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Warm-Up Exercises
1. What congruence postulate shows that
ABE = CDE ?
ANSWER
2. If
E=
ANSWER
SAS
G, find m
118º
E.
Warm-Up Exercises
Target
Use properties of parallelograms.
You will…
Use properties to identify
parallelograms.
Warm-Up Exercises
Vocabulary
• Proving Quadrilaterals are Parallelograms
A quadrilateral is a parallelogram if …
both pairs of opposite sides are parallel – Definition
both pairs of opposite sides are congruent – Theorem 8.7
both pairs of opposite angles are congruent – Theorem 8.8
one pair of opposite sides is congruent and parallel
– Theorem 8.9
its diagonals bisect each other – Theorem 8.10.
Warm-Up1Exercises
EXAMPLE
Solve a real-world problem
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.
Warm-Up1Exercises
EXAMPLE
Solve a real-world problem
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.
Warm-Up
Exercises
GUIDED
PRACTICE
for Example 1
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.
Warm-Up2Exercises
EXAMPLE
Identify a parallelogram
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.
Warm-Up3Exercises
EXAMPLE
Use algebra with parallelograms
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 .
FN = DN Set the segment lengths equal.
5x – 8 = 3x
Substitute 5x –8 for FN and 3x for DN.
2x – 8 = 0
2x = 8
x= 4
Subtract 3x from each side.
Add 8 to each side.
Divide each side by 2.
When x = 4,
FN = 5(4) –8
= 12
DN = 3(4)
= 12
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
What theorem can you use to show that the
quadrilateral is a parallelogram?
2.
ANSWER Theorem 8.9
If one pair of opposite sides is parallel and congruent,
then the quadrilateral is a parallelogram.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
What theorem can you use to show that the
quadrilateral is a parallelogram?
3.
ANSWER Theorem 8.7
If both pair of opposite sides are congruent, then the
quadrilateral is a parallelogram.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
What theorem can you use to show that the
quadrilateral is a parallelogram?
4.
ANSWER Theorem 8.8
If both pair of opposite angles are congruent, then
the quadrilateral is a parallelogram.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
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.
Warm-Up4Exercises
EXAMPLE
Use coordinate geometry
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 =
CD =
[2 – (–3)]2 + (5 – 3)2 =
(5 – 0)2 + (2 – 0)2
=
29
So, AB
29
CD.
Warm-Up4Exercises
EXAMPLE
Use coordinate geometry
Then use the slope formula
to show that AB
CD.
5 – (3)
2
Slope of AB = 2 – (–3) =
5
Slope of CD = 25 –– 00 = 2
5
Because AB and CD have the same slope, they are parallel.
ANSWER
AB and CD are congruent and parallel.
So, ABCD is a parallelogram by Theorem 8.9.
Warm-Up
EXAMPLE
4Exercises
GUIDED
PRACTICE
for Example 4
6. Explain how other methods
can be used to show that
quadrilateral ABCD in Example
4 is a parallelogram.
ANSWER
Find the slopes of all four sides and show that
opposite sides are parallel.
A second way is to find the lengths of each side and
show that opposite sides are congruent.
A third way is to find the point of intersection of the
diagonals and show the diagonals bisect each other.