Download parallelogram round robin proofs

August 28, 2014
August 28, 2014
August 28, 2014
Name:_______________________!
B
A
E
!
Given: ABCD is
a rectangle.
Prove: AC = BD.
ABCD is a
parallelogram. Prove:
AE = EC and
D
C
BE = ED.
!
Statement
ABCD is a
parallelogram
given
AB || DC, AD || BC
definition of a
parallelogram
angle EAB congr. to
angle ECD, angle ABE
congr. to angle EDC
AIAs congruent
triangle AEB congr. to
triangle CED
ASA postulate
AE congr. to CE, DE
congr. to BE
CPCTC
!
!
!
!
!
!
!
!
!
!
!
!
!
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Reason
C
D
Statement
Reason
A
Given: ABCD is
a rhombus.
B
A
Date: _____________________
Prove: AC BD.
!
B
E
D
C
!
Statement
Reason
ABCD is a rectangle
given
ABCD is a rhombus
given
angles BAD, ADC,
DCB, and CBA are right
definition of a rectangle
AD congr. BC congr.
CD congr. BA
definition of a rhombus
AD congr. to BC
opposite sides of a
parallelogram are
congruent (this is the
first theorem we proved
on parallelograms)
DE congr. EB, AE
congr. EC
diagonals of a
parallelogram bisect one
another (see leftmost
proof on this page)
angle ADC congr. to
angle BCD
all right angles are
congruent
triangle ABE congr.
triangle CBE
SSS postulate
DC congr. to DC
reflexive property of
congruence
angle AEB congr.
angle CEB
CPCTC
triangle ADC congr. to
triangle CBD
SAS postulate
m(angle AEB) +
m(angle CEB) = 180º
linear pair postulate
AC congr. to DB
CPCTC
m(angle AEB) =
m(angle CEB)
definition of congruence
m(angle AEB) +
m(angle AEB) = 180ª
substitution
2m(angle AEB) =
180ª
combining like terms
m(angle AEB) = 90ª
division property of
equality
AC perp. to BD
definition of perpendicular
lines
!
!
!
!
!
!
!
!
!
~
(x +would
3y) ABCD still
be a parallelogram?
Explain.
Q(-2, A-4)D
Q(3, 6)
R(2, -4)
R(- 1, - 1)
B C _____________________
! S(2, 3) Date:
S(3, - a)
y-1
Name:_______________________!
Write a two-column or a paragraph proof usingWrite
each amethod.
two-column or a paragraph proof.
15. Given: ~MJK’~ ~KLM
a. By
Theorem
6.6:
If both
14.
Given:
AB -~
CD,
BC ~pairs
AF of opposite
sides of a quadrilateral are congruent, then
!-AFD ~ !-ADF
the quadrilateral is a parallelogram.
Prove: 6.10:
ABCD isIfa one
parallelogram.
b. By Theorem
pair of opposite
Prove: MJKL is a parallelogram,
K
Write a two-column or a paragraph proof.
15
Given:
~ ~,ONPHHK
17.
Given:ARQP
Parallelogram
~ ~JOI of ~QQ.
R GHOI
is the midpoint
Prove:MRON
H1JKisisa aparallelogram.
rhombus.
Prove:
sides of a quadrilateral
are congruent
and parF
C
D II
M
L
!
!
Geometry
Statement
Reason
Chapter 6 Resource Book
triangle MJK congr.
triangle KLM
given
angle JKM congr. angle
LMK, angle JMK congr.
angle LKM
CPCTC
JK || ML, JM || KL
AIAs congruent implies
lines are parallel
MJKL is a parallelogram
!
!
!
definition of a
parallelogram
A
!
Cop~,right © McDougal Littel~ Inc.
All nghts reserved,
Statement
!
given
triangle AFD is
isosceles
base angles of
isosceles triangle are
congruent
AD congr. AF
definition of isosceles
triangle
BC congr. AD
ABCD is a
parallelogram
!
M
Copyright © McDougal Littell h~c,
NI rights
reserved.
Reason
AB congr. CD, BC
congr. AF; angle AFD
congr. angle ADF
transitive property of
congruence
opposite sides of a
parallelogram are
congruent (this is the
first theorem we proved
on parallelograms)
R
H
Q
allel, then the quadrilateral is a parallelogram.
Statement
N
Geometry
Chapter 6 Resource
Book
Reason
HIJK is a parallelogram;
triangle HOI congr.
triangle JOI
!
given
Copyright@ McDougal Li~eII Inc.
HIrights
congr.
KJ, HK congr.
opposite sides of a
All
reserve&
IJ
parallelogram are
congruent (this is the
first theorem we proved
on parallelograms)
HI congr. JI
CPCTC
HI congr. HK
transitive property
HI congr. HK congr. JI
congr. JK
restated for clarity
HIJK is a rhombus
definition of a rhombus
!
!
18. Given
Prov
14. P(7,-1)
15. P(-4, 0)
Q(3, 6)
R(- 1, - 1)
S(3, - a)
16. P(1, 1)
Q(3, 7)
Q(-a, 4)
R(6, 4)
S(- 1, - 3)
R(-5, 1)
Name:_______________________!
S(- 2, - ~)
ragraph proof.
18. Given: Rectangle RECT
Prove: GART ~- ~ACE
HK
R
!
Statement
Reason
Geometry
Chapter 6 Resource Book
!
!
!
!
!
!
Date: _____________________