sss/sas/asa proofs

SSS/SAS/ASAProofs

21

C

A BD

Given: CD AB; 1 2

Prove: ADC BDC

!

Quiz Question

43

W

Y ZX

Given: WX YZ; 3 4

Prove: YXW ZXW

!

HG I

JKF L

Given:

;

Prove:

GKF ILJ

GK IL FL KJ

GF IJ

!

Given: ; 1 2

Prove:

AB AC

B C

1

2

A

C

D

B

!

N R O

MGiven: MR NO;MR bisects NMO

Prove:R is midpoint of NO

!

1 2C

A

B E

D

Given: ; 1 2

Prove:

AC CD

AB ED

!

4 5O

N

M P

Q

Given: ; 4 5

Prove:

NO OQ

MO PO

!Quiz Question

C

BEDA

1 2

Given: ; ;

Prove: 1 2

AC BC AE BD CD CE

H

D

G

B

CE

F 1 2

Given: 1and aresuppplementary

;

Prove: ||

B

BC EG DB HE

GH CD

1 2C

A

B E

D

Given: Cis the midpoint of AE and BD

Prove: 1 2

1 23 4A

B

C

D

E

F

Given: AB||EF; BC||DE; AD CF

Prove: ABC FED

A

XY

12

34

Given: 1 3;AB AC

Prove: ABY ACX

B C

A

XY

12

34

Given: AB AC;AX AY

Prove: 2 4

R

S

U

T12

3

4

X

Given: RS RU; 1 2

Prove: 3 4

D C

A B Y

1 3 4 2

Given: 1 2;AD BC

Prove: AC BD

E

DA

GC

F

B

Given : ABC ADE,DE BC

E and ABCaresupplementary to DCF

Prove: AD bisects EAB

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Q

R S

TP

Given: RP||ST, RQ ST

Prove: S RPT

A

DC

BE

Given: C is the midpoint of BD

AB BD and BD DE

Prove: = ,and that Cisa midpointAC CE

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Quiz Question

Given as homework

CA

B

D

Given: AB BC;Dis midpoint of AC

Prove: BD AC

Hw 11.3

CA

B D

EGiven: E is midpoint of AD and BC

Prove: AB || CD

Hw 11.3

F

ED

A CB

GGiven: F G, FB GB, DG FE, AD CE

Prove: AB CB

Hw 11.4

VT

WR

SU

Given: S V, RS UV, ST VW

Prove: RW UT

Hw 11.4

M S P Q

N R

Given: NM MQ; RQ MQ; MN QR; MS QP

Prove: NP RS

Hw 11.5

CA

B

D

Given: ;

Prove: bisects

AB CB AD CD

BD ABC

Hw 11.5

1 2 3 4

A

B M C

Given: 1 4; AB AC

M is midpoint of BC

Prove: AMB AMC

Hw 11.6

XW

Z Y3

41

2Given: WX YZ; ZW XY

Prove: WX||ZY

QUIZ

Triangle Congruence and Similarity Proofs

Augustus VitugMelissa Moize

Proof Title:

Sketch

• AB AC• BD DC • AD AD• BAD CAD • • • • • • • • • • •

• Given • Given • Reflexive property • SSS• • • • • • • • •

GivenAB ACBD DC

ProveBAD CAD

Statement Reason

Proof Title:

Sketch

• ABC DCB• CB BC• DBC ACB• ABC DCB• CA DB• AB CD• • • • • • • • •

• Alt. Int. ’s • reflexive• Alt. Int. ’s • ASA • CPCTC• CPCTC• • • • • • • •

Given ABCD

ProveAC BDCD AB

Statement Reason

Proof Title: Parallelogram Diagonals Bisect Each Other

Sketch

• ABC DCB• CB BC• DBC ACB• ABC DCB• AB CD• ADC BAD• ABE DCF• AE DE• BECE• • •

• Alt. Int. ’s • Reflexive property • Alt. Int. ’s • ASA• CPCTC • Alt. Int. ’s• ASA • CPCTC• CPCTC• • • • • •

Given

Prove

Statement Reason

ABCD

AEDE

BECE

Proof Title:

Sketch

• ABC DCB• CB BC• ACB DBC• ABC DCB• A D• ABC + DBC DCB +ACB

• • • • • • • • •

• Alt. Int. ’s• reflexive• Alt. Int. ’s• ASA• CPCTC• 2nd Postulate • • • • • • • •

