G E O M E T R Y
CHAPTER 6
QUADRILATERALS
Notes & Study Guide
CHAPTER 6 NOTES
2
TABLE OF CONTENTS
POLYGONS .......................................................................................................... 3
PARALLELOGRAMS ........................................................................................... 5
PROVING QUADRILATERALS ARE PARALLELOGRAMS .............................. 6
RHOMBUSES ....................................................................................................... 7
RECTANGLES...................................................................................................... 8
SQUARES............................................................................................................. 8
TRAPEZOIDS ....................................................................................................... 9
KITES .................................................................................................................. 11
SUMMARY OF QUADRILATERALS ................................................................. 12
AREA FORMULAS ............................................................................................. 13
< end of page >
CHAPTER 6 NOTES
1
1
Polygons
POLYGONS
GOAL 1
uld learn
3
DESCRIBING POLYGONS
INTRODUCTION
Polygons
y, name, and In Chapter
q
6, we is
move
fromfigure
triangles
shapes
four. We will
A polygon
a plane
thattomeets
thewith more sides, specifically
R
s such as look following
vertex
at the five conditions.
main 4-sided shapes and their characteristics.
es in
P
side
1. It is formed by three or more segments
called sides, such that no two sides with a
RE
FE
e sum of the
vertex
1 endpoint
GOAL
DESCRIBING
POLYGONS
common
are collinear.
S
nterior
T
uld
learn  POLYGONS
Any closed,
plane
figure exactly
that is built
fromsides,
threeone
or at each endpoint.
ilateral.
2. Each side
intersects
two other
, name, and
q
polygon is
A segments
a plane
figure
that meets
the
more
(no
curves!)
is
called
a
polygon.
sldsuch
as it
Each endpoint
of a side is a vertex of the polygon. The plural of vertexR vertex
is
learn
following
conditions.
es in
vertices. You can name a polygon by listing its vertices consecutively.
For
P
al-life
side
1.means
It is formed
or more
segments
“poly”
many by
and
“-gon”
means
sides.
instance,
PQRST
andthree
QPTSR
are two
correct names for the polygon above.
he parachute
called sides, such that no two sides with a
sumL of
the
vertex
A LI
are collinear.
S
Closed common
– no gapsendpoint
or “crossovers”
in the segments
nterior
T
Aflat
MP
L two-dimensional
E 1intersects
lateral.
Identifying
Polygons
PlaneE2.X–Each
orside
exactly
two other sides, one at each endpoint.
d learn it
al-life
he parachute
RE
FE
AL LI
Each endpoint of a side is a vertex of the polygon. The plural of vertex is
State whether the figure is a polygon. If it is not, explain why.
vertices. You can name a polygon by listing its vertices consecutively. For
instance, PQRST and QPTSR are two correct names for the polygon above.
B
A
EXAMPLE 1
C
F
D
E
Identifying Polygons
State
whether
is apolygons.
polygon. If
explain
why.
A, Bthe
& figure
C are all
D,itEis&not,
F are
not polygons.
SOLUTION
Figures A, B, and C are polygons.
• Figure D is not a polygon because it has a side that is not a segment.
F
D
 CLASSIFYING
(NAMING)
POLYGONS
B
C
E
A
•
Figure
E
is
not
a
polygon
because
two
of
the
sides
intersect
only one other
side.
Polygons get their names from how many sides they have.
In general,
we put
the
FigureinF front
is notofa polygon
because
of are
its sides
more than two
correct• prefix
the “-gon”
suffix.some
There
someintersect
exceptions.
other sides.
.Sides
. . . . . . . Name
..
Sides Name
Figures A, B, and C are polygons.
3
TRIANGLE
8
OCTAGON
Polygons
the number
of sides
they
have.
• Figureare
D named
is not abypolygon
because
it has
a side
that is not a segment.
SOLUTION
HELP
gon not
ble, use
sides. For
ygon with
-gon.
4
QUADRILATERAL
9
NONAGON
• Figure E is not a polygon because two of the sides intersect only one other side.
of sides
Type of polygon
Number
of sides
Type of polygon
5 Number
PENTAGON
10
DECAGON
• Figure F is not a polygon because some of its sides intersect more than two
3
Triangle
8
Octagon
6other HEXAGON
12
DODECAGON
sides.
4
Quadrilateral
9
Nonagon
or HEPTAGON
n
n–gon
. . .7. . . . .SEPTAGON
..
