16-1: Prove a Quadrilateral is a Parallelogram
Objectives:
1. To prove that a
quadrilateral is a
parallelograms
Assignment:
• P. 226: 11-18
• P. 235: 1-6
• 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
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Property 2
If one pair of opposite sides of a quadrilateral
are congruent and parallel, then the
quadrilateral is a parallelogram.
Property 3
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 3
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 3
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 3
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 3
If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram.
Property 4
If the diagonals of a quadrilateral bisect each
other, 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
16-1: Prove a Quadrilateral is a Parallelogram
Objectives:
1. To prove that a
quadrilateral is a
parallelograms
Assignment:
• P. 226: 11-18
• P. 235: 1-6
• Challenge Problems