Transcript Rectangle, Rhombus, and Square

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?
opposite
parallel
Both _____________
sides are___________
Both _____________
opposite
sides are
congruent
________________
opposite angles are
Both _____________
congruent
________________
consecutive angles are
Both _____________
supplementary
________________
bisect
The diagonals ________________
each
other
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?
2
8
+
2
15
2
x
2
x
=
289 =
17 = x
2
8
+
2
15
2
x
2
x
=
289 =
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?
opposite
Both _____________
parallel
sides are ________________
Both _____________
opposite
sides are
congruent
________________
opposite angles are
Both _____________
congruent
________________
consecutive angles are
Both _____________
supplementary
________________
x  y  180
y
x
y
x
bisect
The diagonals ________________
each
other
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.
2
5
5
5
3
4 3
5
4
5
2
3
2
4
= +
25 = 9 + 16
25 = 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 ANGLES
Ms. 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 PARALLELOGRAMS
Name the type of parallelogram. Explain
how you know using only the markings.
parallelogram
rectangle
rhombus
rhombus
rectangle
rhombus
square
rhombus
6.12 – MISSING PARTS
Find 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.
mQPR  _____
30°
90°
mQTP  _________
mPQT _________
60°
12
RP = _________
SP = _________
15
15
RS = _________
15
60°
90°
30°
15
15
b. The diagonals of rectangle WXYZ
intersect at P. Given that XZ = 12, find
the indicated measure.
40°
WXZ  _________
40°
PYX  _________
50°
80°
50°
XPY 
_________
80°
6
WP = _________
c. The diagonals of square DEFG
intersect at H. Given that EH = 5, find the
indicated measure.
GHF 
90°
HGF 
45°
45°
HFG 
HF =
5
90°
45° 45°
6.13 – AREA
Find the area of the rhombus by finding
the area of each triangle and then adding.
22 275
275
25 275
275
A = 1100 ft2
1.5 7
42 = x2 + 32
16 = x2 + 9
7
1.5 7
1.5 7
7
3
1.5 7
7 = x2
7x
A  6 7cm
2
8 4 8
4
84 8
A = 32 m2
4.5 3
3 3
3
4.5 3
4.5 3
3 3 4.5
3
3
6
A  18 3 ft
2
Parallelogram
Trapezoid
Rectangle
Isosceles
Trapezoid
Rhombus
Kite
Square
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 s
2
or
A  bh