Given

ProveA D

ABD DCA

Statement Reason

Proof Title: Compound Polygon #1

Sketch

• ABC DCB• CBBC• CBD BCA• ABC DCB• FBD+BFD+BDF

ECA+CEA+CAE• BDF CAE• AEFD • ACBD• BFD CEA• CE BF• EF FE• EF+FB FE+EC• EB CF • AB CD• AEFD • CFD BEA

• Alt. int. ’s• reflexive property • Alt. int. ’s• ASA • Triangle Sums • Subtraction Property • Given • CPCTC • ASA• CPCTC • Reflexive Property • Sums of equals • Sums of equals• CPCTC • Given • SSS

Given

Prove

Statement Reason

ABCD

AEFD

AEC=90

BFD=90

CFD BEA

Proof Title: Compound # 2

Sketch

• AC BD• ACD BDC• CD DC• ADC BDC• CE+BE DE+AE• CE DE• CED is isos• • • • • • • •

• Given • Def. isos • Reflexive• SAS• Sums of Equals • Subtraction property • def. isos. • • • • • • •

Given

ProveCED is isos

Statement Reason

ABCD is iso.

Proof Title: ?

Sketch

• AB CD• B D• BE DE• BEA DEC• G H• EC EA• FE XE• FEA XEC• AF CX• • • • • • •

• Given • Given• Def. of Midpoint • SAS • CPCTC • CPCTC • Def. of Midpoint • SAS • CPCTC • • • • •

GivenAB CD B DE is the midpoint of

BD & FX

ProveAF CX

Statement Reason

Proof Title: Simple Bowtie

Sketch

• AECE• DEC BEA• BE DE• AEB CED• ABE CDE• ABE,CDE are alt. int. ’s• AB || DC• • • • • • •

• Def. Bisector • Vertical ’s• Def. Bisector . • SAS • CPCTC • Def. Alt. Int. ’s• Alt. Int. ’s are congruent • • • • • • •

Given

Prove AB || CD

Statement Reason

Line AC bis. Line BD

Proof Title:

Sketch

• A D• BC FE• B = 90- A E = 90- D

• B E• ABC DEF• BA DE• • • • • • • • •

• given• given• difference of equals• - property• ASA• CPCTC• • • • • • • •

GivenABC is a Right

BC FEA D

ProveBA DE

Statement Reason

Proof Title: Compound Bowtie #1

Sketch

• • AGE+EGB CGF+ FGD• • AGB CGD• FGD EGB• • ABGCDG• DFG BEG• • • • • •

• Given• Vertical ’s• Given• SAS • Vertical ’s• Given • Alt. Int. ’s• ASA• CPCTC • • • • •

Given

Prove

Statement Reason

CGAG

DGBG

FGEG

DGBG

CGAG

DGBG

FGEG

Proof Title:

Sketch

• AED BEC • DE CE • ADC-EDC BCD ECD • ADE BCE • ADE BCE • AD BC • ADC BCD • DC CD • ADC BCD • • • • •

• Vertical ’s • Def. iso. • Dif. Of equals • Sub. Prop. • ASA • CPCTC • Sum of Equals • Reflexive Prop. • SAS • • • • •

GivenEDC is isosceles

ProveADC BCD

Statement Reason

Proof Title:

Sketch

• AE BE • AED BEC • DE CE • ADE BCE • FDE GCE • DECE • FED GEC • FD GC• EDC ECD• ADE+EDC BCE+ ECD • ADCBCD • FGDC • •

• Given • Vert. ’s• def. iso. • SAS• CPCTC• def. iso. • alt. int. ’s• CPCTC • Def. iso. • Addition prop. Of equality • • • •

GivenAEBEEDC is isosceles

ProveFG DC

Statement Reason

Proof Title: Diagonals of a Rectangle

Sketch

• AC BD • ACD BDC • CD DC• ACD BDC • AD BC• • • • • • • • • •

• Dfn. Rectangle • Dfn. Rectangle• Reflexive• SAS• CPCTC• • • • • • • • •

Given

Rectangle ABCD

Prove AD BC

Statement Reason

Proof Title:

Sketch

• BAM CAM • ABAC • B C• BAM CAM • BM CM • BMA CMA • BMC 180 • BMA = 1/2180=90=

CMA • AM bis. Of BC • • • •

• bis. • Given • Def. iso. • ASA • CPCTC • CPCTC • straight ’s• Angle Addition • • • • • •

GivenABC w/AM bis. ABAC

ProveAM bis.

