Aim: What are the Properties of a Parallelogram?

1
Aim What are the Properties of a Parallelogram?
Do Now
Describe the properties of an isosceles right
triangle.
2
What is a parallelogram?
B
A
A parallelogram is a quadrilateral with special
features
C
D
Both pairs of opposite sides are parallel
AB CD AC BD
Opposite angles are congruent
?A ? ?D ?B ? ?C
Consecutive angles are supplementary
?A ?B 1800, ?B ?D 1800 etc.
The sum of the four angles is 3600
?A ?B ?C ?D 3600
Opposite sides are congruent
AB ? CD AC ? BD
3
Definition of a Parallelogram
B
A
A parallelogram is a quadrilateral in which two
pairs of opposite sides are parallel.
C
D
Opposite sides are parallel.
Opposite sides are congruent.
Opposite angles are congruent.
Consecutive angles are supplementary.
Diagonals bisect each other.
A diagonal divides a parallelogram into two
congruent triangles.
4
Model Problem
In the diagram below, ABCD is a parallelogram,
If m?C 1060, find the m?A, m?B and m?D
B
C
1060
740
1060
740
A
D
?A and ?C are opposite angles in a parallelogram
and must be equal measures. ?A 1060
?B is supplementary to ?C. Therefore ?B
740
?B and ?D are opposite angles and must be equal
measures. ?D 740
5
Model Problem
In parallelogram ABCD, m?A 3x 33, and ?B
2x 28. Find the value of x and the measure of
all four angles of the parallelogram.
B
C
2x 28
780
3x 33
1020
A
D
?A and ?B are consecutive angles of a
parallelogram and are therefore supplementary.
(2x 28) (3x 33) 180
5x 5 180
?B ?D 2(37) 28 1020
5x 185
?C ?A 3(37) 33 780
x 37
6
Model Problem
In parallelogram PQRS, PQ 12y 14 and SR
3y 49. Find the value of y, PQ and SR.
R
S
3y 49
3(7) 49 70
12(7) 14 70
12y 14
Q
P
Opposite sides of a parallelogram are congruent.
Therefore
3y 49 12y 14
-3y -3y
49 9y 14
63 9y
7 y
7
Model Problem
Find the value of the variables.
y
45
45
3x
3y
135
In a parallelogram
Consecutive angles are supplementary.
Opposite angles are congruent.
3y y 180
3x y
4y 180
3x 45
y 45
x 15
3y 135
8
Informal Proofs with Parallelograms
DB AC are diagonals of ABCD.
B
C
E
A
D
Explain how DDEA ? DBCE?
?BCE ? ?EAD CB DA forming alternate
interior angles that are congruent
(A ? A)
?CEB ? ?AED Vertical angles
(A ? A)
CB ? AD Opposite sides of parallelogram are
congruent
(S ? S)
?DEA ? ?BCE because of AAS ? AAS
9
More about Diagonals of Parallelograms
B
C
E
A
D
Name the corresponding parts of DDEA DBCE
CE ? AE DE ? BE DA ? BC
?CEB ? ?AED ?BCE ? ?DAE ?CBE ? ?EDA
What is true about the diagonals of a
parallelogram based on the congruence of the two
triangles?
The diagonals of a parallelogram bisect each
other.
DE ? EB and CE ? AE
10
Diagonals of a Parallelogram
The diagonals of a parallelogram bisect each
other.
In parallelogram ABCD with diagonals AC and BD,
AC 2x 14 and EC 20. Find the value of x.
AC 2(EC)
check
2x 14 2(20)
40
AC 2x 14
AC 2(27) 14
2x 54
x 27
AC 40
11
Model Problem
Parallelogram ABCD is cut by a diagonal AC. The
m?CAD 30 and the m?B 115. Find the measure
of the following angles. m?D m?BAD m?BCD
m?BCA m?BAC m?ACD
65o
115o
B
C
30o
115o
65o
35o
35o
65o
115o
30o
A
D
65o
30o
35o
35o
12
Model Problem
Find the values for x and y in parallelogram
QRST
R
S
y 6 3y - 4
4x 1 3x 4
Q
T
Since the diagonals of a parallelogram bisect
each other the following is true
y 6 3y - 4
4x 1 3x 4
-y -y
-3x -3x
6 2y - 4
x 1 4
4 4
- 1 - 1
10 2y
x 3
5 y