104cm2

1/2bh 1/2(3*4) 1/2(12) *4 6cm2

A=6-(6*4) A=104-24 A=80cm2

7cm

23cm 5 - 1

A=

A=

bh 5*5 25cm2

5

4cm

2

bh A= 23*15

A=25-4=21cm2

bh

6-

7

23

bh

- 1*4

15 -

A=

4

bh

- 7*6 - 5*2

2 2 345cm2 42cm 10cm

5

2

A=345-42-10 A=293cm2

4 5

4

12 - 5

A=

6in -

21

bh A= 21*12 252cm2

9in

6

3

1/2bh bh A=252-10-18 1/2(5*4)- 6*3 2 A=224cm2 10cm 18cm2

4in

4in A=

21

bh A= 21*16

336in2

16 - 6

1/2bh

bh

12in

80in2

4

10

8

A=336-12-80

- 1/2(6*4)- 8*10 A=244in2 2

8in

6cm 3-4-5 rt triangle 9in 12 -

A=

20

9

12-

4

8

A=

9 15

-

bh 1/2bh 1/2bh A=240-54-16 bh bh bh A= 20*12 - 1/2(9*12) - 1/2(4*8)A=170in2 A= 15*9 - 4*5 - 6*3 2 2 240in2 54in2 16in2 135cm2 20cm 18cm

4

5-

6

3

A=135-20-18 A=97cm2

B=bh B=4*6 B=24

V=Bh V=24*7 V=168in3

V=r2h V=π(22)9 V=π4*9 V=36π V≈113.10in3

B=1/2bh B=1/2(5*8) B=1/2(40) B=20 V=Bh V=20*6 V=120yd3

V=Bh V=86*17 V=1,462in3

B=bh B=10*9 B=90 V=Bh V=90*15 V=1,350ft3

B=1/2bh B=1/2(11*13) B=1/2(143) B=71.5 V=Bh V=71.5*21 V=1,501.5m3

B=1/2bh B=1/2(19*6) B=1/2(114) B=57 V=Bh V=57*17 V=969yd3

V=r2h V=π(92)18 V=π81*18 V=1458π V≈4580.44cm3

7.14m

4m a2+b2=c2 72+b2=102 49+b2=100 -49 -49 √b2=√51 b=7.14

B=bh B=7*4 B=28 V=Bh V=28*2 V=56m3

8cm 6cm 3-4-5 right triangle scale factor 2. 3*2=6

B=1/2bh B=1/2(6*8) B=1/2(48) B=24 V=Bh V=24*7 V=168cm3

V=πr2h D=2r 18=2r 2 2 9=r

8.49cm B=S2 B=82 B=64 V=Bh V=64*23 V=1,472mm3

B=1/2bh B=1/2(7.14*7) B=1/2(49.98) B=24.99 V=Bh V=24.99*5 V=124.95ft3

V=π(92)6 V=π81*6 V=486π V≈1,526.81yd3

a2+b2=c2 72+b2=112 49+b2=121 -49 -49 √b2=√72 b=8.49 B=1/2bh B=1/2(7*8.49) B=1/2(59.43) B=29.72 V=Bh V=29.72*11 V=326.92in3

B=bh B=8*16 B=128 V=Bh V=128*9 V=1,152ft3 11cm D=2r 22=2r 2 2 11=r

V=πr2h V=π(112)20 V=π121*20 V=2420π V≈7,602.65in3

9cm

B=1/2bh B=1/2(9*12) B=1/2(108) B=54 V=Bh V=54*13 V=702ft3

3-4-5 right triangle scale factor 3. 3*3=9

4.73m

sinx=

O h

O 6* sin38°= 6 *6 6(sin38°)=O 6(.6257)=O 3.69=O

B=1/2bh B=1/2(4*4√3) B=1/2(27.71) B=13.86 V=Bh V=13.86*8 V=110.88cm3 30-60-90 triangle . 4, 4√3

a2+b2=c2 3.692+b2=62 13.62+b2=36 -13.62 -13.62 2 √b =√22.38 b=4.73 3.69m

B=1/2bh B=1/2(4.73*3.69) B=1/2(17.45) B=8.73 V=Bh V=8.73*10 V=87.3m3

B=bh B=16*31 B=496 V=Bh V=496*84 V=41,664in3

B=1/2bh B=1/2(8*6) B=1/2(48) B=24 V=Bh V=24*9 V=216ft3

B=bh B=12*7 B=84 V=Bh V=84*11 V=924m3

V=πr2h V=π(18 )21 V=π324*21 V=6804π V≈21,375.4in3 2

B=bh B=23*28 B=644 V=Bh V=644*11 V=7,084cm3

B=S2 B=212 B=441 V=Bh V=441*21 V=9,261yd3

V=Bh V=45*13 V=585mm3

B=S2 B=92 B=81 V=Bh V=81*10 V=810yd3


B=1/2bh B=1/2(6*8) B=1/2(48) B=24 V=Bh V=24*7 V=168ft3

B=bh B=3*6 B=18 V=1/3(Bh) V=1/3(18*8) V=1/3(144)=48cm2

V=1/3πr2h V=1/3π(52)11 V=1/3π25*11 V=1/3(275π) V=91.67π V≈287.99in3

B=1/2bh B=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

B=s2 B=32 B=9 V=1/3(Bh) V=1/3(9*4) V=1/3(36) V=12ft2

B=1/2bh B=1/2(7*9) B=1/2(63) B=31.5 V=1/3(Bh) V=1/3(31.5*8) V=1/3(252) V=84ft3

B=s2 B=62 B=36 V=1/3(Bh) V=1/3(36*10) V=1/3(360) V=120ft2

V=1/3πr2h V=1/3π(122)24 V=1/3π144*24 V=1/3(3456π) V=1152π V≈3,619.11cm3

V=1/3(Bh) V=1/3(35*4) V=1/3(140) V=46.67ft2

3ft


6cm

4.9cm


a2+b2=c2 52+h2=72 25+h2=49 -25 -25 √h2=√24 h=4.9

B=1/2bh B=1/2(6*8) B=1/2(48) B=24 V=1/3(Bh) V=1/3(24*8) V=1/3(192) V=64cm3

B=s2 B=52 B=25 V=1/3(Bh) V=1/3(25*4.9) V=1/3(122.5) V=40.83cm2

5.67ft

B=1/2bh B=1/2(3*4) B=1/2(12) B=6 V=1/3(Bh) V=1/3(6*6) V=1/3(36) V=12ft3

a2+b2=c2 72+h2=92 49+h2=81 -49 -49 √h2=√32 h=5.67

d=2r 18=2r 2 2 9=r

6m d=2r 12=2r 2 2 6=r

B=bh B=7*8 B=56 V=1/3(Bh) V=1/3(56*5.67) V=1/3(317.52) V=105.84ft3

V=1/3πr2h V=1/3π(92)23 V=1/3π81*23 V=1/3(1,863π) V=621π V≈1,950.93in3

V=1/3πr2h V=1/3π(62)12 V=1/3π36*12 V=1/3(432π) V=144π V≈452.39m3

10.25in

a2+b2=c2 42+h2=112 16+h2=121 -16 -16 √h2=√105 h=10.25

B=bh B=8*9 B=72 V=1/3(Bh) V=1/3(72*10.25) V=1/3(738) V=246in3

6in 45-45-90 right triangle b=6

a2+b2=c2 62+h2=132 36+h2=169 11.53cm -36 -36 √h2=√133 h=11.53

B=bh B=7*10 B=70 V=1/3(Bh) V=1/3(70*11.53) V=1/3(807.1) V=269.3cm3

24.39m

a2+b2=c2 92+h2=262 81+h2=676 -81 -81 √h2=√595 h=24.39

B=s2 B=172 B=289 V=1/3(Bh) V=1/3(289*24.39) V=1/3(7048.71) V=2,349.57m3

8ft d=2r 16=2r 2 2 8=r

12

ft


B=1/2bh B=1/2(6*6) B=1/2(36) B=18 V=1/3(Bh) V=1/3(18*7) V=1/3(126) V=42in3

V=1/3πr2h V=1/3π(82)14 V=1/3π64*14 V=1/3(896π) V=144π V≈2,814.87m3

B=1/2bh B=1/2(9*12) B=1/2(108) B=54 V=1/3(Bh) V=1/3(54*11) V=1/3(594) V=198ft3

10.25cm

a2+b2=c2 82+h2=132 64+h2=169 -64 -64 √h2=√105 h=10.25

4cm

V=1/3πr h V=1/3π(42)14 V=1/3π16*14 V=1/3(224π) V=74.67π V≈234.58m3 2

d=2r 8=2r 2 2 4=r

8c m

a2+b2=c2 4.12cm 82+h2=92 64+h2=81 -64 -64 2 √h =√17 h=4.12

16=8 bisect 2

B=bh B=12*11 B=132 V=1/3(Bh) V=1/3(132*4.12) V=1/3(543.84) V=181.28cm3

a2+b2=c2 (4√2)2+h2=112 (16*2)+h2=121 32+h2=121 -32 -32 2 4√ √h =√89 2f t h=9.43 B=s2 B=82 B=64 V=1/3(Bh) V=1/3(64*9.43) V=1/3(603.52) V=201.17ft3 8√2=4√2 bisect 2 a2+b2=c2 42+h2=72 16+h2=49 -16 -16 √h2=√33 h=5.74 9.43ft

8in

8in

B=1/2bh B=1/2(8*8) B=1/2(64) B=32 V=1/3(Bh) V=1/3(32*8) V=1/3(256) V=85.33in3

45-45-90 right triangle a2+b2=c2 32+h2=62 5.2in 9+h2=36 -9 -9 √h2=√27 h=5.2

3in

45-45-90 right triangle

B=s2 B=(3√2)2 B=9*2 B=18 V=1/3(Bh) V=1/3(18*5.2) V=1/3(93.6) V=31.2in3

4√2in square

4√2in

45-45-90 right triangle

B=s2 B=(4√2)2 B=16*2 B=32 V=1/3(Bh) V=1/3(32*5.74) V=1/3(183.68) V=61.23in3

13.64cm

8.57in

tan65= h 6.36

h *6.36 6.36 6.36(tan65)=h 6.36(2.1445)=h 13.64≈h

6.36* tan65=

h=4.95√3 h≈8.57 4.95in

B=s2 B=(3√2)2 B=9*2 B=18 V=1/3(Bh) V=1/3(18*5.2) V=1/3(93.6) V=31.2in3

45-45-90 right triangle digonal =7√2

7√2=3.5√2≈4.95 2

6.36 45-45-90 right triangle diagonal =9√2

9√2=4.5√2≈6.36 2

B=s2 B=92 B=81 V=1/3(Bh) V=1/3(18*13.64) V=1/3(1104.84) V=368.28cm3

h 4.23m tan38= 5.41

5.1ft

h *5.41 6.36 5.41(tan38)=h 5.41(.7813)=h 4.23≈h

5.41* tan38= 5.66 45-45-90 right triangle diagonal =9√2 8√2=4√2≈5.66 2 tan42= h 5.66

B=s2 B=82 B=64 V=1/3(Bh) V=1/3(64*5.1) V=1/3(326.4) V=108.8ft3

h *5.66 6.36 5.66(tan42)=h 5.66(.9004)=h 5.1≈h

5.66* tan42=

5.41m

a2+b2=c2 62+92=d2 36+81=d2 √117=√d2 10.82=d 10.82=5.41 2

B=bh B=6*9 B=56 V=1/3(Bh) V=1/3(56*4.23) V=1/3(236.88) V=78.96m3

V=1/3πr2h V=1/3π(52)14 V=1/3π25*13 V=1/3(325π) V=108.33π V≈340.34ft3

B=bh B=9*8 B=72 V=1/3(Bh) V=1/3(72*11) V=1/3(792) V=264ft3

B=s2 B=72 B=49 V=1/3(Bh) V=1/3(49*2) V=1/3(98) V=32.67in3

B=1/2bh B=1/2(5*14) B=1/2(70) B=35 V=1/3(Bh) V=1/3(35*15) V=1/3(525) V=175cm3

5-12-13 right triangle scale factor 1 12*1=12

B=1/2bh B=1/2(5*12) B=1/2(60) B=30 V=1/3(Bh) V=1/3(30*11) V=1/3(330) V=110cm3

d=2r 12=2r 2 2 6=r V=1/3πr2h V=1/3π(62)14 V=1/3π36*12 V=1/3(432π) V=144π V≈1,021.02ft3

9in

9in

9in

6in

11in 9in

6in

11in

11in

d=2r 8=2r 2 2 4=r

6in

6in

11in

4cm

4cm

4cm

4cm 10m

3-4-5 right triangle scale factor=2 5*2=10

a2+b2=c2 a2+72=122 a2+49=144 -49 -49 √a2=√95 a=9.75

10m 6m

10m

8m

10m 8m

10m 6m

10m

8m

7ft 9.75ft

13ft

12ft

7ft

12ft

10 5√3 10in

5√3in 10in 5√3in 10in

10m

12ft

13ft

13ft 7ft

9.75ft 30-60-90 right triangle

6m

5in

8in

8in

5in

9.75ft

5√3in

5in 8in

10in

11-4=7

7cm

4+6=10

4cm 8cm

11cm 4cm

8cm

4cm 4cm 8cm

11cm 6cm 27-13=14

14ft

19ft

13ft

8ft 14ft 27ft

11ft

8ft 8ft

8ft

4in

4in

4in

13ft

8ft

7.75in

4in

8ft

8ft

8ft

7.75in

7.75in

a2+b2=c2 a2+22=82 a2+4=64 -4 -4 √a2=√60 a=7.75

14ft

19ft

13ft 8ft 14ft

4/2=2

6cm

27ft 19ft

11ft

8ft

8cm

4cm

4cm

27ft

7cm

8cm

11ft

19-11=8

7.75in

10cm

6cm

8cm

11cm

7.75in

10cm

4in 4in

18-8=10 12-5=7

18cm

7cm 5cm

7cm

6√2in

45-45-90 triangle 6√2 a2+b2=c2 3-4-5 rt. 2 2 2 triangle. scale a +3√2 =5 a2+18=25 factor =1 -18 -18 h=4 √a2=√7 a=2.65

