SSS/SAS/ASA Proofs
description
Transcript of SSS/SAS/ASA Proofs
![Page 1: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/1.jpg)
SSS/SAS/ASAProofs
![Page 2: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/2.jpg)
21
C
A BD
Given: CD AB; 1 2
Prove: ADC BDC
!
![Page 3: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/3.jpg)
Quiz Question
43
W
Y ZX
Given: WX YZ; 3 4
Prove: YXW ZXW
!
![Page 4: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/4.jpg)
HG I
JKF L
Given:
;
Prove:
GKF ILJ
GK IL FL KJ
GF IJ
!
![Page 5: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/5.jpg)
Given: ; 1 2
Prove:
AB AC
B C
1
2
A
C
D
B
!
![Page 6: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/6.jpg)
N R O
MGiven: MR NO;MR bisects NMO
Prove:R is midpoint of NO
!
![Page 7: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/7.jpg)
1 2C
A
B E
D
Given: ; 1 2
Prove:
AC CD
AB ED
!
![Page 8: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/8.jpg)
4 5O
N
M P
Q
Given: ; 4 5
Prove:
NO OQ
MO PO
!Quiz Question
![Page 9: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/9.jpg)
C
BEDA
1 2
Given: ; ;
Prove: 1 2
AC BC AE BD CD CE
![Page 10: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/10.jpg)
H
D
G
B
CE
F 1 2
Given: 1and aresuppplementary
;
Prove: ||
B
BC EG DB HE
GH CD
![Page 11: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/11.jpg)
1 2C
A
B E
D
Given: Cis the midpoint of AE and BD
Prove: 1 2
![Page 12: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/12.jpg)
1 23 4A
B
C
D
E
F
Given: AB||EF; BC||DE; AD CF
Prove: ABC FED
![Page 13: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/13.jpg)
A
XY
12
34
Given: 1 3;AB AC
Prove: ABY ACX
B C
![Page 14: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/14.jpg)
A
XY
12
34
Given: AB AC;AX AY
Prove: 2 4
![Page 15: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/15.jpg)
R
S
U
T12
3
4
X
Given: RS RU; 1 2
Prove: 3 4
![Page 16: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/16.jpg)
D C
A B Y
1 3 4 2
Given: 1 2;AD BC
Prove: AC BD
![Page 17: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/17.jpg)
E
DA
GC
F
B
Given : ABC ADE,DE BC
E and ABCaresupplementary to DCF
Prove: AD bisects EAB
��������������
![Page 18: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/18.jpg)
Q
R S
TP
Given: RP||ST, RQ ST
Prove: S RPT
![Page 19: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/19.jpg)
A
DC
BE
Given: C is the midpoint of BD
AB BD and BD DE
Prove: = ,and that Cisa midpointAC CE
���������������������������������������� ���
Quiz Question
![Page 20: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/20.jpg)
Given as homework
![Page 21: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/21.jpg)
CA
B
D
Given: AB BC;Dis midpoint of AC
Prove: BD AC
Hw 11.3
![Page 22: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/22.jpg)
CA
B D
EGiven: E is midpoint of AD and BC
Prove: AB || CD
Hw 11.3
![Page 23: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/23.jpg)
F
ED
A CB
GGiven: F G, FB GB, DG FE, AD CE
Prove: AB CB
Hw 11.4
![Page 24: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/24.jpg)
VT
WR
SU
Given: S V, RS UV, ST VW
Prove: RW UT
Hw 11.4
![Page 25: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/25.jpg)
M S P Q
N R
Given: NM MQ; RQ MQ; MN QR; MS QP
Prove: NP RS
Hw 11.5
![Page 26: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/26.jpg)
CA
B
D
Given: ;
Prove: bisects
AB CB AD CD
BD ABC
Hw 11.5
![Page 27: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/27.jpg)
1 2 3 4
A
B M C
Given: 1 4; AB AC
M is midpoint of BC
Prove: AMB AMC
Hw 11.6
![Page 28: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/28.jpg)
XW
Z Y3
41
2Given: WX YZ; ZW XY
Prove: WX||ZY
QUIZ
![Page 29: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/29.jpg)
![Page 30: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/30.jpg)
Triangle Congruence and Similarity Proofs
Augustus VitugMelissa Moize
![Page 31: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/31.jpg)
Proof Title:
Sketch
• AB AC• BD DC • AD AD• BAD CAD • • • • • • • • • • •
• Given • Given • Reflexive property • SSS• • • • • • • • •
GivenAB ACBD DC
ProveBAD CAD
Statement Reason
![Page 32: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/32.jpg)
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
![Page 33: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/33.jpg)
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
![Page 34: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/34.jpg)
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
![Page 35: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/35.jpg)
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
![Page 36: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/36.jpg)
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.
![Page 37: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/37.jpg)
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
![Page 38: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/38.jpg)
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
![Page 39: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/39.jpg)
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
![Page 40: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/40.jpg)
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
![Page 41: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/41.jpg)
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
![Page 42: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/42.jpg)
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
![Page 43: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/43.jpg)
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
![Page 44: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/44.jpg)
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
![Page 45: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/45.jpg)
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
![Page 46: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/46.jpg)
Proof Title: Simple butterfly
Sketch
• AEBE • BEDAEC • CE DE • AC BD • • • • • • • • • • •
• Given • Vertical angels • Given • CPCTC • • • • • • • • • •
GivenAEBECEDE
ProveAC BD
Statement Reason
![Page 47: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/47.jpg)
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
![Page 48: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/48.jpg)
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
![Page 49: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/49.jpg)
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
![Page 50: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/50.jpg)
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
![Page 51: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/51.jpg)
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
![Page 52: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/52.jpg)
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
![Page 53: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/53.jpg)
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
![Page 54: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/54.jpg)
Proof Title:
Sketch
• DAB CAB• DBA CBA• AB AB• ABD ABC• • • • • • • • • • • • •
• Given• Given • Reflexive Property• ASA• • • • • • • • • •
GivenDAB CABDBA CBA
Prove ABD ABC
Statement Reason
![Page 55: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/55.jpg)
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
![Page 56: SSS/SAS/ASA Proofs](https://reader036.fdocuments.net/reader036/viewer/2022062410/56815759550346895dc50376/html5/thumbnails/56.jpg)
Proof Title:
Sketch
• • • • • • • • • • • • • • •
• • • • • • • • • • • • • •
Given
Prove
Statement Reason