created by chris
markstrum © 2005
6.3 Proving Quadrilaterals
are Parallelograms
California State Standards for Geometry
4: Prove basic theorems involving congruence & similarity.
7: Prove & use theorems involving parallel lines and
properties of quadrilaterals.
12: Find & use side and angle measures in triangles
and polygons
17: Prove theorems using coordinate geometry
created by chris
markstrum 2005
theorem
If a quadrilateral has both pairs of opposite sides
are congruent, then it is a parallelogram
A
B
DC
AB ||DC
AD ||BC
BC
D
C
created by chris
markstrum 2005
theorem
If both pairs of opposite angles of a quadrilateral
are congruent, then it is a parallelogram
A
B

A
C
AB
|| DC

B || BC
D
AD
D
C
created by chris
markstrum 2005
theorem
If a quadrilateral has one angle that is
supplementary to both of its consecutive angles,
then it is a parallelogram.
A
B
mAAB
 m||
B  180
DC
mAAD
 m||BC
D  180
D
C
created by chris
markstrum 2005
theorem
if a quadrilateral’s diagonals bisect
each other, then it is a parallelogram
A
B
AM
CM
AB 
|| DC
M
D
BM
DM
AD || BC
C
created by chris
markstrum 2005
theorem
If a quadrilateral has one pair of opposite sides
parallel and congruent, then it is a parallelogram
A
B
AD
AB || DC
BC
BC
AB 
DC
AD
||BC
D
C
created by chris
markstrum 2005
Example
Decide whether you are given enough information to determine that the
quadrilateral is a parallelogram. Explain your reasoning.
YES
If a quadrilateral has one angle that is
supplementary to both of its consecutive angles,
then it is a parallelogram.
created by chris
markstrum 2005
Example
Decide whether you are given enough information to determine that the
quadrilateral is a parallelogram. Explain your reasoning.
YES
If both pairs of opposite angles
of a quadrilateral are congruent,
then it is a parallelogram.
created by chris
markstrum 2005
Example
Decide whether you are given enough information to determine that the
quadrilateral is a parallelogram. Explain your reasoning.
YES
Definition of a parallelogram
Example
created by chris
markstrum 2005
Describe how you would prove that ABCD is a parallelogram.
Congruent alternate interior angles
make parallel lines. BC || AD
If a quadrilateral has one pair of opposite sides
parallel and congruent, then it is a parallelogram.
created by chris
markstrum 2005
Example
Describe how you would prove that ABCD is a parallelogram.
Congruent alternate interior angles
make parallel lines. BC || AD
AB || CD
Definition of parallelogram
Example
Describe how you would prove that ABCD is a parallelogram.
Congruent alternate interior angles
make parallel lines. AB || CD
Congruent corresponding angles
make parallel lines. BC || AD
Definition of parallelogram
created by chris
markstrum 2005
Given: P is supplementary to Q and
Prove: PQRS is a parallelogram
Statement
1.
2.
3.
P is supplementary to Q
PS || QR
P is supplementary to S
S
created by chris
markstrum 2005
Reason
1. Given
2.
Supplementary consecutive interior
angles means the lines are parallel.
3. Given
Supplementary consecutive interior
angles means the lines are parallel.
4.
PQ || SR
4.
5.
PQRS is a parallelogram
5. Definition
of a parallelogram
Given: AB  CD, AB || CD
Prove: ABCD is a parallelogram
A
D
Statement
created by chris
markstrum 2005
Reason
1.
AB  CD , AB || CD
1.
Given
2.
AC  AC
2.
Reflexive
3.
CAB  ACD
3.
||  a.i.  ' s
4.
CAB  ACD
4.
SAS 
5.
ACB  CAD
5.
C.P.C.T.C.
6.
AD || BC
6.
7.
ABCD is a parallelogram
7.
 a.i.  ' s ||
Def parallelogram
B
C
P
Q
T
Given : PQT  RST
Prove : PQRS is a parallelogram
S
Statement
1.
PQT  RST
2.
PQ  RS , PQT  RST
R
Reason
1.
Given
2.
C.P.C.T.C.
3.
PQ || SR
3.
4.
PQRS is a parallelogram
4.
 a.i.  ' s ||
opp. sides  and ||
created by chris
markstrum 2005
created by chris
markstrum 2005
Show that quadrilateral ABCD is a parallelogram
Noty on the coordinate
( – 1, 2)
A
( 3, 2)
B
1
–4
D
( – 3, – 2)
x
1
–3
C
(1,
–
2)
To get a parallelogram
 Show opposite sides ||
 Show opposite sides 
 Show opposite angles 
plane.  Show consecutive angles
are supplementary
 Show diagonals bisect
each other
 Show one pair of opposite
sides are both || and 
created by chris
markstrum 2005
Show that quadrilateral ABCD is a parallelogram
Show opposite sides ||
mAB  0
mDC  0
mAD  2
mBC  2
AB || DC
AD || BC
Show opposite sides
AB 
4

AB  DC
DC  4
y
AD  2 20
425 22     1   3 
AD  BC2
2
22
BC  2 20
23514   2   2  
2
( – 1, 2)
A
( 3, 2)
B
1
–4
D
( – 3, – 2)
x
1
–3
C
(1,
–
2)
2
2
2
Summary
created by chris
markstrum 2005
Six ways to show a quadrilateral is a parallelogram
1) Show two pairs of opposite parallel sides.
2) Show two pairs of opposite congruent sides.
3) Show two pairs of opposite congruent angles.
4) Show one angle is supplementary to both consecutive
angles.
5) Show the diagonals bisect each other.
6) Show that one pair of opposite sides are both congruent
and parallel.