6.2

What Are Special Parallelograms?Pg. 9

Properties of Rhombi, Rectangles, and Squares

6.2 – What Are Special Parallelograms?___Properties of Rhombi, Rectangles, and Squares

In the previous lesson, you learned that parallelograms have both pairs of opposite sides parallel. You also discovered many different properties of parallelograms. Today you are going to continue your investigation with parallelograms with even more special properties.

6.8–PARALLELOGRAMS WITH RIGHT ANGLES

a. Rectangles are special parallelograms. Since they are parallelograms, what do you already know about rectangles?

Both _____________ sides are___________ 

opposite parallel

Both _____________ sides are

________________ 

opposite

congruent

Both _____________ angles are

________________ 

opposite

congruent

Both _____________ angles are

________________ 

consecutive

supplementary

The diagonals ________________ each other

bisect

b. Mark wanted to learn more about this shape. He noticed that the diagonals seem to have a special relationship beyond just being bisected. He decided to investigate. He drew a rectangle twice, adding one diagonal. Find the length of AC and BD. Show all work. What do you notice?

82 + 152 = x2

289 = x2

17 = x

82 + 152 = x2

289 = x2

17 = x

Diagonals are congruent

c. List the two special properties Rectangles have that general Parallelograms don’t have.

4 right angles

Diagonals are congruent

6.9–PARALLELOGRAMS WITH EQUAL SIDES a. A rhombus is another type of special parallelogram. Since they are parallelograms, what do you already know about rhombuses?

Both _____________

sides are ________________ 

opposite

parallel

Both _____________ sides are

________________ 

opposite

congruent

Both _____________ angles are

________________ 

opposite

congruent

Both _____________ angles are

________________ 

consecutive

supplementary x y

xy x y 180

The diagonals ________________ each other

bisect

c. Audrey wanted to learn more about her shape. She noticed that the diagonals seem to have a special relationship as well. She measured the sides of the rhombus and all were 5 units long. Then she measured AC = 6 units and BD = 8 units. Mark these lengths on the picture below. Is there a way to tell if ∆AEB is a right triangle? Explain.

5

5

5

5 334

4

52 = 32 + 42

25 = 9 + 1625 = 25

The diagonals are perpendicular

d. Audrey noticed something else with the angle in the rhombus. Using the given lines symmetry, mark any angles congruent. What do you notice?

Diagonals bisect the angles

c. List the two special properties Rhombuses have that general Parallelograms don’t have.

4 congruent sides

Diagonals are perpendicularDiagonals bisect angles

6.10 – PARALLELOGRAMS WITH EQUAL SIDES AND RIGHT ANGLESMs. Matthews has a favorite quadrilateral. It is a rhombus combined with a rectangle. a. What is the name of Ms. Matthews' shape? Draw a picture to support your answer.

square

 b. This shape has more properties than any other quadrilateral. Why do you think this is?   

It is a parallelogram, a rectangle, and a rhombus

6.11 – SPECIAL PARALLELOGRAMSName the type of parallelogram. Explain how you know using only the markings.

parallelogram rectangle

rhombus rhombus

rectangle rhombus

square rhombus

6.12 – MISSING PARTSFind the missing information based on the type of shape and its special properties.

a. The diagonals of rhombus PQRS intersect at T. Find the indicated measure.   _____

_________

_________ 

RP = _________

SP = _________

RS = _________

1530°

30°

90°90°

60°60°

121515

mQPR

mQTP

mPQT

1515

b. The diagonals of rectangle WXYZ intersect at P. Given that XZ = 12, find the indicated measure.

_________    _________  _________ WP = _________

40° 40°

50° 50°

80°80°

6

WXZ

PYX

XPY

c. The diagonals of square DEFG intersect at H. Given that EH = 5, find the indicated measure.

 

   HF =

90°

90°45°

45°45°

45°5

GHF

HGF

HFG

6.13 – AREAFind the area of the rhombus by finding the area of each triangle and then adding.

22

25

275

275

275

275A = 1100 ft2

3

42 = x2 + 32

16 = x2 + 97 = x2

7 x

7

7

1.5 7

1.5 7 1.5 7

1.5 7

A 6 7cm2

4

4

4

8

88

8A = 32 m2

6

33

3 3

3 3 4.5 3

4.5 3

4.5 3

4.5 3

A 18 3 ft 2

Parallelogram

Rectangle

Rhombus

Square

Trapezoid

IsoscelesTrapezoid

Kite

Triangle

Rectangle

• All the properties of a parallelogram

• 4 right angles• Diagonals are congruent

A bh

Rhombus

• All the properties of a parallelogram

• Diagonals are perpendicular• Diagonals bisect angles

Add area of each triangle

Square

• All the properties listed above

A s2or A bh