Parallelograms
5-1
EXAMPLE 1
ALGEBRA
Use properties of parallelograms
Find the values of x and y.
ABCD is a parallelogram by
the definition of a
parallelogram. Use Theorem
8.3 to find the value of x.
AB = CD
x + 4 = 12
x=8
By Theorem 8.4,
ANSWER
Opposite sides of a
.
Substitute x + 4 for AB and 12 for CD.
Subtract 4 from each side.
A
In
are
C, or m
A=m
C. So, y ° = 65°.
ABCD, x = 8 and y = 65.
GUIDED PRACTICE
1. Find FG and m
for Example 1
G.
SOLUTION
FG = HE
Opposite sides of a
are
.
x=8
By Theorem 8.4,
ANSWER
E
In
G, or m
E=m
FEHG, FG = 8 and m
G. So, G ° = 60°.
G = 60°.
GUIDED PRACTICE
for Example 1
2. Find the values of x and y.
SOLUTION
JK = ML
Opposite sides of a
18 = y + 3
Substitute 18 for JK and y + 3 for ML.
15 = y
Subtract 3 from each side.
By Theorem 8.4,
J
2x = 50
J=m
L.
Substitute
x = 25
ANSWER
L, or m
are
Divide 2 from each side.
In
JKLM, x = 25 and y = 15.
.
EXAMPLE 2
Use properties of parallelograms
Desk Lamp
As shown, part of the extending arm of a desk lamp is
a parallelogram. The angles of the parallelogram
change as the lamp is raised and lowered.
Find m BCD when m ADC = 110°.
SOLUTION
By Theorem 8.5, the consecutive
angle pairs in ABCD are
supplementary.
So, m ADC + m BCD = 180°.
Because m ADC = 110°,
m BCD =180° –110° = 70°.
EXAMPLE 3 Standardized Test Practice
SOLUTION
By Theorem 8 .6, the diagonals of a parallelogram
bisect each other. So, P is the midpoint of diagonals
LN and OM . Use the Midpoint Formula.
Coordinates of midpoint P of OM = ( 7 +2 0 , 4 +2 0 ) = ( 7 ,2)
2
ANSWER The correct answer is A.
GUIDED PRACTICE
for Examples 2 and 3
Find the indicated measure in
3. NM
JKLM.
SOLUTION
By Theorem 8 .6, the diagonals of a parallelogram
bisect each other. So, N is the midpoint of diagonals
KM .
KN = NM
2 = NM
Substitute
GUIDED PRACTICE
for Examples 2 and 3
Find the indicated measure in
4. KM
JKLM.
SOLUTION
KM = KN + NM
By theorem 8.6
KM = 2 + 2
Substitute
KM = 4
Add
GUIDED PRACTICE
for Examples 2 and 3
Find the indicated measure in
5. m JML
JKLM.
SOLUTION
By Theorem 8.5, the consecutive angle pairs
in JKLM are supplementary.
So, m
KJM + m
Because m
JML = 180°.
KJM = 110°, m
JML =180° –110° = 70°.
GUIDED PRACTICE
for Examples 2 and 3
Find the indicated measure in
6. m KML
JKLM.
SOLUTION
m
JML = m
KMJ + m
70° = 30° + m
40° = m
KML
KML
KNL
Substitute
Subtract
EXAMPLE 2
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.
EXAMPLE 3
ALGEBRA
Use algebra with parallelograms
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 .
EXAMPLE 3
FN = DN
5x – 8 = 3x
2x – 8 = 0
2x = 8
x=4
Use algebra with parallelograms
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.
ANSWER
Quadrilateral CDEF is a parallelogram when x = 4.
GUIDED PRACTICE
for Examples 2 and 3
What theorem can you use to show that the
quadrilateral is a parallelogram?
2.
ANSWER
In the graphic, two opposite sides are equal, i.e, 30m
each and parallel, Therefore, the quadrilateral is a
parallelogram. By theorem 8.9.
GUIDED PRACTICE
for Examples 2 and 3
What theorem can you use to show that the
quadrilateral is a parallelogram?
3.
ANSWER
Two pairs of opposite sides are equal.
Therefore, the quadrilateral is a parallelogram. By
theorem 8.7
GUIDED PRACTICE
for Examples 2 and 3
What theorem can you use to show that the
quadrilateral is a parallelogram?
4.
ANSWER
By theorem 8.8, if the opposite angles are Congruent,
the quadrilateral is a parallelogram.
GUIDED PRACTICE
for Examples 2 and 3
5. For what value of x is
quadrilateral MNPQ a
parallelogram?
Explain your reasoning.
SOLUTION
2x = 10 – 3x
5x = 10
x=2
By Theorem 8.6
[ Diagonals in
bisect each other ]
Add 3x to each side
Divide each side by 5
EXAMPLE 2
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.
EXAMPLE 3
ALGEBRA
Use algebra with parallelograms
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 .
EXAMPLE 3
FN = DN
5x – 8 = 3x
2x – 8 = 0
2x = 8
x=4
Use algebra with parallelograms
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.
ANSWER
Quadrilateral CDEF is a parallelogram when x = 4.
GUIDED PRACTICE
for Examples 2 and 3
What theorem can you use to show that the
quadrilateral is a parallelogram?
2.
ANSWER
In the graphic, two opposite sides are equal, i.e, 30m
each and parallel, Therefore, the quadrilateral is a
parallelogram. By theorem 8.9.
GUIDED PRACTICE
for Examples 2 and 3
What theorem can you use to show that the
quadrilateral is a parallelogram?
3.
ANSWER
Two pairs of opposite sides are equal.
Therefore, the quadrilateral is a parallelogram. By
theorem 8.7
GUIDED PRACTICE
for Examples 2 and 3
What theorem can you use to show that the
quadrilateral is a parallelogram?
4.
ANSWER
By theorem 8.8, if the opposite angles are Congruent,
the quadrilateral is a parallelogram.
GUIDED PRACTICE
for Examples 2 and 3
5. For what value of x is
quadrilateral MNPQ a
parallelogram?
Explain your reasoning.
SOLUTION
2x = 10 – 3x
5x = 10
x=2
By Theorem 8.6
[ Diagonals in
bisect each other ]
Add 3x to each side
Divide each side by 5