8.3 Show a Quadrilateral is a Parallelogram
Objectives:
1. To prove that a
quadrilateral is a
parallelograms
Assignment:
• P. 526-529: 1-11, 1521, 33-36, 38, 41, 43
• Challenge Problems
Objective 1
You will be able
to prove that a
quadrilateral is a
parallelogram
Exercise 1
Write the converse of the statement below:
If a quadrilateral is a parallelogram, then its
opposite sides are congruent.
Parallelogram
A parallelogram is a
quadrilateral with
both pairs of
opposite sides
parallel.
Written
𝑃𝑄𝑅𝑆
𝑃𝑄 ∥ 𝑅𝑆
and
𝑄𝑅 ∥ 𝑃𝑆
Proving Lines Parallel
Proving Triangles Congruent
Proving Triangles Congruent
Investigation 1
In this lesson, we will find ways
to show that a quadrilateral
is a parallelogram.
Obviously, if the opposite
sides are parallel, then the
quadrilateral is a
parallelogram. But could we
use other properties besides
the definition to see if a
shape is a parallelogram?
Property 1
We know that the opposite sides of a
parallelogram are congruent. What about
the converse? If we had a quadrilateral
whose opposite sides are congruent, then
is it also a parallelogram?
B
Step 1: Draw a
quadrilateral with
congruent opposite
sides.
A
D
C
Property 1
Step 2: Draw
diagonal 𝐴𝐷.
Notice this creates
two triangles. What
kind of triangles are
they?
B
A
D
C
ABD  DCA
by SSS 
Property 1
Step 3: Since the two
triangles are
congruent, what
must be true about
BDA and CAD?
B
A
D
C
BDA  CAD
by CPCTC
Property 1
Step 4: Now consider
𝐴𝐷 to be a
transversal. What
must be true about
𝐵𝐷 and 𝐴𝐶?
B
A
D
C
BD || AC
by Converse of
Alternate Interior
Angles Theorem
Property 1
Step 5: By a similar
argument, what
must be true about
𝐴𝐵 and 𝐶𝐷?
B
A
D
C
AB || CD
by Converse of
Alternate Interior
Angles Theorem
Property 1
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Property 2
We know that the opposite angles of a
parallelogram are congruent. What about
the converse? If we had a quadrilateral
whose opposite angles are congruent, then
is it also a parallelogram?
B
Step 1: Draw a
quadrilateral with
congruent opposite
angles.
A
D
C
Property 2
Step 2: Now assign
the congruent
angles variables x
and y. What is the
sum of all the
angles? What is
the sum of x and y?
B
x
y
D
y
x
A
x  y  x  y  360
2 x  2 y  360
x  y  180
C
Property 2
Step 3: Consider 𝐴𝐵
to be a transversal.
Since x and y are
supplementary,
what must be true
about 𝐵𝐷 and 𝐴𝐶?
B
x
y
D
y
x
A
C
BD || AC
by Converse of
Consecutive Interior
Angles Theorem
Property 2
Step 4: By a similar
argument, what
must be true about
𝐴𝐵 and 𝐶𝐷?
B
x
y
D
y
x
A
C
AB || CD
by Converse of
Consecutive Interior
Angles Theorem
Property 2
If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Property 3
We know that the diagonals of a
parallelogram bisect each other. What
about the converse? If we had a
quadrilateral whose diagonals bisect each
other, then is it also a parallelogram?
B
Step 1: Draw a
quadrilateral with
diagonals that
bisect each other.
D
E
A
C
Property 3
Step 2: What kind of
angles are BEA
and CED? So
what must be true
about them?
B
D
E
A
C
BEA  CED
by Vertical Angles
Congruence
Theorem
Property 3
Step 3: Now what
must be true about
𝐴𝐵 and 𝐶𝐷?
B
D
E
A
C
AB  CD
by SAS  and CPCTC
Property 3
Step 4: By a similar
argument, what
must be true about
𝐵𝐷 and 𝐴𝐶?
B
D
E
A
C
BD  AC
by SAS  and CPCTC
Property 3
Step 5: Finally, if the
opposite sides of
our quadrilateral are
congruent, what
must be true about
our quadrilateral?
B
D
E
A
C
ABDC is a
parallelogram by
Property 1
Property 3
If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a
parallelogram.
Property 4
The last property is not a converse, and it is
not obvious. The question is, if we had a
quadrilateral with one pair of sides that are
congruent and parallel, then is it also a
parallelogram?
Step 1: Draw a
quadrilateral with
one pair of parallel
and congruent sides. A
B
D
C
Property 4
Step 2: Now draw in
diagonal 𝐴𝐷.
Consider 𝐴𝐷 to be a
transversal. What
must be true about
BDA and CAD?
B
A
BDA  CAD
by Alternate
Interior Angles
Theorem
D
C
Property 4
Step 3: What must be
true about ABD
and DCA? What
must be true about
𝐴𝐵 and 𝐶𝐷?
B
A
D
C
AB  CD
by SAS and
CPCTC
Property 4
Step 4: Finally, since
the opposite sides
of our quadrilateral
are congruent, what
must be true about
our quadrilateral?
B
A
D
C
ABDC is a
parallelogram by
Property 1
Property 4
If one pair of opposite sides of a quadrilateral
are congruent and parallel, then the
quadrilateral is a parallelogram.
Example 2
In quadrilateral WXYZ, mW = 42°, mX =
138°, and mY = 42°. Find mZ. Is
WXYZ a parallelogram? Explain your
reasoning.
Example 3
For what value of x is the quadrilateral below
a parallelogram?
Example 4
Determine whether the following
quadrilaterals are parallelograms.
Example 5
Construct a flowchart to prove that if a
quadrilateral has congruent opposite sides,
then it is a parallelogram.
Given: AB  CD
BC  AD
Prove: ABCD is a
A
parallelogram
B
C
D
Summary
8.3 Show a Quadrilateral is a Parallelogram
Objectives:
1. To prove that a
quadrilateral is a
parallelograms
Assignment:
• P. 526-529: 1-11, 1521, 33-36, 38, 41, 43
• Challenge Problems