polygons ,quadrilaterals and its theorems.

1. Today, we will learn to…
> identify, name, and describe
polygons
> use the sum of the interior
angles of a quadrilateral

2. # of Sides Name
3
4
5
6
7
8
9
10
12
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
dodecagon

3. Theorem 6.1
Interior Angles of a
Quadrilateral
The sum of the measures
of the interior angles of a
quadrilateral is ______360°

4. Section 6.1 Vocabulary
Convex
Concave
Equilateral
Equiangular
Regular
Diagonal

5. Sides:
Vertices:
Diagonals:
S
T
U
DY
ST TU UD DY YS
S, T, U, D, Y
SU SD TD TY UY

6. S
T
U
DY
There are 10 possible
names of this pentagon.
STUDY
SYDUT
TUDYS
TSYDU
UDYST
UTSYD
DYSTU
DUTSY
YSTUD
YDUTS

7. How many diagonals can
be drawn from N?
N M
O
PQ
R

8. Starting with N, give 2
names for the hexagon.
N M
O
PQ
R
NMOPQR NRQPOM

9. Is this a polygon?
If not, explain. If so, is it
convex or concave?
Yes, it’s a
convex
pentagon

10. Is this a polygon?
If not, explain. If so, is it
convex or concave?
No, polygons must
be made of
segments

11. Is this a polygon?
If not, explain. If so, is it
convex or concave?
Yes, it’s a
concave
dodecagon

12. Is this a polygon?
If not, explain. If so, is it
convex or concave?
No, polygons must
be closed figures

13. Find x.
90 + 87 + 93 + x = 360
x = 90

14. Find x.
3x + 3x + 2x + 2x = 360
x = 36

15. Lesson 6.2
Properties of
Parallelograms
RULERS AND PROTRACTORS
Today, we will learn to…
> use properties of parallelograms

16. A quad is a parallelogram
if and only if two pairs of
opposite sides are parallel
parallelogram

17. Draw a Parallelogram.
Measure each angle.
Measure each side in
centimeters.

18. Theorems 6.2-6.5
If a quadrilateral is a
parallelogram, then…
1) 6.2
2) 6.3
3) 6.4
4) 6.5

19. … opposite sides are
__________congruent

20. … opposite angles are
__________.congruent

21. … consecutive angles are
__________.supplementary
1 2
34
m m m m
m m m m
         
         
1 2 180 1 4 180
3 2 180 3 4 180

22. … diagonals __________
each other.
bisect

23. ABCD is a parallelogram. Find the
missing angle and side measures.
1.
A B
CD
105˚
10
66
10
75˚
75˚
105˚

24. ABCD is a parallelogram.
Find AC and DB.
2. A
CD
8
85
B
5
AC = 10 DB = 16

25. 3. In ABCD, m C = 115˚.
Find mA and mD.
4. Find x in JKLM.
J K
LM
(4x-9)˚
(3x+18)˚
mA = 115˚ mD = 65˚
x = 27

26. ABCD is a parallelogram.
EC =
m BCD =
m ADC =
AD =
5
8
70°
110°

27. The figure is a parallelogram.
x = y =5 4
2x – 6 = 4 2y = 8

28. The figure is a parallelogram.
x = y =30 6
4x + 2x = 180 2y + 3 = y + 9

29. The figure is a parallelogram.
x = y =3 6
y
y
3x + 1 = 10 2y – 1 = y + 5

30. The figure is a parallelogram.
x = y =40 8
3x – 9 = 2x + 31 4y + 5 = 2y + 21

32. Lesson 6.3
Proving that Quadrilaterals
are Parallelograms
What is a converse?
Today, we will learn to…
> prove that a quadrilateral is a
parallelogram