5
Pentagon
10
Decagon
Polygons are6named by the
number
of
sides
they
have.
Hexagon
12
Dodecagon
< end of page >7
Number of sides
HELP
apter 6 Quadrilaterals
gon not
e, use
ides. For
gon with
Heptagon
n
n-gon
Type of polygon
Number of sides
3
Triangle
8
4
Quadrilateral
9
5
Pentagon
10
Decagon
6
Hexagon
12
Dodecagon
Type of polygon
Octagon
CHAPTER
6 NOTES
Nonagon
contains both convex and
concave polygons.
or
and
ng
ebra
b. The polygon has 5 sides, so it is a pentagon.
When extended, none of the sides intersect
POLYGONS
the interior, so the polygon is convex.
4
 CONCAVE v. CONVEX
. . “dented”.
........
CONCAVE polygons are
means
if one
sidethatwas
AThat
polygon
is convex
if no line
contains
a side
the polygon
contains
extended,
it of
would
go through
the is equilateral if all of its sides are congruent. A
A polygon
interior
a interior.
point in the interior of the polygon.
equiangular if all of its interior anglesinterior
are congruent. A po
A polygon that is not convex is called
is
equilateral
and
equiangular.
nonconvex
or concave.
CONVEX polygons
are not dented.
convex polygon
concave polygon
None of the sides, when extended,
will enter the interior.
E X A M P L E 2 Identifying
Convex
PolygonsRegular Polygons
EX
A M P Land
E 3Concave
Identifying
INTERIOR
ANGLES
OF
QUADRILATERALS
Identify
the polygon
and state
whether it
is convex
or concave.
GOAL 2
 REGULAR POLYGONS
Decide whether the polygon is regular.
a.
b.
When
a
polygon
has
all
sides
congruent (equilateral)
and
A diagonal of a polygon is a.
a segment
q
alljoins
angles
(equiangular)
at the same time, it is b. R
that
twocongruent
nonconsecutive
vertices.
referredPQRST
to as regular.
Polygon
has 2 diagonals from
Æ
Æ
point Q, QT and QS.
c
S
P
All regular polygons must be drawn in the Tnice “circular”diagonals
for lots of tick and angle marks.
Sway.
OLUTION
As usual, look
a. The
polygon
has 8 sides, so
it isboth
an octagon.
Like
triangles,
quadrilaterals
have
interior and exterior angles. If you draw a
When
some of
thedivide
sides intersect
diagonal
inextended,
a quadrilateral,
you
it into two triangles, each of which has
S
OLUTION
the angles
interior,with
so the
polygon
is concave.
NTERIOR
NGLES
OF conclude
QUADRILATERALS
2
interior
measures
that
add Iup
to 180°.ASo
you can
that the
GOAL
 DIAGONALS
a. The polygon is an equilateral quadrilateral, but not equia
sum of the measures of the interior angles of a quadrilateral is 2(180°), or 360°.
A diagonal
ofH a polygonAis diagonal
a it
segment
joinsis any
two
is not
regular
polygon.
S
of aathat
polygon
a segment
q
R
B
B nonconsecutive vertices.
nonconsecutive
vertices.
that
joins
two
b. The
polygon
has 5 sides, so it is a pentagon.
Study
Tip
b. This
pentagon
is equilateral
Polygon
has 2 diagonals
from C and
C PQRST
C equiangular. So, it is a r
Two vertices that are
TUDENT
ELP
When extended, none of the sides
Æ intersect
Æ
S
endpoints of the same
point Q, QT and QS.
theare
interior,
polygon
convex.
Non-consecutive
means
NOT
next
to each other.
c. isThis
heptagon
is equilateral, but
not equiangular.
So, it i
diagonals
side
called so the
T
P
consecutive vertices. For
example, P and Q are
.consecutive
. . . . . . . vertices.
Like triangles, quadrilaterals have both interior and exterior angles. If you draw
in aDquadrilateral,
each of which has
D
A A you divide it into two triangles,
 INTERIOR ANGLESdiagonal
OF QUADRILATERALS
interior angles with measures that add up to 180°. So you can conclude that the
A quadrilateral is a polygon
with
four sides.
“quad-”angles
means
and “-lateral”
the interior
of a four
quadrilateral
is 2(180°), or 360°.
A polygon is equilateral sum
if allofofthe
itsmeasures
sides areofcongruent.
A polygon
is
..
A
THEOREM
means
existsBangles
with theareinterior
angles
of
B all quadrilaterals.
T H E Osides.
