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Section 9:Rhombuses, Rectangles, and Squares

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Goals• Use properties of diagonals of rhombuses,

rectangles, and squares• Use properties of sides and angles of rhombuses,

rectangles, and squares

Anchors• Apply appropriate techniques, tools, and formulas to

determine measurements.• Analyze characteristics and properties of two and three

dimensional geometric shapes and demonstrate understanding of geometric relationships.

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Properties of Special ParallelogramsRhombus• All the properties of a parallelogram• All four sides are congruent• The diagonals are perpendicular• The diagonals bisect the interior angles

909090

90

xx

xx

yy

yy

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• All the properties of a parallelogram• All sides meet at 90• Each of the diagonals are congruent

Properties of Special ParallelogramsRectangle

A B

CD

EAC = BD

Makes two pairs of isosceles triangles. AE = EC = ED = EB

xx

x xy

y

y

y

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Properties of Special ParallelogramsSquare

• All the properties of a parallelogram• All the properties of a rhombus• All the properties of a rectangle

A B

CD

4545

454545

45

45

45

90 9090

90

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Parallelogram

Rectangles Squares Rhombuses

7

E

B C

DA

Rectangle ABCD

ABE = 10x - 5

BCA = 4x – 3

Find all angles.

ABE + BCA = 90

X = 7

25

25 25

2565

6565

65

130

130

5050

10x - 5 + 4x - 3 = 90

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U

R S

TQ

Rectangle QRST

QRU = 4x + 4

RUQ = 15x - 12

Find all angles.

QRU + RQU + RUQ = 180

X = 8

4x + 4 + 4x + 4 + 15x - 12 = 18036

36 36

36

5454

54 54

108108

72

72

9

U

RO

DP

Rhombus PROD

PRU = 9x - 4

PDU = 5x + 20

Find all angles.

PRU = PDU

9x - 4 = 5x + 20

x = 6

90

90

9090

50

50

50

50

40

40

40

40

10

U

E

L

Y

J

Rhombus JELY

UEJ = 2x + 6

YLU = 4x

Find all angles.

UEJ + YLU = 90

x = 14

34

2x + 6 + 4x = 90

34

3434

56

56

56

56

90

90

90

90

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What special type of quadrilateral is ABCD?A ( -4 , 7 ) , B ( 6 , 9 ) , C ( 8 , 16 ) D ( -2 , 14 )

Slope of AB =

Slope of BC =

Slope of CD =

Slope of AD =

1 / 5

7 / 2

1 / 5

7 / 2

Length of AB =

Length of BC =

Length of CD =

Length of AD =

2√26

√53

2√26

√53

ABCD is a parallelogram – b/c it has two sets of parallel and congruent sides.

How do we do that? What do we need to know?

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What special type of quadrilateral is EFGH?E ( 4 , -8 ) , F ( 7 , -3 ) , G ( 12 , -6 ) H ( 9 , -11 )

Slope of EF =

Slope of FG =

Slope of GH =

Slope of EH =

5 / 3

-3 / 5

5 / 3

-3 / 5

Length of AB =

Length of BC =

Length of CD =

Length of AD =

√34

√34

√34

√34

EFGH is a square – b/c it has two sets of parallel sides, all sides are congruent, and the slopes are negative reciprocals.

How do we do that? What do we need to know?

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What special type of quadrilateral is IJKL?I ( -9 , -9 ) , J ( -6 , -2 ) , K ( -3 , -9 ) L ( -6 , -16 )

Slope of IJ =

Slope of JK =

Slope of KL =

Slope of IL =

7 / 3

-7 / 3

7 / 3

-7 / 3

Length of IJ =

Length of JK =

Length of KL =

Length of IL =

√58

√58

√58

√58

IJKL is a rhombus– b/c it has two sets of parallel sides and all sides are congruent,

How do we do that? What do we need to know?

Diagonals are perpendicular. Slope of IK = 0 and slope of JL is undefined.

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What special type of quadrilateral is MNOP?M ( 2 , 2 ) , N ( 5 , 11 ) , O ( 14 , 14 ) P ( 11 , 5 )

Slope of MN =

Slope of NO =

Slope of OP =

Slope of MP =

3

1 / 3

3

1 / 3

Length of MN =

Length of NO =

Length of OP =

Length of MP =

3√10

3√10

3√10

3√10

MNOP is a rhombus– b/c it has two sets of parallel sides and all sides are congruent,

How do we do that? What do we need to know?

Diagonals are perpendicular. Slope of MO = 1 and slope of NP = -1.