33. Theorem 6.6
If both pairs of opposite
sides are __________,
then it is a parallelogram.
congruent

34. Theorem 6.7
If both pairs of opposite
angles are __________,
then it is a parallelogram.
congruent

35. Is ABCD a parallelogram? Explain.
1. 2.
A B
CD
10
6
10
6
A B
CD
yes
no

36. Theorem 6.8
If an angle is
_______________ to both
of its consecutive angles,
then it is a parallelogram.
supplementary
1
2
3
m1 + m3 = 180˚
m1 + m2 = 180˚

37. Theorem 6.9
If the diagonals
__________________,
then it is a parallelogram.
bisect each other
AE = EC
and
DE = EB
A
D
B
C
E

38. Is ABCD a parallelogram? Explain.
3. 4. A B
CD
A B
CD
104˚
86˚ 104˚
no yes

39. Theorem 6.10
If one pair of opposite
sides are ___________
and __________, then it
is a parallelogram.
congruent
parallel

40. 5.
8.
7.
6.
No Yes
Yes No

41. 9. List 3 ways to prove that a
quadrilateral is a parallelogram
1) prove that both pairs of opposite
sides are __________
2) prove that both pairs of opposite
sides are __________
3) prove that one pair of opposite sides
are both ________ and ________
parallel
congruent
parallel congruent

42. A ( , ) B ( , ) C ( , ) D ( , )
Prove that this is a parallelogram…
slope of AB is
slope of BC is
slope of CD is
slope of AD is
0
4
-2/5
-2/5
AB =
BC =
CD =
AD =
4.1
5.4
4.1
5.4
2 3 4 -2 6 -3 2
4

43. Lesson 6.4
Special
Parallelograms
Today, we will learn to…
> use properties of a rectangle,
a rhombus, and a square

44. A square is a parallelogram with
four congruent sides and four right angles.
A rhombus is a
parallelogram with
four congruent sides.
A rectangle is a
parallelogram with
four right angles.four congruent sides. four right angles.
four congruent sides four right angles

45. parallelograms
rhombuses rectangles
squares

46. Sometimes, always, or never true?
1. A rectangle is a parallelogram.
2. A parallelogram is a rhombus.
3. A square is a rectangle.
4. A rectangle is a rhombus.
5. A rhombus is a square.
always true
sometimes true
always true
sometimes true
sometimes true

47. Geometer’s Sketchpad
mAEB = 90
CD = 4.48 cm
BC = 4.48 cm
AD = 4.48 cm
AB = 4.48 cm
E
C
A B
D
What do we know about the
diagonals in a rhombus?

48. The diagonals of a rhombus are
_____________.perpendicular
Theorem 6.11

49. What do we know about the
diagonals in a rhombus?
mECD = 40
mEDA = 50
mEDC = 50
mEAD = 40
mEAB = 40
mECB = 40
mEBC = 50
mEBA = 50
E
C
A B
D

50. The diagonals of a rhombus
_____________________.bisect opposite angles
Theorem 6.12

51. What do we know about the
diagonals in a rectangle?
ED = 4.51 cm
EB = 4.51 cm
EC = 4.51 cm
EA = 4.51 cm
E
C
A B
D

52. The diagonals of a rectangle are
_____________.congruent
Theorem 6.13

53. 6. In the diagram, PQRS is a
rhombus. What is the value of y?
2y + 3
5y – 6
P Q
RS
y = 3