REM
equiangular
if A
allpattern
of its interior
congruent.
A polygon
is regular if
C
is equilateral and equiangular.
THEOREM 6.1 Interior Angles of a Quadrilateral
THEOREM
The sum ofofthe
angles
of any
The sum of6.1
the– measures
theinterior
interior
angles
of a
quadrilateral
isIdentifying
360°.
quadrilateral
ALWAYS
be
360°. APolygons
E
X A M P L E 3will
Regular
D
it
C
C
2
1
A
A
m™1 + m™2 + m™3 + m™4 = 360°
3
D
4
TH
OREM
Decide whether the polygon
isEregular.
THEOREM
< end of page>
a.
EXAMPLE 4
b.
THEOREM 6.1
c.
Interior Angles of a Quadrilateral
m™1 + m™2 + m™3 + m™4 = 360°
Find m™Q and m™R.
SOLUTION
2
The sum of the measures of the interior angles of a
1
CHAPTER 6 NOTES
quadrilateral
is 360°.
Interior Angles
of a Quadrilateral
P
80!
3
4
6.2
6.2
Properties of Paralle
PARALLELOGRAMS
5
Properties of
Parallelograms
1 PROPERTIES OF PARALLELOG
GOAL
6.2
6.2
6.2
What
Polygons
INTRODUCTION
you should learn
that have four sides are called quadrilaterals. There are six different
GOAL 1 GOAL
UseSome
some
quadrilaterals.
unique, butInmost
oflesson
them share
In this
chapter
we willyou will
1 are
PROPERTIES
OF
ARALLELOGRAMS
thisP
and features.
in the rest
of the
chapter
u should learn
properties
parallelograms.
study
each ofofthese
six shapes.
A parallelogram is a quadrilateral with both pairs
Properties of Parallelograms
Properties
Properties
Properties of
of Parallelograms
Parallelograms
In this lesson and in the rest of the chapter you will study special quadrilaterals.
Use some
GOAL 2 Use properties of
of parallelograms.
When OF
you
mark
diagrams
of quadrilaterals,
GOAL 11 isPP
ROPERTIES
OF
PARALLELOGRAMS
ARALLELOGRAMS
 PARALLELOGRAMS
A parallelogram
aROPERTIES
quadrilateral
with
both pairs
of opposite
sides parallel.
P
real-life
Whatyou
youparallelograms
shouldlearn
learn inGOAL
1shape)
GOAL
1
What
should
P
ROPERTIES
OF
P
ARALLELOGRAMS
GOAL
P
ROPERTIES
OF
P
ARALLELOGRAMS
use
matching
arrowheads
to
indicate which
A
quadrilateral
(4-sided
that
has
both
pairs
Use properties of
What
should
learn
youasmark
What
you
shouldWhen
learn
situations,
such
the diagrams of quadrilaterals,
1you
GOAL
q
R
Usesome
some
Inthis
thislesson
lessonand
inthe
the
rest of
of
the chapter
chapter you
you will
will study
study
special quadrilaterals.
quadrilaterals.
1 Use
GOAL
ams in
real-life
sides
are
parallel.
For
example,
in the diagram
In
in
rest
the
special
ofofUse
opposite
sides
parallel
isand
called
aquadrilateral
parallelogram.
use
matching
arrowheads
to
which
properties
parallelograms.
1 drafting
GOAL
GOAL
some
table
shown
inlesson
some
parallelogram
AInparallelogram
isthe
aindicate
with
both
pairs
of opposite
opposite
sides
parallel.
Æ
Æ
Æ
Æ
parallelograms.
In
this
rest
you
will
special
quadrilaterals.
this
lessonand
andinis
in
the
restofofthe
thechapter
chapter
you
willstudy
study
specialsides
quadrilaterals.
suchproperties
as
the 1ofUse
A
a
quadrilateral
with
both
pairs
of
parallel.
∞ RS
QR ∞sides
SP parallel.
. The symbol
the
right,
PQ
properties
of of
parallelograms.
properties
parallelograms.
sides
For example,
in the
diagram
GOAL
A
isisat
a aquadrilateral
with
both
pairs
of
Use
properties
parallelogram
Aparallelogram
quadrilateral
with
both
pairsand
ofopposite
opposite
sides
parallel.
