Ch 6 quadrilaterals

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

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

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

Section 6.1 Vocabulary
Convex
Concave
Equilateral
Equiangular
Regular
Diagonal

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

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

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

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

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

Is this a polygon?
convex or concave?
No, polygons must
be made of
segments

Is this a polygon?
convex or concave?
Yes, it’s a
concave
dodecagon

Is this a polygon?
convex or concave?
No, polygons must
be closed figures

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

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

Lesson 6.2
Properties of
Parallelograms
RULERS AND PROTRACTORS
> use properties of parallelograms

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

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

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

… opposite sides are
__________congruent

… opposite angles are
__________.congruent

… 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

… diagonals __________
each other.
bisect

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

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

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

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

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

x = y =30 6
4x + 2x = 180 2y + 3 = y + 9

x = y =3 6
y
y
3x + 1 = 10 2y – 1 = y + 5

x = y =40 8
3x – 9 = 2x + 31 4y + 5 = 2y + 21

Lesson 6.3
Proving that Quadrilaterals
are Parallelograms
What is a converse?
> prove that a quadrilateral is a
parallelogram

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

Theorem 6.7
If both pairs of opposite
angles are __________,
congruent

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

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

Theorem 6.9
If the diagonals
__________________,
bisect each other
AE = EC
and
DE = EB
A
D
B
C
E

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

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

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

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

Lesson 6.4
Special
Parallelograms
> use properties of a rectangle,
a rhombus, and a square

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

parallelograms
rhombuses rectangles
squares

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

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?

The diagonals of a rhombus are
_____________.perpendicular
Theorem 6.11

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

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

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

The diagonals of a rectangle are
_____________.congruent
Theorem 6.13

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

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

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

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

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

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

Lesson 6.5
Trapezoids
& Kites
> use properties of trapezoids
and kites

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

B A
D
C
Compare leg angles.
mC = 65
mD = 115
mA = 90
mB = 90

In ALL trapezoids,
leg angles are
_______________supplementary

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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)

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

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

Rectangle
1) equiangular
2) diagonals are congruent

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

Lesson 6.6
Identifying Special
Quadrilaterals
Complete the chart of characteristics of special quadrilaterals.
> identify special quadrilaterals
with limited information

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

(0, 0)
(4, 0)
(3, 7)
(1, 7)
trapezoid

rectangle
(-1, -3)
(4, -3)
(4, 3)
(-1, 3)

rhombus
(-2, 0)
(3, 0)
(6, 4)
(1, 4)

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

Lesson 6.7
Areas of Triangles
and Quadrilaterals
> find the area of triangles and
quadrilaterals

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

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

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

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
?
?
?

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

Theorem 6.22
Area of a Triangle
A=½ bh

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

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

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

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

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

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

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

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

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

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

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.

Ch 6 quadrilaterals

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