10cm

6cm

12cm

7cm 6cm 18cm 6cm

4in

5cm

8cm

10cm

4in

18cm

8cm

6in 6in

6in 6in

6cm

6cm

8cm

6cm 5cm 10cm

2.65in

2in

6√2in

30in2 5in

30in2

5in

9in

6in

6in

6in

54in2

5in

45in2

5in

45in2

9in

9in

6in

54in2 9in

2

cm 0 8 4

8

cm

2

cm 4

2

8 3

16cm

24cm


24cm

12cm

96+96+384+288+480=1,344cm2

20cm

π52 5ft 78.54ft2

π52 5ft

78.54ft2

15ft

2π(5)(15) 471.24ft2 5ft

ft 2 8 2

16ft

8ft

8ft

ft 2 8 2

16ft

1

ft 2 8 2

16ft

1

8 2

8ft

8ft

8ft

64ft2 8ft

1

64ft2 8ft

ft 2

78.54+78.54+471.24=628.32ft2

1

20cm

96cm2 16cm 12cm 16cm

24cm

12cm

96cm2

20cm

8

20cm

2

30+30+45+45+54+54=258in2

8ft

64+64+128+128+128+128=640ft2

16ft

7in

3

5

in 2 0

8

in 2

0

8

16in

14in

in 2 9 4

7in

7in

112in2

7in

in 2

9in 8 1 in 2

in 2

5in

7in 7in

3

98in

2

7in

5in

6

in 2 3

5in

6

9in

9in

16in

7in

9in

16in

16-9=7

9in

9in 8 1 in 2

14-5=9

9in

81+80+81+80+35+63+49+98+112+63=742in2

8cm

8cm

64cm2 8cm

9.17cm

a2+b2=c2 a2+42=102 a2+16=100 -16 -16 √a2=√84 a=9.17

9.17cm

9.17cm

10/2=5

9.17cm

36.68cm2 36.68cm236.68cm236.68cm2

8cm 8cm 36.68+36.68+36.68+36.68 +64=210.72cm2

8cm

6/2=3 10m 32.24m2 a2+b2=c2 2 a2+32=92 6m 24m 2 a +9=81 8m -9 -9 2 8m √a =√72 8/2=4 25.47m2 10m 2 2 2 a=8.49 37.4m2 10/2=5 a +b =c a2+b2=c2 a2+42=92 a2+52=92 a2+16=81 10m 6m a2+25=81 -16-16 24+32.24+25.47+37.4=119.11m2 -25 -25 √a2=√65 √a2=√56 a=8.06 a=7.48 22-13=9 6in 66in2 11-4=7 11in 4in 4in 6in 78in2 7in 7in 9in 13in 6in 54in2 6in 42in2 6in 7in 6in 132in2 7in 4in 9in 22in 154+52+154+52+66+24+54+42+78+132=808in2

in 2

4

2

2

52in

22in

154in2

52in2

154in2

13in

13in

22in

8.49m

7.48m

8.06m


9/2=4.5

π4.52 4.5cm 63.62cm2

π4.52 4.5cm 63.62cm2

20ft

2

4ft

4.5ft2

8.65ft

8.65ft

18ft2 6ft

3ft

3ft

4.5+4.5+18+18+25.46=70.46ft2

6in

64in

2

2in2 1in 2in 7in

2in2 1in 2in 7in

8in

8in 60in2

64in2

6in

12in2

8in

8in 8in 7in 8in

2 6ft 18ft

2

42in2 6in

48in2

5ft 17.54+21.63+21.63+ 17.54+20=98.34cm2

3ft

10-2=8 7+1=8

17.54ft2

8in

6ft 25.46ft

5ft

4ft

21.63ft2

4.5ft2

3ft

3ft

21.63ft2

6in

6in

6in2

3ft 3√2ft

8in

4.5cm

17.54ft2

4ft

5ft 45-45-90 triangle 3

8.77ft

a2+b2=c2 a2+22=92 a2+4=81 -4 -4 √a2=√77 a=8.77

8.77ft

5/2=2.5 a2+b2=c2 a2+2.52=92 a2+6.25=81 -6.25-6.25 √a2=√74.75 a=8.65

4/2=2

3ft

311.02cm2

63.62+63.62+311.02=438.26cm2

4.5cm

3ft

11cm

2π(4.5)(11)

48in2

1in 6in

10in 2in 8in 64+2+64+2+6+42+60+12+46+48 =348in2

2

11cm 132cm2 11cm

1

1

1

1

0

cm

2

0

cm

11cm

10cm

10cm 132cm

110+110+132+132=484cm2

11cm

2

12cm

12cm

8m

4.47m

a2+b2=c2 a2+42=62 a2+16=36 -16 -16 √a2=√20 a=4.47

4.47m

4.47m

8/2=4

4.47m

2 17.88m2 17.88m2 17.88m 17.88m2

8m 8m 8m 17.88+17.88+17.88+17.88=71.52m2

2π(3)(5)

94.25cm2

5cm 94.25cm2

1

6

in 2

7in

4in

4in

5in 32in2 8in

28+20+12+12+16+32=120in2 4in

3in

4in

in 2

4in

2

20in2 4in

1

5+3=8

8in

4in

2

28in2

1

7-3=4

in 2

3cm

3in

4in

51°+38°=89° m ABD=89°

163°-44°=119° m CBD=119°

42°+35°=77° m ABD=77°

31°+43°=74° m ABD=74°

78°-21°=57° m CBD=57°

125°-36°=89° m CBD=89°

97°-48°=49° m CBD=49° 112°-29°=83° m ABD=83°

119°-67°=52° m CBD=52° 45°+49°=96° m ABD=94°

123°-67°=56° m CBD=56° 115°+65°=180° m ABC=180°

58°+37°+59°=154° m ABE=154°

61°+25°+15°=101° m ABE=101°

22°+28°+78°=128° m ABE=128°

25°+41°+36°=102° m ABE=102°

111°-31°-52°=28° m CBD=28°

136°-35°-42°=59° m ABC=59°

170°-21°-19°-23°=107° m CBD=107°

58°+48°+31°+43°=180° m 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°=51° m CBD=51°

Two angles whose measures add to 90°. Two angles whose measures add to 180°. Two angles who are supplementary and adjacent.

x+33°=90° .

y+134°=180°

x+33°=90° -33°-33° x=57°

-134°-134° y=46°

z+67°=180° -67° -67° z=113°

x+64°=90° -64°-64° x=26°

z+90°=180° -90° -90° z=90°

y+44°=180° -44° -44° y=136°

x+12°=90° -12° -12° x=78°

x+64°=180° -64° -64° x=116°

x+2x=180° 3x=180° 3 3 x=60°

(x+21°)+(x+58°)=180° x+21°+x+58°=180° 2x+79°=180° -79° -79° 2x=101° 2 2 x=50.5°

x+(3x+18°)=90° x+3x+18°=90° 4x+18°=90° -18°-18° 4x=72° 4 4 x=18°

(2x+8°)+(x-24°)=180° 2x+8°+x-24°=180° 3x-16°=180° +16° +16° 3x=196° 3 3 x=65.3°

131°+y=180° -131° -131° y=49° (x+15°)+(x-12°)=180° x+15°+x-12°=180° 2x+3°=180° -3° -3° 2x=177° 2 2 x=88.5°

49°+x=180° -49° -49° x=131° The value of x is the same as the measure of the angle on the other side of the intersecting lines, 131°.

x

x+43°=180° -43° -43° x=137° m DEF=137°

43°

x

143°

143°+x=180° linear pair -143° -143° x=37° m TRS=37°

59°+x=90° -59° -59° x=31°

x

m H=31° 59°

x 78°+x=180° -78° -78° x=102°

78°

m O=102°

x+27°

x

(x+27°)+x=180° linear pair x+27°+x=180° 2x+27°=180° -27° -27° 2x=153° 2 2 x=76.5° m ROK=76.5°

x

2x+11

(2x+11°)+x=180° linear pair 2x+11°+x=180° 3x+11°=180° -11° -11° 3x=169° 3 3 x=56.3° check your work

x=121°


x=28°


x=48°


x=37°

x=165°

x=90°




x+45°=2x+19° -x -x 45°=x+19° -19° -19° 26°=x

x+117°=3x+9° -x -x 117°=2x-9° +9° +9° 126°=2x 2 2 63°=x

x+91°=3x-37° -x -x 91°=2x-37° +37° +37° 128°=2x 2 2 64°=x




x+13°=2x-23° -x -x 13°=x-23° +23° +23° 36°=x


2x-17°=x+12° -x -x x-17°=12° +17° +17° x=29°


x+33°=2x-23° -x -x 33°=x-23° +23° +23° 56°=x


63°


m SOT=m NOE

138°

m EAM=m IAL 63°=m IAL


138°=m NOE

m TKE=m AKS


87°

147°


147°=m AET

87°=m AKS

51°

m IKS=m HKE


51°=m HKE

m GER=m AET

153° m DCE=m ACB


153°=m ACB

x+24°

2x-8°

m HAE=m DAS


x+24°=2x-8° -x -x 24°=x-8° +8° +8° 32°=x m DAS=2x-8° m DAS=2(32)-8° m DAS=64-8° m DAS=56°

m ABD=m CBE


2x-37°=x+65° m -x -x m x-37°=+65° m +37° +37° m x=102°

m GET=m AEN

3x-51°

2x-17°


3x-51°=2x-17° -2x -2x x-51°=-17° +51° +51° x=34° m GET=3x-51° m GET=3(34°)-51° m GET=102°-51° m GET=51°

x+65°

2x-37°

CBE=x+65° CBE=102°+65° CBE=102°+65° CBE=167°

2x+89° x+89° m BED=2X+89°

m BED=2(0°)+89° m BED=0°+89°

m BED=89°

m LEN=m BED


X+89°=2x+89° -x -x 89°=x+89° -89° -89° 0 °=x

m BOE=m NOL


5x+36°=3x+42° -3x -3x 2x+36°=42° 3x+42° 5x+36° -36°-36° 2x=6° 2 2 x=3 m NOL=3x+42° m NOL=3(3°)+42° m NOL=9°+42° m NOL=51°

4x-51° 3x° m PIT=m LIO


m LIO=3x m LIO=3(51°) m LIO=153°

4X-51°=3X° -3x -3x x-51°=0° +51° +51° x=51°

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; 6,7; 7,8 8,1

1,2; 3,4; 5,6; 7,8

1,2;2,3;3,4;4,5;5,6;6,7;7,8;8,1

1,2; 2,3; 3,4; 4,5; 5,6; 6,1 1,4; 2,5; 3,6

1,5; 2,6; 3,7; 4,8

linear pair

complementary

vertical angles

linear pair

adjacent angles

vertical angles

supplementary

complementary

linear pair supplementary

complementary 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

1, 5;

3, 7;

3, 6;

4, 5;

1, 8;

2, 7;

3, 5;

4, 6;

2, 6;

4, 8


consecutive angles

corresponding angles



alternate interior angles



consecutive angles


vertical angles

a linear pair, or supplementary

vertical angles





vertical angles

G

D

I

E

C

B

F

A

H

Corresponding, m 1=m 2

Alternate interior, m 1=m 2 Alternate interior, m 1=m 2

Corresponding, m 1=m 2 Alternate exterior, m 1=m 2

Alternate exterior, m 1=m 2 Corresponding, m 1=m 2

Consecutive, m 1+m 2=180°

Consecutive, m 1+m 2=180°

Consecutive x+68°=180° -68° -68° x=112° Alternate interior 46°=x

Alternate exterior x=37°

Corresponding 111°=x Alternate interior x=58°

Corresponding 175°=x

Corresponding x=26°

Corresponding

Alternate interior

x=31°

x=83°

Linear Pair

Vertical

x+113°=180° -113° -113° x=67°

x=126°

Alternate interior x=132°

Alternate exterior x=96°

Corresponding x=119°

Consecutive x+21°=180° -21° -21° x=159°

Corresponding Alternate exterior 2x+19°=x+23° -x -x x+19°=23° -19°-19° x=4°

2x-61°=x+3° -x -x x-61°=3° +61°+61° x=64°

Consecutive

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°

(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° Corresponding 3x-35°=2x+52° -2x -2x x-35°=52° +35° +35° x=87°

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°

Consecutive (x+23°)+(x+54°)=180° x+23°+x+54°=180° 2x+77°=180° -77° -77° 2x=103° 2 2 x=51.5

Corresponding

3x+71°=5x+37° -3x -3x 71°=2x+37° -37° -37° 34°=2x° 2 2 17°=x 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°

x 85°

68°

Corresponding

x

m ACB=m EFC 68°=m EFC

Alternate interior m DCH=m BFE 85°=m BFE

103°

x 91°

Alternate exterior

x

m HFG=m BCD 103°=m BCD

Consecutive m EFB+m DCH=180° 91°+x=180° -91° -91° x=89° m ACH=89°

x 47°

Vertical m GEH=m BEF 47°=m BCD

x

63° Linear Pair m DCH+m ACH=180° 103°+x=180° -103° -103° x=77 m ACH=77°

x+24°

Alternate Exterior

2x=164° 2 2 x=82° x-8°

m ACH=x+24° m ACH=82+24° m ACH=106°

2x-45°

x+23°

Linear Pair m ACH+m DCH=180° (x+24°)+(x-8°)=180° x+24°+x-8°=180 2x+16°=180° -16° -16°

Consecutive m GF+m DCH=180° (x+32°)+(x+24°)=180° x+32°+x+24°=180 2x+56=180° -56 -56 2x=124 2x=164° 2 2 x=62°

Corresponding

Corresponding

Alternate Interior

m BCD=m BFG x+26°=2x-24° -x -x 26°=x-24° +24° +24° 50°=x m BFG=2x-24° m BFG=2(50)-24° m BFG=100°-24° m BFG=76°

m ADF=m HEC 2x+4°=4x-14° -2x -2x 4°=2x-14° +14° +14° 18°=2x 2 2 9°=x m HEC=4x-14° m HEC=4(9)°-14° m HEC=36-14° m HEC=22°

4x-14°

2x-24°

m DCH=x+24° m DCH=62+24° m DCH=86°

2x+4°

2x-124°

x+26°

° 24 x+ 32° x+

m EFH=m ACH 2x-142°=x+16° -x -x x-142°=x-16° +142° +142° x=126° m ACH=x-16° m ACH=126°-16° m ACH=110°

x+16°

m ACB=m HFG 2x-45°=x+23° -x -x x-45°=23° +45°+45° x=68° m HFG=x+23° m HFG=68°+23° m HFG=91°

AB CD 1 and 2 are corresponding so m 1=m 2

AB is not false.

to CD 1 m 1=m 2 is

AB CD 1 and 2 are corresponding so m 1=m 2

AB CD 1 and 2 are Alternate exterior so m 1=m 2

AB is not to CD 1 so m 1=m 2 is false.

AB CD 1 and 2 are Alternate interior so m 1=m 2

AB is not to CD 1 and 2 are m 1=m 2 is false.

This is true because vertical angles are always equal.

AB is not to CD so m 1+m 2=180° is false.

AB CD, 1 and 2 are consecutive so m 1+m 2=180° is true.

1 and 2 are Corresponding if corresponding angles are equal then AB CD.

1 and 2 are Alternate exterior if Alternate exterior angles are equal then AB CD.