Statement Reason

Proof Title:

Sketch

• ¾=5/6.6666• AD/AB=AE/AC• A A • ADE ABC• • • • • • • • • • •

• 36&2/3=4 5= 20 • CPSTP• Reflexive • SAS • • • • • • • • • •

GivenDE || BC

ProveADE ABC

Statement Reason

Proof Title: Simple butterfly

Sketch

• AEBE • BEDAEC • CE DE • AC BD • • • • • • • • • • •

• Given • Vertical angels • Given • CPCTC • • • • • • • • • •

GivenAEBECEDE

ProveAC BD

Statement Reason

Proof Title:

Sketch

• BCA EFD • FC CF • AF CD • AF + FC DC + CF • ABC DEF • • • • • • • • • •

• Alt. Int. ’s • Reflexive • Given • Addition • ASA • • • • • • • • •

GivenAF DCFE || CBAB || DE

ProveABC DEF

Statement Reason

Proof Title:

Sketch

• BG EG • EG + GF=BG+GC • BC EF• FC FC• AF+FC=DC+CF• GFC GCF • AC=CD• ABC DEF • • • • • • •

• Given • Sum’s of =‘s• Segment addition • reflexive • Sum’s of =‘s • def. isosceles • Segment Addition • SAS• • • • • •

GivenFGC Isosceles AF DCBG EG

ProveABC DEF

Statement Reason

Proof Title: Isos. Triangle to bisector

Sketch

• BM CM • AMB = 90 = AMC• AB AC • B C• BAM CAM• BAM CAM• AM is bis. of A• • • • • • • •

• Def. bis. • Def. bis.• given• def. isos. triangle• SAS• CPCTC• • • • • • • •

GivenD ABC is isos.

AB ACAM is

Prove bis. of BC is bis.

Of A

Statement Reason

Proof Title:

Sketch

• BMA=90= CMA• B C• B+ BMA+ BAM=180 • C+ CMA+ CAM =180• B+ BMA+ BAM= C+ CMA+ CAM

• BAM CAM• AM is bis. of A• • • • • • • •

• Def. Alt. • isos triangle• sums• sums• transitive • - property• • • • • • • •

Given ABC is isos.

AC ABAM is Alt.

Prove Alt. AM is bis. of

A

Statement Reason

Proof Title:

Sketch

• AB AC• C B• AMB=90= AMC• BAM CAM• BM CM• Alt. AM is bis. of BC• • • • • • • • •

• given • def. of isos.• def. Alt.• sums• ASA• CPCTC• • • • • • • •

Given ABC is isos.

AB ACAM is Alt.

Prove Alt. AM is bis. of

BC

Statement Reason

Proof Title: Bisectors Of An Isosceles Trapezoid

Sketch

• ACD BDC• CD DC • CEA DEB • AC BD • C D • CE DE • ACE BED • AE BE • • •

• Base ’s • Reflexive Property • Alternate Interior ’s • Definition Isosceles • Base ’s • Definition of Midpoint• SAS • • • •

GivenTrapezoid ABCD is

and isosceles E is midpoint of CD

ProveAE BE

Statement Reason

Proof Title: Pythagorean Theorem

Sketch

• ABC is a Rt. Triangle• CD is the altitude to hyp.

• c/a=a/e

• a2=c e• c/b=b/d

• b2=cd• a2 + b2 = ce + cd• a2+b2=c2

•

• Given • Through a pt. there is 1 line

to a given line • Either leg is geom. Mean

bet. Hyp. And adj. seg. • Multi proportion prop. • Either leg is geom. Mean

bet. Hyp and adj. seg. • Multi. Propportion prop. • Distributive Prop. • Substitution• • •

GivenRt. ABC

Provea2+b2=c2

Statement Reason

Proof Title:

Sketch

• DAB CAB• DBA CBA• AB AB• ABD ABC• • • • • • • • • • • • •

• Given• Given • Reflexive Property• ASA• • • • • • • • • •

GivenDAB CABDBA CBA

Prove ABD ABC

Statement Reason

Proof Title: Compound Butterfly #1

Sketch

• AE FE• AED FEJ• DE JE• ADE FJE• D J• AEB+BEC+CED

FEG+GEH+HEJ• CED HEJ• CED HEJ• BEC GEH• CE HE• BCE +DCE=180• GHE +JHE=180• BCE+DCE GHE+ JHE• BCE GHE• BEC GHE

• given • vert. ’s• given• SAS• CPCTC • sums ’s

• - prop • ASA• given• CPCTC• supp. ’s• supp. ’s• sums ’s• - prop• ASA

Given AE FE, DE JE

BEC GEHAEB FEG

Prove BEC GEH

Statement Reason

Proof Title:

Sketch

• • • • • • • • • • • • • • •

• • • • • • • • • • • • • •

Given

Prove

Statement Reason