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Parallelograms & Rectangles
The student is able to (I can):
Prove and apply properties of parallelograms.
Use properties of parallelograms to solve problems.
Prove and apply properties of special parallelograms.
Use properties of special parallelograms to solve problems.
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parallelogram
Properties of Parallelograms
A quadrilateral with two pairs of parallel sides.
Therefore, if a quadrilateral is a parallelogram, then it has
two pairs of parallel sides.
>>
>>
T I
ME
TI ME, TE IM
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Properties of Parallelograms
If a quadrilateral is a parallelogram, then opposite sides are
congruent.
If a quadrilateral is a parallelogram, then opposite angles are
congruent.
KI NG, GK IN
K
NG
I
>>
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K
NG
O
K N, O G
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Properties of Parallelograms
If a quadrilateral is a parallelogram, then consecutive angles
are supplementary.
If a quadrilateral is a parallelogram, then its diagonals bisect
each other.
1 2
34
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T U
NE
STS NS, ES US
+ =
+ =
+ =
+ =
m 1 m 2 180
m 2 m 3 180
m 3 m 4 180
m 4 m 1 180
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Examples Find the value of the variable:
1. x =
2. x =
3. y =
5x + 3 2x + 15
(3x)
(x + 84)
y
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Examples Find the value of the variable:
1. x =
2. x =
3. y =
5x + 3 2x + 15
4
(3x)
(x + 84)
y
5x + 3 = 2x + 153x = 12
3x = x + 842x = 84
42
3(42) = 126y = 180 126
54
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rectangle A parallelogram with four right angles.
If a parallelogram is a rectangle, then its diagonals are
congruent (checking for square).
F I
SH
FS IH
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Because a rectangle is a parallelogram, it also inherits all of
the properties of a parallelogram:
Opposite sides parallel
Opposite sides congruent
Opposite angles congruent (actually allallallallangles are
congruent)
Consecutive angles supplementary
Diagonals bisect each other
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Example Find each length.
1. LW
2. OL
3. OW
F O
WL
30
17
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Example Find each length.
1. LW
LW = FO = 30
2. OL
OL = FW = 2(17) = 34
3. OW
OWL is a right triangle, so
OW = 16
F O
WL
30
17
+ =2 2 2OW LW OL
+ =2OW 900 1156
=2OW 256
+ =2 2 2OW 30 34
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rhombus A parallelogram with four congruent sides. (Plural is
either rhombi or rhombuses.)
If a parallelogram is a rhombus, then its diagonals are
perpendicular.
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If a parallelogram is a rhombus, then each diagonal bisects a
pair of opposite angles.
1 2
3 4
5 6
7 8
1 234
567
8
Since opposite angles are also congruent:1 2 5 63 4 7 8
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Examples 1. What is the perimeter of a rhombus whose side length
is 7?
2. Find the value of x
3. Find the value of y
x
8
Perimeter = 40
(3y+11)
(13y9)
10
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Examples 1. What is the perimeter of a rhombus whose side length
is 7?
4(7) = 28
2. Find the value of x
The side = 10
Pyth. triple: 6, 8, 10
x = 6
3. Find the value of y
13y 9 = 3y + 11
10y = 20
y = 2
x
8
Perimeter = 40
(3y+11)
(13y9)
10
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square A quadrilateral with four right angles and four congruent
sides.
Note: A square has all of the properties of bothbothbothboth a
rectangle andandandand a rhombus:
Diagonals are congruent
Diagonals are perpendicular
Diagonals bisect opposite angles.