1 and 2 are Vertical angles. Vertical angles don't require parallel lines so we don't know if AB CD so probably false.

1 and 2 are Alternate interior if Alternate interior angles are equal then AB CD





Triangle Sum Theorem 60°+90°+x=180° 150°+x=180° -150° -150° x=30° Triangle Sum Theorem 25°+26°+x=180° 51°+x=180° -51° -51° x=129°




Triangle Sum Theorem 32°+x+x=180° 32°+2x=180° -32° -32° 2x=148° 2 2 x=74°

Triangle Sum Theorem 24°+x+x=180° 24°+2x=180° -24° -24° 2x=156° 2 2 x=78° Equilateral Triangle x=60°

Triangle Sum Theorem 25°+37°+x=180° 62°+x=180° -62° -62° x=118° Triangle Sum Theorem 46°+71°+x=180° 117°+x=180° -117° -117° x=63°


Triangle Sum Theorem 30°+x+x=180° 30°+2x=180° -30° -30° 2x=150° 2 2 x=75°

Triangle Sum Theorem 30°+90°+x=180° 120°+x=180° -120° -120° x=60° m G=60°

30°

18° 94°

x

x

Triangle Sum Theorem 18°+94°+x=180° 112°+x=180° -112° -112° x=68° m A=68°

x

Triangle Sum Theorem 138°+17°+x=180° 138° 155°+x=180° 17° -155° -155° x=25° m M=25°

47° x 43°

Triangle Sum Theorem 48°+x+x=180° 48° 48°+2x=180° -48° -48° 2x=132° 2 2 x x=66° m T=66°

x

x

x

24°

Triangle Sum Theorem 13°+24°+x=180° 37°+x=180° -37° -37° x=143° m M=143°

x

x

13° x

118°

Triangle Sum Theorem 47°+43°+x=180° 90°+x=180° -90° -90° x=90° m H=90°

Equilateral Triangle x=60° m O=60°m T=60° m P=60°

Triangle Sum Theorem 118°+26°+x=180° 144°+x=180° -144° -144° x=36° 26° m H=36°

Triangle Sum Theorem x+2x+(x-16°)=180° x+2x+x-16°=180° 4x-16°=180° +16° +16° 4x=196° 4 4 x=49°

Triangle Sum Theorem x+(x+43°)+(x-31°)=180° x+x+43°+x-31°=180° 3x+12°=180° -12° -12° 3x=168° 3 3 x=56°

Triangle Sum Theorem x+x+(x-27°)=180° x+x+x-27°=180° 3x-27°=180° +27° +27° 3x=207° 3 3 x=69°

Triangle Sum Theorem x+3x+(x+15°)=180° x+3x+x+15°=180° 5x+15°=180° -15° -15° 5x=165° 5 5 x=33°

Triangle Sum Theorem x+(x+55°)+(x+38°)=180° x+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°)=180° x+21°+x+23°+x+61°=180° 3x+105°=180° -105°-105° 3x=75° 3 3 x=25°

Triangle Sum Theorem x-23° 2x+x+(2x+30°)=180° 2x+x+2x+30°=180° 5x+30°=180° 2x 2x+30° -30° -30° x-17° 5x=150° 5 5 Triangle Sum Theorem x=30° 2x+(x-23°)+(x-17°)=180° m B=2x° 2x+x-23°+x-17°=180° m B=2(30°) 4x-40°=180° m B=60° +40° +40° 4x=220° 4 4 x=55° m S=2x° m S=2(55)° x+5° m S=110° 2x

x

Triangle Sum Theorem 90°+(x-7°)+(x+5°)=180° 90°+x-7°+x+5°=180° 2x+88°=180° -88° -88° 2x=92°m Z=46°-7° 2 2 m Z=39° x=46° m Z=x-7°

x-7°

x-22°

3x+19°

x-17°

Equilateral Triangle x=60° m R=60°m Q=60° m S=60°

x x x

Triangle Sum Theorem (x-22°)+(3x+19°)+(x-17°)=180° x-22°+3x+19°+x-17°=180° 5x-20°=180° m X=3x+19° +20° +20° m X=3(40°)+19° 5x=200° m X=120°+19° 5 5 m X=139° x=40°

m 1+m 2=m 3 114°+38°=x°

m 1+m 2=m 3

152°=x

54°+63°=x

117°=x

m 1+m 2=m 3

m 1+m 2=m 3

18°+42°=x

48°+68°=x

60°=x

116°=x

m 1+m 2=m 3 43°+106°=x 149°=x

m 1+m 2=m 3 60°+90°=x 150°=x

m 1+m 2=m 3 21°+138°=x 159°=x

m 1+m 2=m 3 35°+90°=x 125°=x

m 1+m 2=m 3 29°+x=87° -29° -29° 58° x=______

m 1+m 2=m 3 142°+x=167° -142° -142° x=25°

m 1+m 2=m 3 42°+x=77° -42° -42° x=35°

m 1+m 2=m 3 78°+x=131° -78° -78° x=53°

m 1+m 2=m 3 39°+x=106° -39° -39° x=67°

m 1+m 2=m 3 x+x=148° 2x=148° 2 2 x=74°

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°

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+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-6°)+x=148° x-6°+x=148° 2x-6°=148° +6° +6° 2x=154° 2 2 x=77°

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+34°)+(x+34°)=3x 2(x+34°)=3x 2x+68°=3x -2x -2x 68°=x

m 1+m 2=m 3 m 1+m 2=m 3 (x-16°)+(x+3°)=x+42° (x-19°)+(x-26°)=x x-19°+x-26°=x x-16°+x+3°=x+42° 2x-45°=x 2x-13°=x+42° -2x -2x -x -x -45°=-x x-13°=42° -1 -1 +13°+13° 45°=x x=55°

x 93°

35° 61°

x

48°

m 1+m 2=m 3 35°+61°=x 96°=x

m 1+m 2=m 3 93°+48°=x 141°=x

x 6x

x+71°

m 1+m 2=m 3 (x+71°)+x=6x x+71°+x=6x 2x+71°=6x -2x -2x 71°=4x 4 4 17.75°=x

m m m m

Z=x+71° 17.75 Z=(_____)+71° 17.75° Z=______+71° 78.75° Z=______

2x+19°

42° x°

m 1+m 2=m 3 x+42°=2x+19° -x -x 42°=x+19° -19° -19° m 23°=x m m m x°

x°

148°

60°

75°

105°

60°

x

m 1+m 2=m 3 60°+60°=x 120°=x CAS=2x+19° CAS=2(23°)+19° CAS=46°+19° CAS=65°

m 1+m 2=m 3 x+x=148° 2x=148° 2 2 x=74° m D=x m D=74°

x

60°

m ATS=x m ATS=120°

4x+9° 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 m m m

2x+51°

x°

HEP=4x+9° HEP=4(42°)+9° HEP=168°+9° HEP=177°

m 1+m 2=m m GRE+m ERA=180° 105°+x=3x+9° -x -x 75°+m ERA=180° 105°=2x+9° -75° -75° -9° -9° m ERA=105° 96°=2x m 2 2 m 3x+9° 48°=x m m

3

HEP=3x+9° HEP=3(48°)+9° HEP=144°+9° HEP=153°

*Note: There are many possible ways to solve some of these problems. What could I use to find x?

What could I use to find x?

B1: 53º+m 1+x=180° triangle sum theorem

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 sum theorem

What could I use to find x? B1: m 2+m 1+x=180° triangle sum theorem

What could I use to find x? B1: m 9=x corresponding angles



B1: m 1+x=180° linear pair

B1: 35º+m 7+x=180° triangle sum theorem

What could I use to find x? B1: x=m 7 corresponding What could I use to find m 7? B2: m 5=m 7 corresponding

What could I use to find x? B1: m 2+61°+x=180° triangle sum theorem What could I use to find m 2? B2: m 1=m 2 vertical angles

What could I use to find x? B1: m 2+m 3+x=180° triangle sum theorem What could I use to find m 3? B2: m 3=53° corresponding angles

What could I use to find x? B1: m 1+m 3=x remote exterior angle theorem What could I use to find m 3? B2: m 3+132°=180° linear pair

*Note: There are many possible ways to solve some of these problems.

What could I use to find x? B1: m 7+68°=x remote exterior angle theorem What could I use to find m 7? B2: m 7+35°=90° complementary

What could I use to find x? B1: m 2+m 5+x=180° angle addition postulate, straight angle What could I use to find m 5? B2: m 5+76°+18°=180° triangle sum theorem.

What could I use to find x? B1: m 2+71°+x=180° triangle sum theorem What could I use to find m 2? B2: m 1=m 2 vertical angles What could I use to find m 1? B3: m 1+48°+53°=180° triangle sum theorem

What could I use to find x? B1: m 9=x corresponding angles What 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° triangle sum theorem What could I use to find m 10? B2: m 10+=42° vertical angles What could I use to find m 13? B3: m 13=78° corresponding angles

What could I use to find x? B1: m 1+93°+x=180° triangle sum theorem What could I use to find m 1? B2: m 1+m 4+83°=180° angle addition postulate What could I use to find m 4? B3: m 4+132°+24°=180° triangle sum theorem

What could I use to find x? B1: m 1+82°=x remote exterior angle theorem What could I use to find m 1? B2: m 1+m 2+82°=180° triangle sum theorem What could I use to find m 2? B3: m 2=32° alternate interior angles

What could I use to find x? B1: m 1+31°=x=180° remote exterior angle theorem What could I use to find m 1? B2: m 1+142°=180° vertical angles

How can I find x? B1: m 3=x alternate interior angles How can I find m 3? B2: m 3+36°+58°=180° triangle sum therorem

How can I find x? B1: m 7+43°=x remote exterior angle theorem How can I find m 7? B2: m 7+68°=90° complementary angles triangle sum therorem

B2: m 3+36°+58°=180° m 3+94°=180° -94° -94° m 3=86° B1: m 3=x=180° 86°=x

B2: m 7+68°=90° -68°-68° m 7=22° B1: m 7+43°=x 22°+43°=x 65°=x

How can I find x? B1: m 3=x vertical angles How can I find m 8? B2: m 3+39°=180° linear pair

B2: m 3+39°=180° -39 -39° m 3=141

How can I find x? B1: m 1+m 2=x remote exterior angle theorem How can I find m 2? B2: m 2+142°=180° linear pair How can I find m 1? B3: m 1+31+87=180 angle addition postulate, straight angle

B3: m 1+31°+87°=180° m 1+118°=180° -118°-118° m 1=62° B2: m 2+142°=180° -142° -142° m 2=38° B1: m 1+m 2=x 38°+62°=x 100°=x

B1: m 3=x 141°=x

How can I find x? B1: m 13=x alternate exterior angles How can I find m 13? B2: m 13=163° corresponding angles

B2: m 13=163° B1: m 13=x 163°=x

How can I find x? B2: m 1+63°+89°=180° B1: (m 1+31°)+(52°+24°) m 1+152°=180° +x=180° triangle sum -152°-152° theorem m 1=28° How can I find m 1? B2: m 1+63°+89°=180° B1: (m 1+31°)+(52°+24)=180° triangle sum theorem 28°+31°+52°+24°+x=180° 135°+x=180° -135° -135° x=45°

How can I find x? B1: m 1=x vertical angles How can I find m 1? B2: m 1+41°+72°=180° triangle sum theorem

B2: m 1+41°+72°=180° m 1+113°=180° -113°-113° m 1=67° B1: m 1=x 67°=x

B3: m 4=52° How can I find x? B2: m 1+72°+68°=180° B1: m 1+m 4=x remote m 1+140°=180° exterior angle theorem -140°-140° How can I find m 1? m 1=40° B2: m 1+72°+68°=180° angle addition postulate, B1: m 1+m 2=x straight angle 52°+40°=x How can I find m 4? 92°=x B3: m 4=52° vertical angles

How can I find x? B1: m 6+m 7=x remote exterior angle theorem How can I find m 6? B2: m 6+48°=90° complementary angles How can I find m 7? B3: m 7=53° corresponding angles.

How can I find x? B1: m 1+55°=x remote exterior angle theorem How can I find m 1? B2: m 1+46°=90° complementary angles

B3: m 7=53° B2: m 6+48°=90° -48°-48° m 6=42° B1: m 6+m 7=x 42°+53°=x 95°=x

B2: m 1+46°=90° -46°-46° m 6=44° B1: m 1+55°=x 44°+55°=x 99°=x

B3: m 10=78° How can I find x? B2: m 11+m 10+20°=180° B1: m 11+x=180° linear m 11+78°+20°=180° pair m 11+98°=180° How can I find m 11? -98°-98° B2: m 11+m10+20°=180° m 11=82° triangle sum theorem How can I find m 10? B1: m 11+x=180° B3: m 10=78° 82°+x=180° corresponding angles. -82° -82° 98°=x

B2: m 1+139°=180° How can I find x? -139° -139° B1: m 1+(38°+26°)+x=180° m 1=41° triangle sum theorem B1: m 1+(38°+26°)+x=180° How can I find m 1? 41°+64°+x=180° B2: m 1+139°=180° linear 105°+x=180° pair -105° -105° x=75°

B1: 47°+21°+m 3=180° How can I find x? 47°+21°+m 3=180° B1: m 5+41°+x=180° 68°+m 3=180° triangle sum theorem -68° -68° How can I find m 5? m 3=112° B2: m 5=m 3 B2: m 3=m 5 How can I find the m 3? 112°=m 5 B3: 47°+21°+m 3=180° B1: 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=180° B3: m 2=91° triangle sum theorem B2: m 1=72° How can I find m 1? B2: m 1=72° B1: m 1+m 2+x=180° corresponding angles 91°+72°+x=180° How can I find the m 2? 163°+x=180° B3: m 2=91° vertical -163° -163° angles x=17°

B3: 69°+90°+m 1=180° How can I find x? 159°+m 1=180° B1: m 2+96°+x=180° -159° -159° triangle sum theorem m 1=21° How can I find m 2? B2: (m1+m 2)+69°+72°=180°B2:(m1+m 2)+69°+72°=180° triangle sum theorem 21°+m 2+69°+72°=180° How can I find the m 1? 162°+m 2=180° B3: -162° -162° 69°+90°+m 1=180° m 2=18° triangle sum theorem B1: m 2+96°+x=180° 18°+96°+x=180° 114°+x=180° -114° -114° x=66°

B3: m 3=44° How can I find x? B1: m 2+m3+x=180° B2: m 2=126° triangle sum theorem How can I find m 2? B2: m 2=126° B1: m 2+m 3+x=180° corresponding angles 126°+44°+x=180° How can I find the m 3? 170°+x=180° B3: m 3=44° alternate -170° -170° exterior angles x=10°

How can I find x? B1: m 8=x corresponding angles How can I find the m 8? B2: m 8=119° alternate interior angles

How can I find x? B1: m 1+90°=x remote exterior angle theorem How can I find m 1? B2: m 1+150°=180°

B2: m 8=119° B1: m 8=x 119°=x

B2: m 1+150°=180° -150° -150° m 1=30° B1: m 1+90°=x 30°+90°=x 120°=x

How can I find x? B1: m 3+47°=x remote exterior angle theorem How can I find m 1? B2: m 3+21°=90° complementary angles

B2: m 3+21°=90° -21°-21° m 3=69° B1: m 3+47°=x 69°+47°=x 116°=x

B2: m 4+125°=180° How can I find x? -125°-125° B1: m 4=x m 4=55° alternate interior angles How can I find the m 4? B1: m 4=x B2: m 4+125°=180° 55°=x consecutive angles

(Sometimes a very hard looking problem can have an easy solution.)

s1+s2=s3 4+7>11 11>11 false! no triangle s1+s2=s3 4+3>6 7>6 true triangle

s1+s2=s3 2+5>9 7>9 False! no triangle

s1+s2=s3 7+7>13 14>13 true triangle



s1+s2=s3 8+9>17 17>17 false no triangle


















max.=4+3=7 min.=4-3=1 1in
max.=15+20=35 min.=20-15=5 5m
max.=9+47=56 min.=47-9=38 38mm
max.=63+47=110 min.=63-47=16 16km
max.=8+11=19 min.=11-8=3 3cm
max.=21+29=50 min.=29-21=8 8ft
max.=13+36=49 min.=36-13=23 23mi
max.=13+47=60 min.=47-13=34 34cm
max.=66+65=131 min.=66-65=1 1yd
max.=59+77=132 min.=77-55=22 22m
max.=12+53=65 min.=53-12=41 41ft
max.=13+86=99 min.=86-13=73 73mm
max.=11+11=22 min.=11-11=0 0in
max.=92+64=156 min.=92-64=28 28km
max.=83+58=141 min.=83-58=25 25m
max.=25+53=78 min.=53-25=28 28ft
SSS AAS

HL SAS

ASA SAS

AAS

ASA

SSS

HL

BC EF given C F given AC DF given So ABC by SAS.

