5.6 Proving That a Quadrilateral
is a Parallelogram
Objective:
After studying this section, you will be able to
prove that a quadrilateral is a parallelogram.
Methods to prove quadrilateral
ABCD is a parallelogram
A
D
C
B
1. If both pairs of opposite sides of a
quadrilateral are parallel, then the quadrilateral is
a parallelogram (reverse of the definition).
2. If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram (converse of a
property).
3. If one pair of opposite sides of a quadrilateral
are both parallel and congruent, then the
quadrilateral is a parallelogram.
4. If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a parallelogram
(converse of a property).
5. If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram (converse of a
property).
F
E
D
Given: ACDF is a parallelogram
AFB  ECD
Prove: FBCE is a parallelogram
A
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
9.
9.
10.
10.
B
C
Given:
CAR is isosceles, with base CR
AC  BK
C  K
Prove: BARK is a parallelogram
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
9.
9.
C
A
B
R
K
Q

Given: Quadrilateral QUAD with
angles as shown
Show that QUAD is a parallelogram

 x2


U

5



 3x
D
x 

3 x x  5 x 


2
10
3
 15 x 2

A
Given: NRTW is a parallelogram
W
NX  TS
WV  PR
V
T
X
S
Prove: XPSV is a parallelogram
N
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
8.
8.
9.
9.
10.
10.
P
R
Summary
Using one of the methods to prove
quadrilaterals are parallelograms,
create your own problem and show
how it is a parallelogram.
Homework: worksheet