6.2 what are special parallelograms? pg. 9 properties of rhombi, rectangles, and squares
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6.2
What Are Special Parallelograms?
Pg. 9Properties 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 perpendicular
Diagonals 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°
12
1515
mQPR
mQTP
mPQT
15
15
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
275
A = 1100 ft2
3
42 = x2 + 32
16 = x2 + 9
7 = 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
3
3 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
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