AB TN given BT ON given AT TO given So BAT by SSS.

NTO

DEF

G J given GH JK given H K given So GHI JKL by ASA.

BY FN given BA FE given AY EN given So BAT by SSS.

EF OW given E O given ED OC given So DEF by SAS.

COW

UW ZX given X W given VW YX given So UVW by SAS.

YXZ

TON

Q T given QR TU given R U given So QRS by ASA.

TUV

C E given A D given AB DF given So ABC by AAS.

DEF

Vertical Angles Shared Line

CA BD given CAD ADB given AD AD shared line So CAD by SAS.

BDA

So PQR P T given by ASA. PR RT given PRQ SRT vertical

Shared Line

HO HE given OP EP given HP HP shared line

Shared Line

So HOP by SSS.

BDA CAD given AD AD shared line BAD CDA given So ABD by ASA.

Shared Line

TSR

HEP

DCA

Vertical Angles

not congruent


ASS

B E BC CE BCA Vertical

Vertical Angles not congruent

ASS

Shared Line

Alt. Int given ECD angles

So ABC by ASA.

DEC

PS PS shared line PQ RS given So PQS by HL.

SRP

AB CD given ABC DCB given CB CB shared line So ABC by SAS.

DCB

Shared line

N M given NO MO given LOM PON Vertical angles So LMO by ASA.

PNO

Vertical Angles Corresponding Angles

TA TA shared line TS TR given

TVU VWX Corresponding U X given TV VW given So TUV by AAS.

VXW

not congruent

ASS

So STA by HL.

TRA

Shared Line

B D Alt. Int BC DC given BCA ECD Vertical angles So ABC by ASA.

DEC

Vertical Angles

AE BF given EC BD given AF AF shared line So ACE by SSS.

Shared line

HFDB

not congruent not enough info

Vertical Angles ACB DCE Vertical angles A E given BC DC given So ABC by AAS.

EDC

ED ID given JI EJ given JD JD shared line So EJD by SSS.

IJD

Shared Line Vertical Angles

PQO QSR Corresponding O R given PO QR given

A E Alt. Int AD ED given VDA EDR Vertical angles

So POQ by AAS.

So VAD by ASA.

QRS

Corresponding Angles DA KA given DAR KAR given AR AR shared line So DAR by SAS.

KAR

Shared Line

not congruent

ASS

RED

Alternate Interior Angles Shared Line SD SD shared line SI SE given So SDI by HL.

SDE

UL EK given ULK LKE given LK LK shared line So ULK by SAS.

EKL

Shared Line

True the triangles are congruent, so BC EF because CPCTC.


True, the triangles are congruent by SAS, so C F because CPCTC.

True, the triangles are congruent by ASA, so ON RQ because CPCTC.

True, the triangles are congruent by ASA. So D B because CPCTC. False.... the triangles are congruent by SSS, but the parts don't correspond. They aren't the same part.


False.... the triangles are congruent, but the parts don't correspond. They aren't the same part.




True the triangles are congruent by SAS. So TX VW because CPCTC.

AC=DF CPCTC x=7

Unsolvable, the triangles may not be congruent.

The triangles are congruent by ASA, and AB=ED CPCTC. x=17


m G=m J CPCTC x=35

VU=YZ CPCTC x=12

The triangles are congruent by SSS, and m GJH=m IJH CPCTC. x=46

Unsolvable, the triangles are congruent by ASA, but the parts are not corresponding.

C=m F CPCTC x+11=2x+4 -x -x 11=x+4 -4 -4 7=x

Unsolvable, the triangles are congruent by ASA, but the parts are not corresponding.

he triangles are congruent by SAS AB=DC CPCTC 2x+5=2x-8 -x -x 5=x-8 +8 +8 12=x

he triangles are congruent by SAS QO=MO CPCTC 4x+7=5x+3 -4x -4x 7=x+3 -3 -3 4=x


he triangles are congruent by SSS m V=m Y CPCTC x+57=2x-29 -x -x 57=x-29 +29 +29 86=x


he triangles are congruent by SSS m V=m T CPCTC 2x-21=3x-47 -2x -2x -21=x-47 +47 +47 26=x

3 5

2 5

4 = 10

6 = 24

1 4

21 = 49

3 7

6 = 8

3 4

7 = 35

1 5

18 = 27

2 3

26 = 52

1 2

2 = 3

8 x

2(x)=3(8) 2x=24 2 2 x=12

x 2 = 18 7 7(x)=2(18) 7x=36 7 7 x=5.14

x 6 = 32 11 11(x)=6(36) 11x=192 11 11 x=17.45

x 4 = 9 7 7(x)=4(9) 7x=36 7 7 x=5.14

x 4 = 12 5 5(x)=4(12) 5x=48 5 5 x=9.6

x 5 = 26 13 13(x)=5(26) 13x=130 13 13 x=10

9 10 = 12 x 9(x)=12(10) 9x=120 9 9 x=13.33

5 2 = x 15 2(x)=15(5) 2x=75 2 2 x=37.5

5 4 = 8 x 5(x)=4(8) 5x=32 5 5 x=6.4

8 7 = x 14 x 1 = 8 2 7(x)=14(8) 7x=112 7 7 x=16

x 2 = 73 5 5(x)=73(2) 5x=146 5 5 x=29.2

3 5 = x+3 20 3 1 = x+3 4 1(x+3)=4(3) x+3=12 -3 -3 x=9

x 6 = 15 18 x 1 = 15 3 3(x)=1(15) 3x=15 3 3 x=5

x 13 = 29 21 21(x)=13(29) 21x=377 21 21 x=17.95

6 4 = 15 x+4 6(x+4)=15(4) 6x+24=60 -24 -24 6x=36 6 6 x=6

8 22 = x 58 8 11 = x 29 11(x)=29(8) 11x=232 11 11 x=21.09

3 9 = 4 x 3(x)=4(9) 3x=36 3 3 x=12in

2 136 = 3 x 2(x)=3(136) 2x=408 2 2 x=204cm

12 4 = x 8 1 13 = 2 x

1(x)=2(13) x=26in

5 10 = 7 x

5(x)=7(10) 5x=70 5 5 x=14in

12 4 = x 7 4(x)=7(12) 4x=84 4 4 x=21m

3 54 = 8 x 3(x)=8(54) 3x=432 3 3 x=144in

x 6 = 46 30 x 1 = 46 5

6 27 = 9 x 2 27 = 3 x

5(x)=1(46) 5x=46 5 5 x=9.2in

2(x)=3(27) 2x=81 2 2 x=40.5in

3 1 AB DF DE EF = = 9 3 So... AC = AB = BC DE 1 BC = 4 = 3 EF 12 So...∆ABC~∆DEF 1 CA = 5 = by SSS 3 15 FD

O M

R ∆MNO~∆PQR by AA P

3 6 DE = = 7 14 AB 3 BC 15 = = 7 EF 35

DE BC So... AB = EF and... E B So...∆ABC~∆DEF by SAS

3 6 SU SU ST TU = = 4 So... VX = VW = WX 8 VX 3 ST = 9 = 4 VW 12 So...∆SUT~∆VWX 3 TU = 12 = by SSS 4 WX 16

8 2 IH IH GI GH = = = = So... 20 5 KL KL JL JK 2 GI = 10 = 5 JL 25 So...∆GHI~∆JKL 2 GH = 12 = by SSS 5 JK 30

M M same angle MON MQP corresponding MNO MPQ corresponding


∆ABC~∆ADE by AAA or... ∆ABC~∆ADE by AA

A A same angle ACB AED corresponding ABC ADE corresponding

3 6 VX = = 4 So... the triangles 8 SU 3 VW = 9 are not similar. = 4 12 ST 1 XW = 12 = 2 TU 24

∆GHI~∆GJK by AAA or... ∆GHI~∆GJK by AA

G G same angle GIH GKJ corresponding GHI GJK corresponding 6 2 ST = = 9 3 SV SU = 8 2 = SW 12 3 ST SU = SV SW and... S S shared angle So...

MN = NP QN = NO

6 1 = 12 2

MN QN So... NP = NO 5 1 and... MNQ ONP = 10 2 vertical angles So... ∆MNQ~∆PON by SAS

So...∆SUT~∆SWV by SAS

C DBE Corresponding ABC E Corresponding G J Alternate Interior K I Alternate Interior GHK IHJ Vertical

∆ABC~∆DEB by AA

∆GHK~∆IHJ by AAA or... ∆GHK~∆IHJ by AA M M same angle MON MQP Corresponding MNO MPQ Corresponding ∆GHI~∆GJK by AAA or... ∆GHI~∆GJK by AA

5 10 AB AB EB = = 11 So... BD = BC 22 BD and... ABE CBD EB = 20 5 = BC 11 Vertical Angles 44 So...∆ABE~∆CBD by SAS

UT = WV WS = UV

5 15 = 8 24 5 20 = 8 32

AB EF So... DE = BC and... U W So...∆ABC~∆DEF by SAS

3 6 LJ LJ JK KL = = 7 So... IG = GH = HI 14 IG 3 JK = 12 = 7 GH 28 So...∆GHI~∆JKL 3 KL = 15 = by SSS 7 HI 35

A D Alternate Interior E C Alternate Interior ABE CBD Vertical ∆ABE~∆CBD by AAA or... ∆ABE~∆CBD by AA


M M same angle MON MQP Corresponding MNO MPQ Corresponding

2 10 DE DE EF CA = = = = So... 9 45 AB AB BC FD 2 EF = 12 = 9 BC 54 So...∆ABC~∆DEF 2 CA = 14 = by SSS 9 FD 63

3 12 LJ = = 5 15 IG 3 JK = 15 = 5 GH 25 4 LK = 24 = 5 IH 30

WT = UV TU = VS

So...the triangles are not similar

5 10 = 8 16 6 8

G J H K I L All angles

=

3 4

So...The triangles are not similar

∆GHI~∆JKL by AAA or... ∆GHI~∆JKL by AA of equilateral triangles are 60°.

2

4 42 16 = 2= 5 25 5

3 2 32 9 = 2= 8 64 8

3

3

27 √27 3 =3 = 64 √64 4

2 2 22 4 = 2= 7 7 49

3 9 √ 9 = = 49 √49 7

3

1 13 1 = 3= 3 3 27

7 3 73 343 = 3= 8 8 512

25 √25 5 = = 64 √64 8 5 3 53 125 = 8 = 83 512

5 2 52 25 = 2= 9 9 81

25 √25 5 = = 36 √36 6

3

3

5 3 53 125 = 3= 6 6 216

3

3

64 √64 =3 = 4 729 √729 9 2

4 42 16 = 2= 9 81 9

3

27 √27 3 =3 = 125 √125 5

5 2 52 25 = = 11 112 121

3 2 32 9 = 2= 5 25 5

3

3

1 √ 1 =3 = 1 343 √343 7

3

27 √27 3 =3 = 512 √512 8 2

3 9 3 = 2= 8 64 8 2

3

3

125 √125 = 5 =3 1728 √1728 12 5 2 52 25 = = 12 122 144

20=2x 2 2 10=x

Corresponding Angles x=57°

x=2(14) x=28

Corresponding Angles x=49°

Corresponding Angles 17=x

x=2(18) x=36

x=71°

13=x

x+7=2(x-4) x+7=2x-8 -x -x 7=x-8 +8 +8 15=x

x+15=2(x-10) x+15=2x-20 -x -x 15=x-20 +20 +20 35=x

x+13=2x-21 -x -x 13=x-21 +21 +21 34=x

2x+17=4x-19 -2x -2x 17=2x-19 +19 +19 36=2x 2 2 18=x

x+13=2(x-7) x+13=2x-14 -x -x 13=x-14 +14 +14 27=x

x+15=2x-27 -x -x 15=x-27 +27 +27 42=x

x+7=2x-9 -x -x 7=x-9 +9 +9 16=x

x+63=2x-19 -x -x 63=x-19 +19 +19 82=x

If you practice then you will gain more skill.

If you click on a hyperlink then you will go to a new web page.

If it is your birthday then you will If two lines intersect at right get presents. angles then they are perpendicular.

If you drink and drive then you will go to jail.

If you know how to use logic then people will think you are smart. If the diagonals of a quadrilateral bisect each other then it is a rhombus.

If angles are vertical then they are always equal.

If parallel lines are cut by a transversal then consecutive angles are supplementary.

If you leave a can of soda open then it will go flat.

If it is a healthy dog then it has 4 If two sides of a right triangle are paws. 3in and 4in then the hypotenuse is 5in. If it is a triangle then its area is base times height divided by two.

If a figure is a square then its perimeter is 4 times its side length.

If it is an isosceles triangle then it has two equal sides and two equal angles.

If there is a single point on a line then only one perpendicular line can be drawn from it.

