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8.3 show that a quadrilateral is a parallelogram
1. 8.3 Show that a Quadrilateral is a Parallelogram
8.3
Bell Thinger
1. What congruence postulate
shows that ABE
CDE ?
ANSWER
2. If
E
ANSWER
SAS
G, find m
118º
E.
3. 8.3
Example 1
Ride An amusement park ride has a moving
platform attached to four swinging arms. The
platform swings back and forth, higher and higher,
until it goes over the top and around in a circular
motion. In the diagram below, AD and BC represent
two of the swinging arms, and DC is parallel to the
ground (line l). Explain why the moving platform AB is
always parallel to the ground.
4. 8.3
Example 1
SOLUTION
The shape of quadrilateral ABCD changes as the
moving platform swings around, but its side lengths do
not change. Both pairs of opposite sides are
congruent, so ABCD is a parallelogram by Theorem 8.7.
By the definition of a parallelogram, AB DC .
Because DC is parallel to line l, AB is also parallel to
line l by the Transitive Property of Parallel Lines.
So, the moving platform is parallel to the ground.
5. 8.3
Guided Practice
1. In quadrilateral WXYZ, m W = 42°, m X = 138°,
m Y = 42°. Find m Z. Is WXYZ a parallelogram?
Explain your reasoning.
ANSWER
138°; yes; the sum of the measures of the interior
angles in a quadrilateral is 360°, so the measure of
Z is 138°. Since opposite angles of the
quadrilateral are congruent, WXYZ is a parallelogram.
7. 8.3
Example 2
ARCHITECTURE The doorway
shown is part of a building in
England. Over time, the building
has leaned sideways. Explain
how you know that SV = TU.
SOLUTION
In the photograph, ST UV and
ST UV. By Theorem 8.9,
quadrilateral STUV is a
parallelogram.
By Theorem 8.3, you know that opposite sides of a
parallelogram are congruent. So, SV = TU.
8. 8.3
Example 3
ALGEBRA For what value of
x is quadrilateral CDEF a
parallelogram?
SOLUTION
By Theorem 8.10, if the
diagonals of CDEF bisect
each other, then it is a
parallelogram. You are given
that CN EN . Find x so that
FN DN .
9. 8.3
Example 3
FN = DN
5x – 8 = 3x
2x – 8 = 0
2x = 8
x= 4
Set the segment lengths equal.
Substitute 5x –8 for FN and 3x for DN.
Subtract 3x from each side.
Add 8 to each side.
Divide each side by 2.
When x = 4, FN = 5(4) –8 = 12 and DN = 3(4) = 12.
Quadrilateral CDEF is a parallelogram when x = 4.
13. 8.3
Guided Practice
5. For what value of x is
quadrilateral MNPQ a
parallelogram?
Explain your reasoning.
ANSWER
2; The diagonals of a parallelogram bisect
each other so solve 2x = 10 – 3x for x.
14. 8.3
Example 4
Show that quadrilateral ABCD
is a parallelogram.
SOLUTION
One way is to show that a pair of
sides are congruent and parallel.
Then apply Theorem 8.9.
First use the Distance Formula to show that AB and CD
are congruent.
AB =
[2 – (–3)]2 + (5 – 3)2 =
29
CD =
(5 – 0)2 + (2 – 0)2
29
Because AB = CD =
=
29 , AB
CD.
15. 8.3
Example 4
Then use the slope formula to show that AB
CD.
5 – (3)
2
Slope of AB = 2 – (–3) =
5
2–0
2
Slope of CD = 5 – 0 =
5
Because AB and CD have the same slope, they are
parallel.
AB and CD are congruent and parallel. So, ABCD is a
parallelogram by Theorem 8.9.
17. Exit
8.3 Slip
Tell how you know that the quadrilateral is a
parallelogram
1.
ANSWER
The diagonals bisect each other.
18. Exit
8.3 Slip
Tell how you know that the quadrilateral is a
parallelogram
2.
ANSWER
Both pairs of opposite angles congruent. You can
also conclude that opposite sides are parallel and
use the definition of a parallelogram.
20. Exit
8.3 Slip
4.
In a coordinate plane, draw the quadrilateral ABCD
with vertices A(–1, 3), B(7, –1), C(6, –4), and D(–2, 0).
show that ABCD is a parallelogram.
ANSWER
AB = CD = 4 5 and BC = AD = 10, so two
pairs of opposite sides are congruent.