54. Find x.
7. rhombus
A
B
C
D
xº
52º
x = 38º

55. Find mCDB.
8. rhombus
A
B
C
D
32º
mCDB =32º

56. Find AB.
9. rectangle
A
B
CD
10
12
AB = 16
?
202 = x2 + 122
10

57. Find x.
10. square
A B
CD
xº xº
x = 45˚

58. Find EA & AB.
11. square
EA =
A B
CD
4
E
AB = 5.7
x2 = 42 + 42
x2 = 16 + 16
x2 = 32
x = 5.7
4
4

60. Lesson 6.5
Trapezoids
& Kites
Today, we will learn to…
> use properties of trapezoids
and kites

61. A trapezoid is a
quadrilateral with only
one pair of parallel sides.
A B
D C
base
base
leg leg

62. B A
D
C
Compare leg angles.
Geometer’s Sketchpad
mC = 65
mD = 115
mA = 90
mB = 90

63. In ALL trapezoids,
leg angles are
_______________supplementary

64. A trapezoid is an
isosceles trapezoid
if its legs are congruent.

65. Geometer’s Sketchpad
Compare base angles.Compare leg angles.How do you know it is isosceles?
mA = 67
mD = 67
mC = 113 
mB = 113 
CD = 3.7 cm
AB = 3.7 cm
A D
B C

66. Theorem 6.14 & 6.15
A trapezoid is isosceles if and
only if base angles are
___________.congruent

67. Base angles are congruent.
A B
CD
AC  BD
The trapezoid is isosceles.
The triangles share CD.
ADC  BCD by SAS
CPCTC

68. Theorem 6.16
A trapezoid is isosceles if
and only if its diagonals
are __________.congruent
AC  BD
A B
CD

69. ABCD is an isosceles trapezoid.
Find the missing angle measures.
1. A B
CD
100°
80° 80°
100°

70. 2. The vertices of ABCD are
A(-1,2), B(-4,1), C(4,-3), and
D(3,0). Show that ABCD is an
isosceles trapezoid.
Figure is graphed on next slide.

71. 3
2
1
-1
-2
-3
-4
-6 -4 -2 2 4 6
D(3, 0)
C(4, -3)
B(-4, 1)
A(-1, 2)
AD || BC ?
AB =
CD =
- ½
- ½
Legs are  ? Diagonals are  ?
AC=
BD =
5010
10 50
OR?
Slope of AD is
Slope of BC is

72. x = 118Find x.

73. The midsegment is a segment
that connects the midpoints of
the 2 legs of a trapezoid.

74. Geometer’s SketchPad
EF = 8 cm
CD = 12 cm
AB = 4 cm
EF = 7 cm
CD = 11 cm
AB = 3 cm
A
EF = 5 cm
CD = 6 cm
AB = 4 cm
EF = 7 cm
CD = 9 cm
AB = 5 cm
FE
A B
D C

75. Theorem 6.17
Midsegment Theorem for
Trapezoids
The midsegment of a
trapezoid is _________ to
each base and its length is
______________ of the
bases.
parallel
the average

76. Find x.
3. 4.
7
11
x
x
17
20
x = 9 x = 23

77. KITE
A kite has two pairs of
consecutive congruent
sides but opposite sides are
not congruent and no sides
are parallel.

78. Kite
What do we know if these points are equidistant
from the endpoint of the segment?

79. Theorem 6.18
In a kite, the longer
diagonal is the
_________________
of the shorter diagonal.
perpendicular bisector

80. Kite
What do we know about congruent
triangles?
How do we know the triangles are
congruent?

81. Kite

82. Theorem 6.19
In a kite, exactly one
pair of opposite angles
are ________.congruent
The congruent angles are formed
by the noncongruent sides.

83. Find x and y.
5. 6.
5
x y
x˚ 125˚
y˚
(y+30)˚
29
x = 2 y = 2
x = 125
y = 40

84. Theorem 6.19*
In a kite, the longer
diagonal
________________.bisects opposite angles

85. mJ =70°
mL = 70°
Find the missing angles.

86. x =35
Find x.