If all three sides of one

triangles are the same lengths as all three from

another then the triangles are congruent..

If you steal vegetables from farmer Maggot then he will chase after you.

C She lost her patients.

Angle A is acute.

38°

Circle A and B are concentric.

Joe eats ice cream.

4x=12

It will sing.

It is cold. Corresponding angles are congruent

Dad will ask you to mow.

I will lose my hearing

an angle that measures 85°

an Emperor Penguin

a rocket

Mathew McConehey

a triangle

x=2

a student at NYU

a sparrow

7 nickles

two 90° angles that are not alternate interior

they could be 50°and 110° someone who lives in Colorado Springs

an isosceles triangle

20°

11 times 21 is 231

a convenience store a square

cherry pie

If a figure is a rhombus then it is

a square.

If a figure is not a square then it is not a rhombus. If a figure is not a rhombus then it is not a square. If we need a cage then there are ten lions. If there are not ten lions then we don't need a cage. If we don't need a cage then there aren't ten lions.

If a triangle has two congruent angles then it is isoceles. If a triangle is not isoceles then it doesn't have two congruent angles. If a triangle doesn't have two congruent angles then it is not Isocleles.

If a figure is a trapezoid then it is a parallelogram. If a figure is not a parallelogram then it is not a trapezoid. If a figure is not a trapezoid then it is not a parallelogram.

If we put on our tuxedo's then it is 6:00. If it is not 6:00 then we don't put on our tuxedo's. If we don't put on our tuxedo's then it is not 6:00.

If x=1 then 5x+6=11. If 5x+6≠11 then x≠1. If x≠1 then 5x+6≠11.

If an angle measures 90° then it is a right angle. If an anlgle is not a right angle then it does not measure 90°. If an angle does not measure 90° then it is not a right angle.

If lines intersect at right angles then they are perpendicular. If lines aren't perpendicular then they don't intesect at right angles. If lines don't intersect at right angles then they are not perpendicular.

If two angles add to 90°then they are complementary. - True If two angles aren't complementary then they don't add to 90°. - True If two angles don't add to 90° then they aren't complementary. -true

If a figure is a triangle then it is a right triangle. -false an obtuse triangle. If a figure is not a right triangle then it is not a triangle -false an acute triangle. If a figure is not a triangle then it is not a right triangle -true

If two lines in a plane are parallel then they never intersect. -true If two lines in a plan intersect then they aren't parallel. -true

If an animal has whiskers then it is cat -False a walrus If an animal is not a cat then it doesn't have whiskers -False a walrus If an animal does not have whiskers then it is not a cat true

If two angles add to 180° then they are supplementary. - True If two angles aren't supplemenetary then they don't add to 180°. - True if two angles don't add to 180° they they aren't supplemenetary - True

If two angles have the same measure then they are congruent. - True If two angles aren't congruent then they don't have the same measure. -True

If two lines in a plane are not parallel then they intersect. -true

If two angles don't have the same measure then they aren't congruent. - True

If two angles are equal then they are vertical. -False corresponding angles If two angles are not vertical then they are not equal. -False corresponding angles If two angles are not equal then they are not vertical. -true

If two segments have the same measure then they are congruent - True If two segments aren't congruent then they don't have the same measure - True If two segments don't have the same measure then they aren't congruent - True

If two angles are adjacent then they are a linear pair. If a triangle has two congruent sides then it is isoceles. Iff a triangle is Isoceles then it has two congruent sides. If a triangle contains one right angle then it is a right tirangle.

If AC+CB=AB on a line then C is between A and B. Iff C is between A and B on a line then AC+CB=AB.

Iff a triangle is a right triangle then it contains one right angle. If a fish eats other fish then it is a shark. a baracuda

If two angles are supplementary then they add to 180°. Iff two angles add to 180° then they are supplementary.

If s1+s2>s3 then the figure is a triangle. If a figure's perimeter is 4 times its side then it is a square.

Iff a figure is a triangle then s1+s2>s3.

Iff a figure is a square then it's perimeter is 4 times it's side.

If a triangle has three congruent sides then it is equilateral. Iff a triangle is equilateral then it has three congruent sides. If a figure is a polygon then it is a rectangle. an octagon

If an animal is a mammal then it is a cow. a goat If two angles are equal then they are vertical. two 30° angles that are not vertical

Working backwards we must ask the key question..."How can we show that two triangles are congruent?" The answer? A triangle congruence theorem like SSS, SAS, ASA, AAS, or HL. This gives us B1 ∆ABC ∆DEF by some property, but which one. Then start working forward. Listing all of the given information gives two pairs of congruent angles B E and C F and included sides AC DF. This gives us ∆ABC ∆DEF by ASA and the proof is complete. (Your answer may read slightly different.)

Working backwards we must ask the key question..."How can we show that two triangles are congruent?" The answer? A triangle congruence theorem like SSS, SAS, ASA, AAS, or HL. This gives us B1 ∆ABC ∆DEF by some property, but which one? Then start working forward. listing all of the given information gives AB BD. BC BC by the reflexive property, because it is a shared line. This gives us ∆ABC ∆DEF by HL which is B1 and the proof is complete. (Your answer may read slightly different.) Working backwards we must ask the key question..."How can we show that two triangles are congruent?" The answer? A triangle congruence theorem like SSS. SAS, ASA, AAS, or HL this gives us B1 ∆ABC ∆DEF by some property, but which one? Then start working forward. Listing all of the given information gives AB is parallel to ED and AC EC. When we see parallel lines we should look for alternate interior, or corresponding angles. In this case A E will give us ASA, because ACB and DCE are vertical angles. This gives us ∆ABC ∆DEF by ASA and the proof is complete.

B

E

AC DF BC DF ΔABC ΔDEF

AE bisects BD DC BC

PQ RS PQS RSQ Given ASA

Given Given Given

L N LOM NMO MO MO

Given Vertical Angles AAS

PQ ST ΔPQR ΔTSR

Reflexive Property SAS

Given AAS

Given Alternate Interior Vertical Angles

M O MPL NPO ΔMPL ΔNPO

PQ SU QR ST PR TU

Given

AB DC

Vertical Angles

Given

Given SSS

C is the M.P. of BE BC EC

Reflexive Property HL

MN ON ΔLMN ΔPON

Given Given

SP bisects QT QRP

SRT

Definition of Bisect SAS

MO MO ΔLMO ΔNOM

PS PS

AB DE B D ΔABC ΔDCE

Given Reflexive Property HL

AB DC AC DF ABC DCE

Given Given

LP Bisects MO LO MP

ASA

L P LNO MNP ΔLNO ΔMNP

Given Definition of Bisect Vertical Angles

SAS

Given

Q is the MP of PR PS QT PQ RQ

Given SSS

Statements 1. AB DE 2. AC DF 3. A D 4. ABC DEF

Reasons 1. Given 2. Given 3. Given 4. SAS

Statements 1. PQ TU 2. P T 3. Q U 4. PQR TUS

Reasons 1. Given 2. Given 3. Given 4. ASA

Statements 1. JK MN 2. KL NO 4. JKL MNO

Reasons 1. Given 2. Given 4. HL

Statements 1. AB DF 2. BC DE 3. AC EF 4. ABC FDE

Reasons 1. Given 2. Given 3. Given 4. SSS

Statements 1. P S 2. Q T 3. QR TU 4. PQR STU

Reasons 1. Given 2. Given 3. Given 4. AAS

Statements 1. J M 2. JK MN 3. K N 4. JKL MNO

Reasons 1. Given 2. Given 3. Given 4. ASA

Statements 1. AB CD 2. ABD CDB 3. BD BD 4. ABD CDB

Reasons 1. Given 2. Given 3. Reflexive Property 4. SAS

Statements 1. LM JM 2. KM KM 4. JKL MNO

Reasons 1. Given 2. Reflexive Propety 4. HL

Statements 1. PS QR 2. PQ SR 3. PR PR 4. PRS RPQ

Reasons 1. Given 2. Given 3. Reflexive Property 4. SSS

Statements 1. PR TR 2. P T 3. PRQ SRT 4. PQR TUS

Statements 1. AB ED 2. A D 3. BCA ECD 4. ABC DCE

Reasons 1. Given 2. Given 3. Alternate Interior 4. ASA

Reasons 1. Given 2. Given 3. Vertical 4. AAS

Statements 1. JN bisects ML 2. M L 3. JKM NKL 4. MK LK 5. MJK LNM

Reasons 1. Given 2. Given 3. Vertical Angles 4. Def. of Bisect 5. AAS

Statements 1. C is the M.P. of AE 2. BA DE 3. A E 4. AC EC 5. ABC EDC

Statements 1. K N 2. JK MN 3. KL NO 4. ABC EDC

Statements 1. PQ RS 2. PR QS 2. RPS QSP 3. RSP QPS 4. PS PS 5. RPS QSP

Reasons Statements 1. Given 1. R is the MP of PT 2. Given 2. P T 3. Given 3. PRQ TRS 4. Def. of Midpoint 4. PR TR 5. SAS 4. PQR TSR

Reasons 1. Given 2. Given 3. Given 4. SAS

Reasons 1. Def of Parallelogram 2. Def. of Parallelogram 2. Alternate Interior 3. Alternate Interior 4. Relexive Property 5. ASA

Statements 1. BA CD 2. ADB CDB 3. ABD CDB 4. BD BD 5. ABD CDB

Statements 1. KN bisects JM 2. JK MN 3. ML JL 4. J M 5. KLJ MLN 5. JKL MNL

Reasons 1. Given 2. Given 3. Given 4. Def. of M.P. 4. ASA

Reasons 1. Given 2. Given 3. Alternate Interior 4. Reflexive Property 5. ASA

Reasons 1. Given 2. Given 3. Def. of Bisect 4. Alternate Interior 5. Vertical Angles 5. ASA

Working backwards, the statement "Prove: M P" begs the key question, "How can we show two angles from two different triangles are congruent?" The answer? CPCTC. This means that if we can prove the triangles are congruent then M P, because they are corresponding parts of congruent triangles. So B1 is M P by CPCTC. This then begs the second key question, "How can I show two triangles are congruent?" This means B2 will be MNO PQR by some congruence property, but which one? Now we work forward and see we have. MN PQ, NO QR, and OM RP Given. This means MNO PQR by SSS which is B2 and the proof is complete.

Working backwards, the statement "Prove: ST VT" begs the key question ,"How can we show two segments from two different triangles are congruent?" The answer? CPCTC. This means that if we can prove the triangles are congruent then ST VT, because they are corresponding parts of congruent triangles. So, B1is ST VT by CPCTC. This then begs the second key question, "How can I show two triangles are congruent?" This means B2 will be STU VUT by some congruence property. Which one? Now we work forward and see we have SU UV given. We also see these are right triangles and that TU is a shared side so... MNO PQR by HL, which is B2, and the proof is complete.

Working backwards, the statement "Prove: BC DC" begs the key question, "How can we show two segments from two different triangles are congruent?" The answer? CPCTC. This means that if we can prove the triangles are congruent then BC DC because they are corresponding parts of congruent triangles. So B1 is BC DC by CPCTC. This then begs the second key question, "How can I show two triangles are congruent?" This means B2 will be STU VUT by some congruence property. Which one? Now we work forward and see we have AB DE, and C is the midpoint of AE Given. We know to look for alternate interior or corresponding angles when we see parallel lines, and we see BAC DEC. and AC EC because C is the midpoint. We also see vertical angles ACB DCE so... This means ABC DEF by ASA, which is B2, and the proof is complete.

MNP G

K

∆GHI ∆KLJ

ST VU TV TV ∆STV ∆UVT

GH JI GHI JIK ∆GHI ∆JIK

Given

MP NO

OPN

Given Reflexive Property SAS

CPCTC

AC CE DC BC CPCTC

Given Given Midpoint CPCTC

MP NP

MO ON

Given Alternate Interior SAS CPCTC

Given Reflexive Property HL CPCTC

TU TU ∆STU ∆VUT ST UV

Given Reflexive Property

AB DE AE bisects BD BC CD

GH IJ G JIK ∆STU ∆VUT GIH IKJ

ST WV ST VW STU WVU ∆STU ∆VUW SU WU

Given Given ASA

Vertical Angles CPCTC

Alternate Interior Vertical ASA CPCTC

AC EC

GH IJ

Given

Given

MN PO NMO

Alternate Interior

POM

∆PMO ∆NOM PM ON

AC CE ∆STU ∆VUT A DCE

Reflexive Property CPCTC

Given Given SSS CPCTC

Statements 1. GH JK 2. HI KL 3. IG LJ 4. ABC EDC 5. I L

Reasons 1. Given 2. Given 3. Given 4. SSS 5. CPCTC

Statements 1. U V 2. T W 3. TU VW 4. STU WVX 5. S X

Reasons 1. Given 2. Given 3. Given 4. ASA 5. CPCTC

Statements 1. GH JI 2. GHJ IJH 3. JH JH 4. GHJ IJH 5. GJ HI

Reasons 1. Given 2. Given 3. Reflexive Property 4. SAS 5. CPCTC

Statements 1. N P 2. M Q 3. MO QR 4. MNO QPR 5. O R

Statements 1. AC EF 2. AB ED 3. STU WVX 4. BC FD

Statements 1. MN NO 2. MP OP 3. NP NP 4. MNP ONP 5. O M

Reasons 1. Given 2. Given 3. Given 4. AAS 5. CPCTC

Reasons 1. Given 2. Given 3. HL 4. CPCTC

Reasons 1. Given 2. Given 3. Reflexive Property 4. SSS 5. CPCTC

Statements 1. TU WU 2. T W 3. TUS WUV 4. GHJ IJH 5. TS WV

Reasons 1. Given 2. Given 3. Vertical Angles 4. ASA 5. CPCTC

Statements 1. HG JI 2. GI IK 3. HIG JKI 4. HGI JIK 4. ABC DEC 5. C F

Reasons 1. Given 2. Given 3. Given 4. Corresponding 4. ASA 5. CPCTC

Statements 1. TS VW 2. TU WU 3. STU VWU 4. TUS WUV 4. STU VWU 5. TS VW

Reasons 1. Given 2. Given 3. Alternate Interior 4. Vertical 4. ASA 5. CPCTC

Statements 1. AB DE 2. B E 3. BCA DCE 4. ABC DEC 5. AC DC

Reasons 1. Given 2. Given 3. Vertical Angles 4. AAS 5. CPCTC

Statements 1. MN PO 2. MP NO 3. PN PN 4. ABC DEC 5. M O

Reasons 1. Given 2. Given 3. Reflexive Property 4. SSS 5. CPCTC

Statements 1. AB DE 2. CBD ADB 3. ABD CDB 4. BD BD 5. STU VWU 6. BC AD

Reasons 1. Given 2. Given 3. Alternate Interior 4. Reflexive Property 5. ASA 6. CPCTC

Working backward we must ask the key question. "How can we show two triangles are similar?" The answer? Use a similarity property such as SSS, SAS, or AA (AAA). That leads us to B1: JGI~ KGH by one of these properties. But which one? We need to start working forward. We see MN, PR and ON,QR are proportional and N R in the given information. This gives us JGI~ KGH by SAS, which is B1, and the proof is complete!

