Download Task on Parallelograms

Geometry Concepts 6.2 Properties of Parallelograms
Name______________________________
A quadrilateral is a polygon with four sides.
Quadrilaterals are named using the four letters of the vertices. The four letters are not given in a random
order. After the first letter of the name is chosen, one must proceed in a clockwise or counter-clockwise
direction around the figure when choosing the other three letters of the name.
------------------------------------------------------------------------------------------------------------------------------1.
H
Provide the name for the quadrilateral to the right if
one starts at vertex H and goes around the figure in a
counter-clockwise direction.
E
F
2.
G
Is "Quadrilateral EHGF" a correct name for the
quadrilateral to the right?
3.
Is "Quadrilateral FGHE" a correct name?
------------------------------------------------------------------------------------------------------------------------------The sum of the measures of the interior angles of a quadrilateral is 360 .
--------------------------------------------------------------------------------------------------------------------4. What is mS in the figure below? 5. What is the value of x in the figure below?
Q
55
P
9x
122
138
S
8x - 4
R
4x
5x
------------------------------------------------------------------------------------------------------------------------------A parallelogram is a specific type of quadrilateral.
B
A
A parallelogram is a quadrilateral in which both pairs of opposite
sides are parallel.
Obviously, ABCD is not a parallelogram because AB and DC are
not parallel. If those sides were extended,
they would eventually intersect.
Quadrilateral IJKL to the right is a parallelogram. Both pairs
of opposite sides are parallel. For example, if one extends
D
C
I
J
IJ and LK indefinitely, the opposite sides would still
never intersect.
L
K
1
For Questions 6-8, is the figure a parallelogram?
6.
7.
8.
M
P
H
R
S
U
T
E
F
N
O
G
------------------------------------------------------------------------------------------------------------------------------Today, we will learn two important characteristics about parallelograms.
First, opposite sides of a parallelogram are congruent.
In other words, opposite sides have the same length.
I
J
In the figure to the right, IJ and LK have the same length.
K
L
Suppose IL  5 inches . What is JK?
Second, opposite angles of a parallelogram are congruent.
In other words, opposite angles have the same measure.
In the figure to the right, I has the same measure as K .
In addition, which angle would be congruent to L ?
------------------------------------------------------------------------------------------------------------------------------For Questions 9-10, find the value of the variable in the parallelograms below.
9.
10.
7c - 26
4y + 10
7y - 11
5c
-------------------------------------------------------------------------------------------------------------------------------
Look at what x and y represent in the diagram - x and y are the
measures of two angles that are "next to each other" inside of
the parallelogram. Interior angles like x and y that share a side
are known as consecutive angles.
x
.
y
y
x
The equation x  y  180 is essentially stating that the measures of consecutive angles in a
parallelogram add to be 180 degrees. This gives us the following property:
Consecutive angles in a parallelogram are supplementary.
Two angles are supplementary if the sum of their measures is 180 .
2
In the parallelogram to the left, mD  64 . What is mC ?
A
B
D
Since C and D are consecutive angles (next to each other), the
sum of their measures must be 180 degrees.
64 
C
Hence, mC  180  64  116
--------------------------------------------------------------------------------------------------------------------EFGH is a parallelogram (pictured to the right). If mF  5x  1 , and
mG  4x 10 , what is the value of x?
E
As you can see in the figure, F and G are consecutive angles, so
they must be supplementary:
F
H
G
5 x  1  4 x  10  180
9 x  9  180
9x
 189
x  21
--------------------------------------------------------------------------------------------------------------------1. A parallelogram has one interior angle
2. In parallelogram JKMN, mJ  6 y ,
that measures 85 . What are the measures
of the other three interior angles?
and mM  8 y  19 . What is the value
of y?
--------------------------------------------------------------------------------------------------------------------A diagonal is a line segment whose endpoints are non-consecutive
P
O
vertices of a polygon. "Non-consecutive vertices" simply means
vertices that are not next to each other.
As you can see, a diagonal for OPRS to the right was constructed
by drawing a line segment from vertex O to vertex R.
S
R
The other diagonal for the parallelogram could be drawn by connecting P to S. Draw it.
--------------------------------------------------------------------------------------------------------------------U
T
The diagonals of a parallelogram bisect each other.
A
Z
W
In other words, the diagonals cut each other into two equal pieces.
In the parallelogram to the left, both diagonals have been drawn. TA is
equal to AZ. In addition, UA is equal to AW.
3
A
F
S
P
I
If AS  3r  5 , and SI  6r  7 , and APIF is a parallelogram, what
is the value of r?
Note that AS and SI are the two equal pieces of diagonal AI ,
because the other diagonal ( FP ) cuts it in half. So:
3r  5  6r  7
12  3r
r4
3.
In the figure above, suppose FS  5x , and FP  12 x  7 . What is the value of x?
Classwork/Homework:
For Questions 1-5, consider the parallelogram below.
S
1.
Provide the two possible four-letter names for the
parallelogram that both begin with the letter U.
2.
Complete the statement: RU is parallel to _________.
3.
Complete the statement: UT is congruent to __________.
4.
True or False: ST must be congruent to UT .
R
T
U
True or False: R must be congruent to T .
--------------------------------------------------------------------------------------------------------------------For Questions 6-7, find the value of the variable in the parallelograms below.
6.
7. mH  5z  8 and mI  9 z  42
5.
H
4x + 18
5z + 8
L
10x + 6
J
9z - 42
I
--------------------------------------------------------------------------------------------------------------------8.
A quadrilateral has three interior angles with measures of 49 , 111 , and 133 . What is
the measure of the fourth interior angle?
4
9.
Parallelogram WYXZ is constructed such that ZW = 31 inches, and WY = 22 inches.
What is XZ? What is YX?
---------------------------------------------------------------------------------------------------------------------
For Questions 1-6, fill in the blank.
11.
According to the definition of a parallelogram, a parallelogram is a quadrilateral in
which both pairs of opposite sides are _____________.
11.
______________ angles of a parallelogram are supplementary.
12.
The sum of the measures of the interior angles of a quadrilateral is __________.
13.
The diagonals of a parallelogram _____________ each other.
14.
Opposite sides of a parallelogram are not only parallel, but also ____________.
15.
Opposite angles of a parallelogram are ______________.
--------------------------------------------------------------------------------------------------------------------For Questions 16-17, consider the quadrilateral to the right.
16.
The name of the quadrilateral is ACBD. Label
the remaining vertices on the figure with the
appropriate letters.
B
17.
Based on the look of the picture, why is the quadrilateral
not a parallelogram?
--------------------------------------------------------------------------------------------------------------------18.
Suppose parallelogram MNOP has the following characteristic: mM  56 .
Find the measures of each of the other interior angles. Be sure to be specific (for
example, write mN = _______, and so on).
5
For Questions 19-20, find the value of the variable for the parallelogram.
19. mO  7 y  20 and mP  3 y
20. CE  2k  1 and EB  5k 19
A
O
7y + 20
R
5k - 19
3y
P
Q
2k - 1
B
E
C
D
--------------------------------------------------------------------------------------------------------------------21.
Quadrilateral JNDG to the right is a parallelogram.
If EN  3x  4 , and GN  38 , what is the value of x?
N
J
E
D
G
--------------------------------------------------------------------------------------------------------------------22.
The teacher says, "In the quadrilateral to the left, the measure of
L
S
angle V is 119 degrees."
T
Marie replies, "That means the measure of angle L is also 119 .
V
What is wrong with Marie's conclusion?
6