87. Find x.
x = 110

88. Find x.
x = 5

90. Based on our theorems, list all of
the properties that must be true
for the quadrilateral.
1. Parallelogram
(definition plus 4 facts)
2. Rhombus (plus 3 facts)
3. Rectangle (plus 2 facts)
4. Square (plus 5 facts)

91. Parallelogram
1) opposite sides are parallel
2) opposite sides are congruent
3) opposite angles are congruent
4) consecutive angles are
supplementary
5) diagonals bisect each other

92. Rhombus
1) equilateral
2) diagonals are perpendicular
3) diagonals bisect opposite angles

93. Rectangle
1) equiangular
2) diagonals are congruent

94. Square
1) equilateral
2) equiangular
3) diagonals are perpendicular
4) diagonals bisect opposite angles
5) diagonals are congruent

95. Lesson 6.6
Identifying Special
Quadrilaterals
Complete the chart of characteristics of special quadrilaterals.
Today, we will learn to…
> identify special quadrilaterals
with limited information

96. Given the following coordinates,
identify the quadrilateral.
(-2, 1)
(-2, 3)
(3, 6)
(0, 1)
kite

97. Given the following coordinates,
identify the quadrilateral.
(0, 0)
(4, 0)
(3, 7)
(1, 7)
trapezoid

98. Given the following coordinates,
identify the quadrilateral.
rectangle
(-1, -3)
(4, -3)
(4, 3)
(-1, 3)

99. Given the following coordinates,
identify the quadrilateral.
rhombus
(-2, 0)
(3, 0)
(6, 4)
(1, 4)

100. In quadrilateral
WXYZ, WX = 15,
YZ = 20, XY = 15,
ZW = 20. What is it?
It is a kite!

102. Lesson 6.7
Areas of Triangles
and Quadrilaterals
Today, we will learn to…
> find the area of triangles and
quadrilaterals

103. Postulate 22
Area of a Square
Area = side2
A=s2

104. Postulate 23
Area Congruence Postulate
If two polygons are congruent,
then they have the same area.

105. Theorem 6.20
Area of a Rectangle
Area = base ( height )
A = bh

106. 1. Find the area of the polygon
made up of rectangles.
4 m
10 m
2 m
9 m
11 m
7 m
11(2) = 22 m2
8(4)
=
32m2
5(4)=
20 m2
74 m2
?
?
?

107. Theorem 6.21
Area of a Parallelogram
Area = base ( height)
A=bh
Do experiment.

108. Theorem 6.22
Area of a Triangle
A=½ bh

109. Area of a Trapezoid
hh
b2
A = ½ h b1 + ½ h b2
b1
A = ½ h (b1 + b2)
A = ½ h b1 + ½ h b2

110. Theorem 6.23
Area of a Trapezoid
A = ½ height (sum of bases)
A=½ h (b1+b2)

111. 2. parallelogram 3. trapezoid
6
4 5
5 5
3
4
9
A = 6(4)
A = 24 units2
A = ½ 4(9+3)
A = 24 units2

112. Area of a Kite
b
b
x
y
A = ½ bx + ½ by
A = ½ b (x + y)
What is b? a diagonal
What is x + y? a diagonal
A = ½ d1 d2

113. Theorem 6.24
Area of a Kite
Area = ½ (diag.)(diag.)
A=½ d1 d2

114. Area of a Rhombus
A = ½ bx + ½ by
A = ½ b(x + y)
What is b? a diagonal
What is x + y? a diagonal
A = ½ d1 d2
b
b
x
y

115. Theorem 6.25
Area of a Rhombus
Area = ½ (diag.)(diag.)
A=½ d1 d2

116. 4. Rhombus 5. Kite
4
3
5
3
4
A = ½ 6(8)
A = 24 units2
A = ½ 6(9)
A = 27 units2

117. 6. Rhombus 7. Trapezoid
8
x
A = 80 units2
x = 5
A = 55 units2
h = 5
h
13
9

118. 8. Find the total area.
15
8 A = ½(10)(8+20)
A = 440 units2
20
25
A = 140
A = 20(15)
A = 300
?10

119. A = 12(11)
blue A = ½ (12)(5)
11
12
A = 132
132 = 122 + x2
x = 513
just blue?
blue A = 30
pink A = 132 – 60
pink A = 72
2 blue regions A = 60
?5
9. Find the areas of the blue and
pink regions.