Working backward we must ask the key question. "How can we show two triangles are similar?" The answer? Use a similarity property such as SSS, SAS, or AA (AAA). That leads us to B1: JGI~ KGH by one of these properties. But which one? We need to start working forward we see S W in the given information. We also see vertical angles SUV TUW This gives us JGI~ KGH by AA, which is B1, and the proof is complete!

Working backward we must ask the key question. "How can we show two triangles are similar?" The answer? Use a similarity property such as SSS, SAS, or AA (AAA). That leads us to B1: JGI~ KGH by one of these properties. But which one? We need to start working forward we see AB is parallel to DC in the given information. This gives us two pairs of corresponding angles CDE, and DCE. This gives us ABE~ DCE by AA, which is B1, and the proof is complete!

I

L

Given AA

Given N

Q

SAS

AB BC AC = = DE EF DF

ST TU US = = WV VX VW

∆STU~∆WVX

GK HJ K HJI

Corresponding Angles AA

MQ OP MNQ

PNO

Alternate Interior AA

S

V

Vertical Angles AA

A DBC E BDC ∆ACE~∆BCD

GH HI = JK KL H

K

SAS

NQM

A

∆STU~∆VWX

Equilateral Triangles Equilateral Triangles

NOP

B

Given Corresponding Angles

Given Alternate Interior AA

Reflexive Property AA

Statements 1.

GH GI = KJ JL

Reasons 1. Given

Statements 1. M P 2. O Q 3. ∆GHJ~∆IJH

Reasons 1. Given 2. Given 3. AA

Reasons

2. G J 3. ∆GHJ~∆IJH

2. Given 3. ASA

Statements

Reasons

ST TU = VW WX

Statements

1. Given

1.

1.

2. T W 2. Given 3. ∆STU~∆VWX 3. SAS

Statements 1. GK HJ 2. G JHI Corresponding 3. K HJI 4. ∆GKI~∆HJI

Reasons 1. Given 2. 3. Corresponding 4. AA

AB BC CA = = FD DE EF

1. Given

2. ∆ABC~∆DEF 2. SSS

Statements

Reasons

NO PO = QO MO

1. Given

1.

2. NOM POQ 3. ∆MNO~∆PQO

2. Vertical Angles 3. SAS

Statements 1. S UTV 2. U U 3. ∆SUW~∆TUV

Reasons 1. Given 2. Reflexive Property 3. AA

Statements

Reasons

1. GH JK 2. HGI JKI 3. GIH JIK 4. ∆GHI~∆KJI

1. 2. 3. 4.

Statements

Reasons

1. S V 2. T W 3. U X 3. ∆STU~∆VWX

1. 2. 3. 3.

Given Alternate Interior Vertical AA

Equilateral Triangle Equilateral Triangle Equilateral Triangle AAA

Statements 1. AB DC 2. ACB E 3. B DCE 4. ∆ABC~∆DCE

Reasons 1. Given 2. Given 3. Corresponding 4. AA

Statements Reasons 1. MQ NP 1. Given 2. M PNO 2. Corresponding 3. Q NPO 3. Corresponding 4. ∆QMO~∆PNO4. AA

Statements 1.

AB AC = DC CE

2. AB CD 2. A DCB 3. ∆GHJ~∆IJH

Reasons 1. Given 2. Given 2. Corresponding 3. SAS

An angle is obtuse if it is less than 90°.

A square doesn't have four sides.

Two angles that are supplementary don't add to 180°. A school is not a place you can learn things.

A right angle doesn't measure 90°.

Parallel lines in a plane intersect.

A polygon has 2 sides.

A parallelogram has only one pair of parallel sides.

Lines in a plane intersect in two points. The area of a triangle is not one half its base times its height.

A triangle has 4 or more sides.

Lines in a plane intersect in two points.

This is false because it violates the definition of a triangle.

This is false because if it were true one of the lines must be curved. This violates the definition of a line.

A trapezoid doesn't have a pair of parallel lines.

An acute angle measures more than 90°.

This is false because it violates the definition of a trapezoid.

This is false because it violates the definition of an acute angle.

Complementary angles don't add to 90°. This is false because it violates the definition of complementary angles.

A triangle has two obtuse angles. This is false because the sum of the angles of a triangle must be 180°. If two obtuse angles are present then the sum would have to be greater than 180°.

A

H

H

O

O

A O

H

O

A

H

A

H

H O A

A

O A

O

H

H

H O

A

H

O A A

O

H

O

sin41°=

O H

A A

cos34°= A H

H O O A

cos30°= A H

H tan28°= O A

O H A O H

cos19°= A H

A

H

A O

cos75°= A H

6*

x sin41°= 6 x sin41°= 6 *6

19 x 19 *x x* cos34°= x x(cos34°)= 19 cos34° cos34° cos34°=


tan28°=

6(sin41°)=x 6(0.6561)=x 3.94≈x

19 cos34° 19 x= 0.8290 x=

x≈22.92 7 cos30° 7 x= 0.8660 x=

x≈8.08

x 7

x 7* tan28°= *7 7 7(tan28°)=x 7(0.5317)=x 3.72≈x 31 x 31 *x x* cos19°= x x(cos19°)= 31 cos19° cos19° cos19°=

x cos75°= 18

x 18* cos75°= 18 *18

18(cos75°)=x 18(0.2588)=x 4.66≈x

31 cos19° 31 x= 0.9455 x=

x≈32.79

A

O tan59°= A

O

H

H

O

tan45°= O A

A A

O

sin37°=

O H

H A

cos19°= A H

H O

O

H

cos32°= A H

A

H

O

tan12°=

A

O A

tan59°=

x 26

x 26* tan59°=26 *26 26(tan59°)=x 26(1.6643)=x 43.27≈x 17 tan45°= x 17 *x x* tan45°= x x(tan45°)= 17 tan45° tan45° x sin37°= 28 x 28* sin37°= 28 *28 28(sin37°)=x 28(0.6018)=x 16.85≈x


x cos32°= 13

x 13* cos32°=13 *13 13(cos32°)=x 13(0.8480)=x 11.02≈x x 64 x 64* tan12°=64 *64 64(tan12°)=x 64(0.2126)=x 13.61≈x tan12°=

x= x=

17 tan45° 17 1

x=17

5 cos19° 5 x= 0.9455 x=

x≈5.29

O A

cos73°= A H

H A O

H

O H

sin48°=

O H

cos26°= A H

A

H

sin50°=

O

O H

A

A

tan25°= O A

O

H

O

A H


sin39°=

O H

16 x 16 *x x* sin48°= x x(sin48°)= 16 sin48° sin48° sin48°=

37 cos73° 37 x= 0.2924 x=

x≈126.54 16 sin48° 16 x= 0.7431 x=

x≈21.53

x cos26°= 9 x 9* cos26°= 9 *9 9(cos26°)=x 9(0.8988)=x 8.09≈x x sin50°= 28

x 28* sin50°= 28 *28 28(sin50°)=x 28(0.7660)=x 21.45≈x 14 tan25°= x 14 *x x* tan25°= x x(tan25°)= 14 tan25° tan25°

14 tan25° 14 x= 0.4663

86 x 86 *x x* sin39°= x x(sin39°)= 86 sin39° sin39°

86 sin39° 86 x= 0.6293

sin39°=

x=

x≈30.02

x=

x≈136.66

O A

sinx°=

H

O H

sinx°=

15 25

-1 sin-1* sinx°=0.6000 *sin

x≈36.87°

x=sin-10.6000

O H

cosx°= A H

A

cosx°= 9 10 -1 cos-1* cosx°=0.9000 *cos

x≈25.84°

x=cos-10.9000

H

tanx°= O A

A

tanx°=

11 21

tan-1*tanx°=0.5238 *tan

-1

x≈27.65°

x=tan-10.5238

O sinx°=

H

O

O H

9 13

-1 sin-1* sinx°=0.6923 *sin

x≈43.81°

x=sin-10.6923

A

A

H

tanx°= O A

H

tanx°=

21 18

tan-1* tanx°=1.1667 *tan

-1

x≈49.4°

x=tan-11.1667

O O

A

sinx°=

cosx°= A H

cosx°= 25 32 -1 cos-1* cosx°=0.7813 *cos

x=cos-10.7813

x≈38.62°

O tanx°= O A

H

tanx°=

5 8

tan-1* tanx°=0.6250 *tan

-1

x≈32.01°

x=tan-10.6250

A

H

cosx°= A H

O

cosx°= 15 23 -1 cos-1* cosx°=0.6522 *cos

x≈49.29°

x=cos-10.6522

A O

cosx°= A H

H

A

cosx°= A H

A

sinx°=

H

O H

cosx°= 35 57 -1 cos-1* cosx°=0.6140 *cos

x≈52.12°

A

sinx°=

12 16

-1 sin-1* sinx°=0.7500 *sin

x≈48.59°

x=sin-10.7500

O

H

x≈53.71°

x=cos-10.6140

H

O

-1 cos-1* cosx°=0.5918 *cos

x=cos-10.5918

A O

cosx°= 29 49

tanx°= O A

tanx°=

13 11

tan-1* tanx°=1.1818 *tan x=tan-11.1818

-1

x≈49.76°

Tanx=

8 6

Tanx=

24 7

Tanx=

9 40

Tanx=

10 24

Tanx=

Tanx=

5 12

15 8

Tanx=

11 60

14 48

Tanx=

5 Tanx= 2√6

Tanx= 2√11 10

Tanx= 1 √3

Tanx=

Tanx= 2√7 6

Tanx=

Tanx=

12 35

16 15

Tanx=

2 2

11 4√3

Tanx= 4√2 7

Tanx=

12 10

sin2x+cos2x=1 sin2x+(6/7)2=1 sin2x+(36/49)=1 -(36/49) -(36/49) 2 sin x=13/49 √sin2x=√13/49 sinx=√13/7

sin x+cos x=1 (5/8)2+cos2x=1 (25/64)+cos2x=1 -(25/64) -(25/64) cos2x=39/64 √cos2x=√39/64 cosx=√39/8 2

2

sin2x+cos2x=1 sin2x+(11/12)2=1 sin2x+(121/144)=1 -(121/144) -(121/144) 2 sin x=23/144 √sin2x=√23/144 sinx=√23/12

sin2x+cos2x=1 sin2x+(4/9)2=1 sin2x+(16/81)=1 -(16/81) -(16/81) sin2x=65/81 √sin2x=√65/81 sinx=√65/9

sin2x+cos2x=1 (5/7)2+cos2x=1 (25/49)2+cos2x=1 -(25/49) -(25/49) cos2x=24/49 √cos2x=√24/49 cosx=2√6/7

sin2x+cos2x=1 sin2x+(9/15)2=1 sin2x+(81/225)=1 -(81/225) -(81/225) sin2x=144/225 √sin2x=√144/225 sinx=12/15

sin2x+cos2x=1 (5/13)2+cos2x=1 (25/169)+cos2x=1 -(25/169) -(25/169) 2 cos x=144/169 √cos2x=√144/169 cosx=12/13

sin2x+cos2x=1 (√2/5)2+cos2x=1 (2/25)+cos2x=1 -(2/25) -(2/25) 2 cos x=23/25 √cos2x=√23/25 cosx=√23/5

sin2x+cos2x=1 sin2x+(3/8)2=1 sin2x+(9/64)=1 -(9/64) -(9/64) 2 sin x=55/64 √sin2x=√55/64 sinx=√55/8 tanx=√55/3

sin x+cos x=1 (6/11)2+cos2x=1 (36/121)+cos2x=1 -(36/121) -(36/121) cos2x=85/121 √cos2x=√85/121 cosx=√85/11 2

2

tanx=6/√85

sin2x+cos2x=1 (2/7)2+cos2x=1 (4/49)+cos2x=1 -(4/49) -(4/49) cos2x=45/49 √cos2x=√45/49 cosx=3√5/7 tanx=2/3√5

sin2x+cos2x=1 (√3/4)2+cos2x=1 (3/16)+cos2x=1 -(3/16) -(3/16) cos2x=13/16 √cos2x=√13/16 cosx=√13/4 tanx=√3/√13

sin2x+cos2x=1 sin2x+(4/9)2=1 sin2x+(16/81)=1 -(16/81) -(16/81) sin2x=65/81 √sin2x=√65/81 sinx=√65/9 tanx=√65/4

sin2x+cos2x=1 sin2x+(11/13)2=1 sin2x+(121/169)=1 -(121/169) -(121/169) sin2x=48/169 √sin2x=√48/169 sinx=4√3/13 tanx=4√3/11

sin2x+cos2x=1 (7/16)2+cos2x=1 (49/256)+cos2x=1 -(49/256) -(49/256) cos2x=207/256 √cos2x=√207/256 cosx=3√23/16 tanx=7/3√23

sin2x+cos2x=1 sin2x+(√5/5)2=1 sin2x+(5/25)=1 -(5/25) -(5/25) 2 sin x=20/25 √sin2x=√20/25 sinx=2√5/5 tanx=2√5/√5

