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Transcript 
Discovering Geometry
Two-Column Proofs
Name________________________
Date________________________
Proof Puzzles
Each two-column proof is mixed up. Put each two-column proof in order.
1.
Given:
MJ  KL
L
M
ML  KJ
Proof:
J  L
J
K
Statements
Reasons
A.
J  L
CPCTC
B.
JKM LMK
SSS
C.
ML  KJ
Given
D.
MK  MK
Reflexive
E.
MJ  KL
Given
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
2.
Page 2
Given:
Quadrilateral RYAN is a kite
Proof:
N  Y
R
Y
N
A
Statements
Reasons
A.
RN  RY
Definition of a kite
B.
Quadrilateral RYAN is a kite
Given
C.
N  Y
CPCTC
D.
RA  RA
Reflexive
E.
RNA  RYA
SSS
F.
NA  YA
Definition of a kite
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
Page 3
B
3.
Given:
AC  BD
D is the midpoint of AC
Proof:
AB  BC
A
D
Statements
C
Reasons
A.
AB  BC
CPCTC
B.
ADB  CDB
Definition of perpendicular lines
C.
ADB  CDB
SAS
D.
BD  BD
Reflexive
E.
AC  BD
Given
F.
AD  DC
Definition of midpoint
G.
D is the midpoint of AC
Given
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
Page 4
C
4.
Given:
AB || CD
A
E is the midpoint of AD
Proof:
AB  CD
E
D
B
Statements
Reasons
A.
AE  ED
Definition of midpoint
B.
AEB  CED
Vertical angles are 
C.
E is the midpoint of AD
Given
D.
EAB  EDC
Alternate interior angles are 
E.
AB || CD
Given
F.
AB  CD
CPCTC
G.
AEB DEC
ASA
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
Page 5
More Proof Puzzles
Each two-column proof is mixed up, also the statements and reasons
are not matched correctly. Put each two-column proof in order.
1.
B
AB  CB
Given:
ABD  CBD
AD  DC
Proof:
D
A
C
Statements
Reasons
A. ADB  CDB
F. Given
B. BD  BD
G. CPCTC
C. AD  DC
H. SAS
D. AB  CB
J. Reflexive
E. ABD  CBD
K. Given
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
2.
Page 6
A  Z
Given:
A
B
C is the midpoint of AZ
CBA  CYZ
Proof:
C
Y
Statements
Z
Reasons
A. AC  CZ
F. ASA
B. A  Z
G. Definition of midpoint
C. C is the midpoint of AZ
H. Given
D. CBA  CYZ
J. Vertical angles are 
E. ACB  ZCY
K. Given
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
3.
Page 7
A
AB  DC
Given:
AC  DB
Proof:
D
E
ACB  DBC
C
B
Statements
Reasons
A. ABC  DCB
F. SSS
B. AC  DB
G. CPCTC
C. BC  BC
H. Given
D. AB  DC
J. Reflexive
E. ACB  DBC
K. Given
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
4.
Page 8
FGH and JHG are rt. s
Given:
FG  JH
FH  JG
Proof:
J
F
K
H
G
Statements
Reasons
A. FGH  JHG
G. Reflexive
B. FGH and JHG are rt. s
H. CPCTC
C. FG  JH
J. All right angles are 
D. FH  JG
K. Given
E. FGH  JHG
L. SAS
F. GH  GH
M. Given
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
Page 9
More and More Proof Puzzles
Each two-column proof is mixed up, also the statements and reasons
are not matched correctly or necessarily on the correct side.
Put each two-column proof in order.
1.
WZ  YZ
Given:
W
VZ  ZX
VW  XY
Proof:
V
X
Z
Y
Statements
Reasons
A. WZ  YZ
F. Given
B. Given
G. VZ  ZX
C. Vertical angles are 
H. CPCTC
D. VZW  XZY
J. VW  XY
E. SAS
K. WZV  XZY
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
Page 10
A
2.
AB  BC
Given:
AE  DE
CAB  EAD
AB  AE
AC  AD
Proof:
D
C
B
Statements
E
Reasons
A. Definition of perpendicular lines
H. CPCTC
B. ABC  AED
J. Given
C. ABC  AED
K. AC  AD
D. AE  DE
L. AB  BC
E. Given
M. Given
F. AB  AE
N. CAB  EAD
G. Given
O. ASA
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
3.
Page 11
S
R and T are rt. s
Given:
VSR  VST
RV  TV
Proof:
T
R
V
Statements
Reasons
A. Reflexive
G. SRV  STV
B. CPCTC
H. SV  SV
C. All right angles are 
J. R and T are rt. s
D. AAS
K. VSR  VST
E. Given
L. RV  TV
F. Given
M. R  T
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
4.
Given:
Page 12
ABD is isosceles
D
E is the midpoint of AD
F is the midpoint of BD
Proof:
E
DBE  DAF
F
G
A
Statements
B
Reasons
A. SAS
J. Given
B. Definition of isosceles triangle
K. Definition of midpoint
C. ABD is isosceles
L. BED  AFD
D. E is the midpoint of AD
M. Reflexive
E. F is the midpoint of BD
N. Given
F. D  D
O. CPCTC
G. AD  DB
P. DE  DF
H. DBE  DAF
Q. Given
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
Page 13
Practice Proofs
Complete a two-column proof and a flowchart proof.
1.
Given:
Prove:
Quadrilateral CTXV is a kite
C
CTV  XTV
V
T
X
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
Discovering Geometry
Two-Column Proofs
2.
Page 14
Given:
E is the midpoint of DB and AC
Prove:
DA  CB
C
D
E
A
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
B
Discovering Geometry
Two-Column Proofs
3.
Given:
Page 15
O
MNO is isosceles
MP  ON
Q
P
QN  MO
Prove:
MPO  NQO
M
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
N
Discovering Geometry
Two-Column Proofs
Page 16
E
4.
Given:
EG  FI
EF  HI
G is the midpoint of EH
Prove:
G
FEG  IHG
F
H
Adapted from Wayne Nirode by Jo Ann Fricker
Key Curriculum Press Consultants
I
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