c=a√2

x=3 y=3√2

a

b=a√3

c=2a

x=7√3 y=14

a b=a c=2a 8=2a 2 2 4=a b=a√3

c=2a

a

a

y=4 x=4√3

x=9 y=18

b=a

a

x=3 y=3 c=a√2

b=a√3

x=7√3 y=14

c=2a

b=a√3

c=2a a

x=5 y=5

a

c=a√2

b=a

c=2a a b=a√3

y=6 x=16

b=a√3

c=a√2

c=2a

b=a a

x=3 y=6

a

y=5√2 x=5√2√2 x=5√4 x=5*2 x=10

a

b=a

c=a√2

c=2a

a

b=a√3

b=a√3

c=2a

a

b=a

c=a√2 y=2√2

y=8√2

x=2√2√2 x=2√4 x=2*2 x=4

x=8√2√2 x=8√4 x=8*2 x=16

c=2a 16=2a 2 2 8=a

a

b=a√3 a c=2a

x=8 y=8√3

c=2a 9=2a 2 2 4.5=a y=4.5 x=4.5√3

c=2a 11=2a 2 2 5.5=a y=5.5 x=5.5√3

b=a

y=7√2

a c=a√2

x=7√2√2 x=7√4 x=7*2 x=14

b=a√3

c=a√2 a

b=a c=a√2 6=a√2 √2 √2 6 =a √2

6 *√2 √2 *√2

6√2 √4

c=2a

a

x=3√2 y=3√2

6√2 2 3√2 =a

b=a√3 a

c=2a

b=a√3

y=2*5√3 y=10√3

x=5√3√3 x=5√9 x=5*3 x=15

b=a

a

c=2a c=2a

b=a√3

y=2*12√3 y=24√3

x=12√3√3 x=12√9 x=12*3 x=36

b=a√3

c=2a

a

c=a√2 c=a√2 9=a√2 √2 √2 9 =a √2

b=a√3

y=2*7√3 y=14√3

x=7√3√3 x=7√9 x=7*3 x=21

9√2 √4

9√2 2 9√2 =a 2 c=a√2

a

c=2a

9 *√2 √2 *√2

x=

9√2 2

y=

9√2 2

c=a√2 12=a√2 √2 √2 12 =a √2

b=a x=6√2 y=6√2

12 *√2 √2 *√2

12√2 √4

12√2 2 6√2

c=2a

c=2a

a b=a√3

b=a√3

a

c=2a

b=a√3

c=2a

b=a√3

16√3=2x 2 2 8√3=x

y=8√3√3 y=8√9 y=8*3 y=24

22√3=2y 2 2 11√3=y

x=11√3√3 x=11√9 x=11*3 x=33

a

a

c=2a

b=a

c=a√2

b=a√3 c=2a

b=a√3

12√3=2x 2 2 6√3=x

y=6√3√3 y=6√9 y=6*3 y=18

c=a√2

x=5√3

y=5√3√2 y=5√6

b=a√3 c=a√2

c=2a a

c=a√2

c=2a

b=a√3

8√3=2y 2 2 4√3=y

x=4√3√3 x=4√9 x=4*3 x=12

a

x=8√3√2 x=8√6 y=8√3

b=a

A

E

F

H

G

B

D

C

P

O

U

V

X

W

B

E

F

C

H

L

K

R

T

S

D

N

J

Q

A

M

I

G

N

I

J

M

L

K

O P m M+m P=180°

FZ HZ x=11 AB DC x=8

IZ KZ x=9

PM MN x=13

QZ TZ 15=x

V

X

x=54°

21°+x+90°=180 111°+x=180° -111° -111° x=69°

FZ GZ 16=x

72°+x°=180 -72° -72° x=106° 120°+x°=180 -120° -120° x=60°

QR RT

ZUV

x=11

∆BCZ is a 45-45-90 triangle x=13

∆BCZ is a 30-60-90 triangle 2x=30 2 2 x=15

XUZ

x=24°

FZ HZ x=28

MN OP x=32

QR TS x+34=2x-17 -x -x 34=x-17 +17 +17 51=x

BZ DZ 3x+5=x+19 -x -x 2x+5=19 -5 -5 2x=14 2 2 x=7

2x°+4x°=180° 6x°=180° 6 6 x=30°

QZ SZ 2x+9=x+46 -x -x x+9=46 -9 -9 x=37

UX VW x+9=2x-7 -x -x 9=x-7 +7 +7 16=x

FG HG x+40=4x+7 -x -x 40=3x+7 -7 -7 33=3x 3 3 11=x

∆POZ is a 45-45-90 triangle x=11√2

UX WX 2x+6=x+24 -x -x x+6=24 -6 -6 x=18

DZ AZ x+1=3x-25 -x -x 1=2x-25 +25 +25 26=2x 2 2 13=x

KIJ LIK x-13=3x-27 -x -x -13=2x-27 +27 +27 14=2x 2 2 7=x

m RQT+m STQ=180° 4x-66+2x+36=180° 6x-30=180° +30 +30 6x=210° 6 6 x=35°

AC BD 2x+31=3x+11 -2x -2x 31=x+11 -11 -11 x=20

x+52°=90° -52° 52° x=38°

MZ OZ 3x+28=x+96 -x -x 2x+28=96 -28-28 2x=68 2 2 x=34

UW VX 2x+5=x+17 -x -x x+5=17 -5 -5 x=12

EHG GFE x+32=4x-16 -x -x 32=3x-16 +16 +16 48=3x 3 3 16=x

∆NZO is a 45-45-90 triangle x°+133°=180° -133°-133° x=47

QZ SZ 5x+18=3x+34 -3x -3x 2x+18=34 -18-18 2x=36 2 2 x=8

4x+52°+2x-12°=180° 6x+40°=180° -40° -40° 6x°=140° 6 6 x=23 1/3°

LZ JZ 2x-39=x-10 -x -x x-39=-10 +39 +39 x=29

17√2=NO NO=PO 17√2=x

UW VX 4x-13=2x+35 -2x -2x 2x-13=35 +13 +13 2x=48 2 2 x=24

x+13°=45° -13°-13° x=32°

∆NZO is a 30-60-90 triangle 2(x-4)=x-4 2x-8=x-4 -x -x x-8=-4 +8 +8 x=4

n=8 (n-2)180° (8-2)180 6*180 1,080°

n=12 (n-2)180° (12-2)180 10*180 1,800°

n=13 (n-2)180° (13-2)180 11*180 1,980°

n=4 (n-2)180° (4-2)180 2*180 360°

n=10 (n-2)180° (10-2)180 8*180 1,440°

n=9 (n-2)180° (9-2)180 7*180 1,260°

n=19 (n-2)180° (19-2)180 17*180 3,060°

n=7 (n-2)180° (7-2)180 5*180 900°

n=9 (n-2)180° n (9-2)180° 8 7*180° 9 1,260° =140° 9

n=8 (n-2)180° n (8-2)180° 8 6*180° 8 1,080° =135° 8

n=4 (n-2)180° n (4-2)180° 4 2*180° 4 360° =90° 4

A Square

n=10 (n-2)180° n (10-2)180° 10 8*180° 10 1,440° =144° 10

n=19 (n-2)180° n (19-2)180° 19 17*180° 19 3,060° =161.05° 19

n=7 (n-2)180° n (7-2)180° 7 5*180° 7 900° =128.57° 7 n=11 (n-2)180° n (11-2)180° 11 9*180° 11 1,620° =147.27° 11

n=12 (n-2)180° n (12-2)180° 12 10*180° 12 1,800° =150° 12

(n-2)180=900° 180 180 n-2=5 +2 +2 n=7 sides Heptagon

(n-2)180=3,060° 180 180 n-2=17 +2 +2 n=19 sides

19-gon

(n-2)180=3,240° 180 180 n-2=18 +2 +2 n=20 sides 20-gon

(n-2)180=3,780° 180 180 n-2=21 +2 +2 n=23 sides 23-gon

(n-2)180=1,080° 180 180 n-2=6 +2 +2 n=8 sides

Octagon

(n-2)180=3,420° 180 180 n-2=19 +2 +2 n=21 sides

21-gon

(n-2)180=2,160° 180 180 n-2=12 +2 +2 n=14 sides

14-gon

(n-2)180=2,340° 180 180 n-2=13 +2 +2 n=15 sides 15-gon

n=8 (n-2)180° (8-2)180 6*180 1,080° 172°+131°+141°+126°+154°+106°+167°+x=1080°

997°+x=1080° -997° -997° x=83°

n=7 (n-2)180° (7-2)180 5*180 900° x+x+113°+124°+169°+148°+89°=900° 643°+2x=900° -643° -643° 2x=257° 2 2 x=128.5 n=6 (n-2)180° (6-2)180 4*180 720°

x+134°+86°+x+120°+94°+x=1,260° 434°+2x=720° -434° -434° 2x=286° 2 2 x=143°

n=9 (n-2)180° (9-2)180 7*180 1,260°

110°+173°+169°+x+168°+155°+122°+175°+157°=1260°

1229°+x=1260° -1229° -1229° x=31°

n=10 (n-2)180° (10-2)180 8*180 1,440°

128°+164°+154°+170°+132°+90°+174°+x+148+161=1260°

1,321°+x=1,440° -1,321° -1,321° x=119°

n=9 (n-2)180° n (9-2)180° 9 7*180° 9 1,260° =140° 9

n=5 (n-2)180° (5-2)180 3*180 540°

x+35+x-4+x+46+2x+1+x-18=540° 6x+60=720° -60 -60° 6x=660° 6 6 x=110°

n=7 (n-2)180° n (7-2)180 7 5*180 7 128.57°

2x+38=128.57° -38 -38 2x=90.57 2 2 x=45.29°

n=8 (n-2)180° n (8-2)180 8 6*180 8 135° 2x-17=135° +17 +17 2x=152 2 2 x=76°

n=6 (n-2)180° (6-2)180 4*180 720° x+81+4x-10+x+35+5x-60+3x+23+x+21=720°

15x+90=720° -90 -90° 15x=630° 15 15 x=42°

n=10 (n-2)180° n (10-2)180 10 8*180 10 144°

n=9 (n-2)180° (9-2)180 7*180 1,260° x+7+x+10+x+18+x-20+x-36+x+23+x+29+x+11+x-5=720°

9x+37=1260° -37 -37° 9x=1223° 9 9 x=135.89°

5x-61=144° +61 +61 5x=205 5 5 x=41°

n=5 360° n 360° 5 72°

n=10 360° n 360° 10 36°

n=9 360° n 360° 9 40°

n=11 360° n 360° 15 24°

n=6 360° n 360° 6 60°

n=7 360° n 360° 7 51.43°

n=11 360° n 360° 11 32.73°

n=12 360° n 360° 12 30°

360° =15° n 360° n* =15° *n n 360°=15°n 15° 15° 24=n 24 sides

360° n 360° n* =20° *n n 360°=20°n 20° 20° 18=n 18 sides

360° n 360° n* =45° *n n 360°=45°n 45° 45° 8=n 8 sides

360° n 360° n* =11.25° *n n 360° = 11.25°n 11.25° 11.25° 32=n 32 sides

360° n 360° n* =36° *n n 360°=36°n 36° 36° 10=n 10 sides

360° n 360° n* =9° *n n 360°=9°n 9° 9° 40=n 40 sides

360° n 360° n* =60° *n n 360°=60°n 60° 60° 6=n 6 sides

360° n 360° n* =30° *n n 360°=30°n 30° 30° 12=n 12 sides

47°+38°+88°+x+36°+97°=360° 306°+x=360° -306° -306° x=54°

75°+65°+45°+x+92°+29°+31°=360° 337°+x=360° -337° -337° x=23°

39°+84°+20°+31°+25°+x+23°+18°=360° 70°+63°+50°+94°+82°+x=360° 240°+x=360° 359°+x=360° -240° -240° -359° -359° x=120° x=1°

7°+x+37°+27°+21°+23°+58+59°+21°+35=360°

288°+x=360° -288° -288° x=72°

23°+71°+64°+34°+56°+x+26°=360° 274°+x=360° -274° -274° x=86°

n=9 360° n 360° 9 40°

x+13°+x+56°+x+13°+2x°+2x+67°+x+18°+x+23°=360°

9x+190°=360° -190° -190° 9x=170° 9 9 x=18.88°

2x+28°=40° -28°-28° 2x=12° 2 2 x=6°

n=10 360° n 360° 10 36°

3x-15°=36° +15°+15° 3x=51° 3 3 x=17°

3x+x+12°+x+31°+2x+x-3°=360° 8x+40°=360° -40° -40° 8x=320° 8 8 x=40° n=6 360° n 360° 6 60°

3x-20+x-15°+2x-24°+3x-9°+2x°=360° 11x-68°=360° +68° +68° 11x=428° 11 11 x=38.91°

2x+22°=60° -22°-22° 2x=38° 2 2 x=19°

D=2r AC=22

ATC+AC=360° 276°+AC=360° -276° -276° AC=84°

mSRU and mSTU=180° They are semicircles.

C=2∏r C=2∏7 C=14∏≈43.98

EF=13 All radii from the same circle are equal.

D=2r 28=2r 2 2 14=AH

TN=12 All radii from the same circle are equal.

C=∏D C=∏32 C=32∏≈100.53

CAN+CN=360° 226°+CN=360° -226° -226° CN=134°

DOW+DW=360° 294°+DW=360° -294° -294° CN=66°

C=2∏r C=2∏13 C=26∏≈81.68

C=∏D C=∏17 C=17∏≈53.41

C=∏D C=∏30 C=30∏≈94.25

C=2∏r C=2∏24 C=48∏≈150.8

OT=43 All radii from the same circle are equal.

D=2r 32=2r 2 2 16=UV

C=∏D 21∏=∏NI ∏ ∏ 21=NI

AN+CN+AF=360° 32°+94°+AF=360° 126°+AF=360° -126° -126° AF=234°

C=2∏r 22∏=2∏IH 2∏ 2∏ 11=IH

C=∏D 16∏=∏HT ∏ ∏ 16=HT

C=2∏r 36∏=2∏r 2∏ 2∏ 18=VA

OP+PS=180° 37°+PS=180° -37° -37° PS=143°

C=2∏r 18∏=2∏RA 2∏ 2∏ 9=RA

FN+UF+UN=360° 168°+34°+UN=360° 202°+UN=360° -202° -202° UN=158°

2(m C)=mAB 2x=124° 2 2 x=62°

m H=mGI 139°=x

2(m N)=mOM 2x=78° 2 2 x=39°

m T=mSU 171°=x

m E=mDF x=167°

2(m K)=mJL 2(96°)=x 192°=x

m H=mGI x=113°

2(m W)=mVY 2(90°)=x 180°=x

2(m D)=mAB 2(32°)=mAB 64°=mAB m C=mAB x=64°

2(m K)=mHI 2(25°)=mHI 50°=mHI m J=1/2mDF x=1/2(50°) x=25°

2(m R)=mQS 2(51°)=mQS 102°=mQS m T=mQS x=102°

m E=mDF 48°=mDF m G=1/2mDF x=1/2(48°) x=24°

m N=mLM 162°=mLM m G=1/2mLM x=1/2(162°) x=81°

2(m W)=mUV 2(41°)=mUV 82°=mUV m Y=mUV x=82°

2(m D)=mAC 2(81°)=mAC 162°=mAC m B=1/2mAC x=1/2(162°) x=81°

2(m F)=mEH 2(89°)=mEH 178°=mEH m G=1/2mEH x=1/2(178°) x=89°

mIJK+mKLI=360° 190°+mKLI=360° -190° -190° mKLI=170°

mPMN+mNOP=360° 182°+mKJL=360° -182° -182° mNOP=178° m M=1/2mNOP x=1/2(178°) x=89°

m J=1/2mKLI x=1/2(170°) x=85° mRST+mTQR=360° 122°+mTQR=360° -122° -122° mTQR=238° m S=1/2mTQR x=1/2(238°) x=119°

2(m C)=mDAB 2(134°)=mDAB 268°=mDAB mDAB+mDCB=360° 268°+mDCB=360° -268° -268° mDCB=92° m A=1/2mDCB x=1/2(92°) x=46°

2(m V)=mUYW 2(68°)=mUYW 136°=mUYW mUYW+mUVW=360° 136°+mUVW=360° -136° -136° mUVW=224° m Y=1/2mUVW x=1/2(224°) x=112°

2(m F)=mEHG 2(123°)=mEHG 246°=mEHG mEHG+mEFG=360° 246°+mEFG=360° -246° -246° mEFG=114° m H=1/2mEFG x=1/2(114°) x=57°

2(m J)=mILK 2(111°)=mILK 222°=mILK mILK+mIJK=360° 222°+mIJK=360° -222° -222° mIJK=138°

2(m N)=mMPO 2(141°)=mMPO 282°=mMPO mMPO+mMNO=360° 282°+mMNO=360° -282° -282° mMNO=78°

m A=1/2mDCB x=1/2(138°) x=69°

m H=1/2mMNO x=1/2(78°) x=39°

m R=1/2mQS m 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°

m Z=1/2mWY m Z=1/2(32°) m Z=16°

m Z+m Y+m W=180° 16°+73°+x=180° 89°+x=180° -89° -89° x=91° m E=1/2mDF m E=1/2(120°) m E=60° m E+m F+m D=180° 60°+90°+m D=180° 150°+m D=180° -150° -150° m D=30° 2(m D)=mFE 2(30°)=x 60°=x

m L=1/2mJK m L=1/2(248°) m L=124°

m L+m K+m J=180° 124°+24°+m J=180° 148°+m J=180° -148° -148° m J=32° 2(m D)=mLK 2(32°)=x 64°=x

m U=1/2mVT m U=1/2(142°) m U=71° m U+m T+m V=180° 71°+65°+x=180° 136°+x=180° -136° -136° x=44°

m A=1/2mCB m A=1/2(146°) m A=73° m A+m C+m B=180° 73°+87°+x=180° 160°+x=180° -160° -160° x=20° m H=1/2mDF m H=1/2(106°) m H=53°

m H+m G+m I=180° 53°+94°+m I=180° 147°+m I=180° -147° -147° m I=33° 2(m D)=mGH 2(33°)=x 66°=x m M=1/2mNO m M=1/2(124°) m M=62° m M+m N+m O=180° 62°+x+x=180° 62°+2x=180° -148° -148° 2x=32 2 2 x=16°

6x=3*4 6x=12 6 6 x=2

9x=6*3 9x=18 9 9 x=2

2x=12*3 2x=36 2 2 x=18

8x=16*3 8x=48 8 8 x=6

20x=10*8 20x=80 20 20 x=4

16x=12*4 16x=48 16 16 x=3

7x=13*9 7x=117 7 7 x≈16.71

21x=13*14 21x=182 21 21 x≈8.67

2(x+1)=5*6 7x+7=30 -7 -7 7x=23 7 7 x≈3.29

2(x+3)=3*6 2x+6=18 -6 -6 2x=12 2 2 x=6

8(x-3)=4(x+5) 8x-24=4x+20 -4x -4x 4x-24=20 +24 +24 4x=44 4 4 x=11

2(x+9)=3*13 2x+18=39 -18-18 2x=21 2 2 x=10.5

2(x-6)=1(x+3) 2x-12=x+3 -x -x x-12=3 +12 +12 x=15

5(x-2)=4(x+3) 5x-10=4x+12 -4x -4x x-10=12 +10 +10 x=22

7(x-7)=6(x+6) 7x-49=6x+36 -6x -6x x-49=36 +49 +49 x=85

5(x+5)=5(2x) 5x+25=10x -5x -5x +25=5x 5 5 5=x

AB=AC 10=AC

DE=DF 12=DF

GH=GI 15=GI

MN=MO x+5=2x-11 -x -x 5=x-11 +11 +11 16=x

ST=SU 2x+12=3x-8 -2x -2x 12=x-8 +8 +8 20=x

JK=JL 4=JL

PQ=PR x+7=2x-9 -x -x 7=x-9 +9 +9 16=x

VW=VX x+27=4x+13 -x -x 27=3x+13 -13 -13 14=3x 3 3 4.67≈x

13=x

72+132=x2 49+169=x2 218=x2 √218=√x2 14.76≈x

52+x2=62 25+x=36 -25 -25 x2=11 √x2=√11 x≈3.32

127=x

9=x

3-4-5 right triangle x=5

18=x

45-45-90 right triangle x=9√2

27°+x=90° -27° -27° x=63°

34°+x=90° -34° -34° x=56°

166°/2=FG 83°=FG

53°+x=90° -53° -53° x=37°

134°/2=mAB 67°=mAB

184°/2=mMO 92°=mMO mMO+mMN=180° 92°+x=180° -92° -92° x=88°

61°+x=90° -61° -61° x=29°

154°/2=mQR 77°=mQR mQR+mRS=180° 77°+x=180° -77° -77° x=103°

62+x2=92 36+x=81 -36 -36 x2=45 √x2=√45 x≈6.71

90°+48°+x=180 138°+x=180° -138° -138° x=42°

m JIG+m GIH=90° 65°+m GIH=90° -65° -65° m GIH=25°

142+x2=192 196+x=361 -196 -196 x2=165 √x2=√165 x≈12.85

90°+38°+x=180° 128°+x=180° -128° -128° x=52°

m BAC+m CAF=90° 23°+m CAF=90° -23° -23° m CAF=67°

m QRT+m TRU=90° 37°+m TRU=37° -37° -37° m TRU=53°

3-4-5 rt. triangle scale factor =3 3*3=9 x=9

312+132=x2 961+169=x2 1130=x2 √1130=√x2 33.62≈x

m BDE=149° central angle m CEB+m BED=90 ° 83°+m BED=90° -83° -83° m BED=7° 149°+7°+x=180 156°+x=180° -156° -156° x=24° m OML=1/2(180°) m OML=90° m NOM+m MOL=90° 32°+m MOL=90° -32° -32° m MOL=58° 90°+58°+x=180 148°+x=180° -148° -148° x=32° x 8* tan50°= 8 *8 8(tan50°)=x 8(1.1918)=x 9.53=x

90°+65°+x=180 155°+x=180° -155° -155° x=25°

m HGJ=1/2(180°) m HGJ=90° m KJG+m GJH=90° 71°+m GJH=90° -71° -71° m GJH=19° 90°+19°+x=180 109°+x=180° -109° -109° x=71° 3-4-5 rt. triangle scale factor =6 5*6=30 QT=30

m EDB=84°central angle m ABE+m EBD=90° 45°+m EBD=90° -45° -45° m EBD=45° 84°+45°+x=180 129°+x=180° -129° -129° x=51°

120° = 1 360° 3 C=2πr C=2π6 C=12π 12∏ 12π* 1 = = 4∏ 3 3 4∏≈12.57

78° 13 = 360° 60 C=πr2 C=π82 C=64π 832∏ 64π* 13 = = 60 60 208∏ ≈43.56 15

135° 3 = 360° 8 C=2πr C=2π10 C=20π 60∏ 20π* 3 = = 8 8 15∏ ≈23.56 2

10° = 1 360° 36 C=2πr C=2π13 C=26π 26∏ 26π* 1 = = 36 36 13∏ ≈2.27 18

272° 34 = 360° 45 C=2πr C=2π18 C=36π 1224∏ 36π* 34 = = 45 45 136∏ ≈85.45 5

200° 5 = 360° 9 C=2πr C=2π34 C=68π 340∏ 68π* 5 = = 9 9 340∏≈118.68 9

20° = 1 360° 18 C=2πr C=2π7 C=14π 14∏ 14π* 1 = = 18 18 7∏ ≈2.44 9

130° 13 = 360° 36 C=2πr C=2π15 C=30π 390∏ 30π* 13 = = 36 36 65∏ ≈34.03 6

90° = 1 360° 4 C=2πr C=2π12 C=24π 24∏ 24π* 1 = = 4 4 6∏ ≈18.85

180° 1 = 360° 2 C=2πr C=2π22 C=44π 44∏ 44π* 1 = = 2 2 22∏ ≈69.12

80° 2 = 360° 9 C=πr2 C=π22 C=4π 8∏ 4π* 2 = 9 9 8∏ ≈2.79ft2 9

=

90° = 1 360° 4 C=πr2 C=π122 C=144π 144π* 1 =144∏ = 4 4 144∏ ≈113.1cm2 4

320° 8 = 360° 9 C=πr2 C=π152 C=225π 225π* 8 =1800∏= 9 9

200∏ ≈628.32m2

276° 23 = 360° 30 C=πr2 C=π232 C=529π 12167∏ 529π* 23 = = 30 30 12167∏ ≈1274.13in2 30

140° 7 = 360° 18 C=πr2 C=π52 C=25π 175∏ 7 25π* = = 18 18 175∏ 2 18 ≈30.54in

130° 13 = 360° 36 C=πr2 C=π192 C=361π 361π* 1 =361∏ = 36 36 361∏ 2 36 ≈31.50mm

157° 157 = 360° 360 C=πr2 C=π312 C=961π 961π*157=7688∏= 360 360 961∏ 2 45 ≈67.09ft

21° 7 = 360° 120 C=πr2 C=π72 C=49π 49π* 7 =343∏ = 120 120 343∏ ≈8.98mm2 120

(4,8) (-6,3) x1 y 1 x2 y 2 x1+x2 y1+y2 2 2 4+(-6) 8+3 2 2 -2 11 2 2 (-1,5.5)=MP

(-3,-9) x1 y 1

(8,-12) x2 y 2

x1+x2 y1+y2 2 2 5+(8) -7+(-12) 2 2 13 -19 2 2 (6.5,-8.5)=MP

(-4,6) x2 y 2

x1+x2 y1+y2 2 2 -3+(-4) -9+(6) 2 2 -7 -3 2 2 (-4.5,-1.5)=MP

(-13,4) (0,9) x1 y 1 x2 y 2 x1+x2 y1+y2 2 2 -13+0 4+9 2 2 -13 13 2 2 (-6.5,6.5)=MP

(12,15) x1 y 1

(5,-7) x1 y 1

(14,-7) x2 y 2

x1+x2 y1+y2 2 2 12+14 15+(-7) 2 2 26 8 2 2 (13,4)=MP

(-7,8) x1 y 1

(-11,13) x2 y 2

x1+x2 y1+y2 2 2 -7+(-11) 8+(13) 2 2 -18 21 2 2 (-9,10.5)=MP

(10,-4) (15,3) x1 y 1 x2 y 2 x1+x2 y1+y2 2 2 10+15 -4+3 2 2 25 -1 2 2 (12.5,-.5)=MP

(19,-12) x1 y 1

(-19,-21) x2 y 2

x1+x2 y1+y2 2 2 19+(-19) -12+(-21) 2 2 0 -33 2 2 (0,-16.5)=MP

A (5,7) B (x2,y2) M (-3,-4) x1 y 1 xm y m x1+x2 2 =xm

y1+y2 =ym 2 7+y 5+x 2* 2 2 =-3*2 2* 2 2 =-4*2 7+y2=-8 5+x2=-6 -7 -77 -5 -5 y2=-15 x2=-11 (-11,-15)=B

L (8,-9) N (x2,y2) M (8,7) x1 y 1 xm y m x1+x2 2 =xm

8+x2 2 =8 *2 8+x2=16 -8 -8 x2=8

2*

y1+y2 =ym 2 -9+y 2* 2 2 =7 *2 -9+y2=14 +9 +9 y2=23

(8,23)=N

J (-12,5) L (x2,y2) K (11,-10) x1 y 1 xm y m x1+x2 2 =xm

y1+y2 =ym 2 5+y -12+x 2* 2 2 =11*2 2* 22 =-10 *2 5+y2=-20 -12+x2=22 -5 -5 +12 +12 y =-25 x2=34 2 (34,-25)=L

R (-7,-4) S (x2,y2) Q (8,-6) x1 y 1 xm y m x1+x2 2 =xm

2*

-7+x2 2 =8 *2 -7+x2=16 +7 +7 x2=23

y1+y2 =ym 2 -4+y 2* 2 2 =-6*2 -4+y2=-12 +4 +4 y2=-8

(23,-8)=S

H (1,0) F (x2,y2) G (-4,-6) x1 y 1 xm y m x1+x2 2 =xm

y1+y2 =ym 2 0+y 1+x 2* 2 2 =-4*2 2* 2 2 =-6*2 y2=-12 1+x2=-8 -1 -1 x2=-9 (-9,-12)=F X (-8,13) Z (x2,y2) Y (-5,-15) x1 y 1 xm y m x1+x2 2 =xm

y1+y2 =ym 2 13+y -8+x 2* 2 2 =-5*2 2* 2 2 =-15 *2 13+y2=-30 -8+x2=-10 -13 -13 +8 +8 y2=-43 x2=-2 (-2,-43)=B

(3,9) x1 y 1

(-8,4) x1 y 1

(-2,-7) x2 y 2

D=√(x2-x1) +(y2-y1) D=√(-2-3)2+(-7-9)2 D=√(-5)2+(-16)2 D=√25+256 D=√281 D≈16.76 2

(7,9) x1 y 1

2

(-6,7) x2 y 2

D=√(x2-x1)2+(y2-y1)2 D=√(-6-7)2+(7-9)2 D=√(-13)2+(-2)2 D=√169+4 D=√173 D≈13.15

(13,7) x1 y 1

(-10,-4) x2 y 2

D=√(x2-x1)2+(y2-y1)2 D=√(-10-13)2+(-4-7)2 D=√(-23)2+(-11)2 D=√529+121 D=√650 D≈25.50 (15,-16) x1 y1

(10,-5) x2 y 2

D=√(x2-x1)2+(y2-y1)2 D=√(10-15)2+(-5-(-16))2 D=√(-5)2+(11)2 D=√25+121 D=√166 D≈12.88

(5,-4) x2 y 2

D=√(x2-x1)2+(y2-y1)2 D=√(5-(-8))2+(-4-4)2 D=√(13)2+(-8)2 D=√169+64 D=√233 D≈15.26

(3,-9) x1 y 1

(8,2) x2 y 2

D=√(x2-x1)2+(y2-y1)2 D=√(8-3)2+(2-(-9))2 D=√(5)2+(11)2 D=√25+121 D=√146 D≈12.08

(-5,-13) x1 y1

(-6,-12) x2 y 2

D=√(x2-x1)2+(y2-y1)2 D=√(-6-(-5))2+(-12-(-13))2 D=√(-1)2+(1)2 D=√1+1 D=√2 D≈1.41 (-2,64) x1 y1

(-28,61) x2 y 2

D=√(x2-x1)2+(y2-y1)2 D=√(-28-(-2))2+(61-64)2 D=√(-26)2+(-3)2 D=√676+9 D=√685 D≈28.16

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Let s Practice Geometry

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