Example
6.ofof are parallel.
le shown
in2 Use
GOAL 2
Æ
Æ
Æ
Æ
R
When
you
mark
diagrams
of
quadrilaterals,
properties
q
R
When
you
mark
diagrams
of.quadrilaterals,
q
∞PQRS
RS
and
QR⁄PQRS
∞ SP
The is
symbol
at
the
PQ
parallelograms
inproperties
real-life
2 2
GOAL
GOAL
Use
of ofright,
read
“parallelogram
PQRS.”
Use
Shown:
parallelogram
or
parallelograms
inproperties
real-life
use
matching
arrowheads
to
indicate
which
RR
When
you
mark
diagrams
of
quadrilaterals,
When
you
mark
diagrams
of
quadrilaterals,
qq
use
matching
arrowheads
to
indicate
which
situations,
suchinasin
the
parallelograms
real-life
parallelograms
⁄PQRS isuse
read
“parallelogram
PQRS.”
P
S
situations,
such as
thereal-life
sides
areparallel.
parallel.
Forexample,
example,
in the
thewhich
diagram
matching
totoindicate
uselearn
matching
arrowheads
indicate
which
you
should
itarrowheads
sides
are
For
in
diagram
drafting
table
shown
in
Æ
Æ
Æ Æ
situations,
such
asas
the
situations,
such
drafting
table
shown
inthe
Æ
Æ
Æ
RS
and
QR ∞∞ SP
SP
Thediagram
symbol
atsides
theright,
right,
PQ∞∞Æ
should
learn
it
sides
are
For
example,
inin..the
areparallel.
parallel.
For
example,
the
diagram
Example
6.table
RS
and
QR
The
symbol
at
the
PQ
drafting
table
shown
in
drafting
shown
in
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Example 6.
⁄PQRS
read
PQRS.”
∞RS
RSand
andQR
QR∞ ∞SP
SP
Thesymbol
symbol
atthe
theright,
right,
∞“parallelogram
. .The
at
PQ
P
S
properties
ofPQ“parallelogram
Example
6.of You can use ⁄PQRS
Example
6.!
isisread
PQRS.”
P
S
use properties
⁄PQRSisisread
read“parallelogram
“parallelogramPQRS.”
PQRS.”
⁄PQRS
PP
SS
Why
you
should
learn
it
parallelograms
to understand
Why
you should
learnT it
TRHAELO
RLE
SA A
B O U T PA R A L L E L O G R A M
ams to
understand
HEO
M S A B O U T PA
LE
OM
GR
MS
Why
youshould
should
learn
Why
learn
itit R EOF

PROPERTIES
Youyou
can
use
properties
of
ors !
lift!You
works
how
a scissors
lift worksPARALLELOGRAMS
in
canin
use properties
of
Why
RE
FE
FEFEFE
RERERE
RE
RE
FE
FE
parallelograms
to understand
can
properties
YouLYou
can
useuse
of of four
T Hsides
E O R Eand
MS A
O U T PAparallel
RALLELOGRA
MS
!
!
LBesides
having
2BBpairs
have many
parallelograms
toproperties
understand
I Exs.
TH
ELO
S A
O U T of
PA R A L L E Llines,
O G R Aparallelograms
MS
LRREREM
Aa scissors
THEOREM
6.2
how
lift51–54.
works
in
parallelograms
to
understand
parallelograms
to
understand
E
E
M
U
PARRAALLLLE6.2
ELLOOG
GRRA
AM
MSS
TT
H
OIO
M
SSAA
BB
OO
U
TTPA
AHE
THEOREM
how
a scissors
lift
works
in
important
properties.
Each
one
is
represented
by
its
own
theorem.
L
Exs.
L
how
a other
scissors
works
in
q
R
I
how
a51–54.
scissors
liftlift
works
in
A If a quadrilateral
THEOREM is
6.2a parallelogram, then its
Exs. 51–54.
ALLLLIL LI
THEOREM 6.2
Exs.
51–54.
Exs.
51–54.
If
a
quadrilateral
is
a
parallelogram,
then
THEOREM
6.2
A A IoppositeTHEOREM
q
6.2
R
sides
are
congruent.
If a quadrilateral is a parallelogram, then its
q
R
If a quadrilateral is a parallelogram, then its
qq
R
Ifa aquadrilateral
quadrilateral
parallelogram,
thenits
its
sides
are congruent.
opposite
are
isiscongruent.
aaopposite
parallelogram,
then
Æ
ÆIf
ÆsidesÆ
sides
are congruent.
PQ £ RSopposite
and
SP
£
QR
opposite
sides
arecongruent.
congruent.
Æ
Æsides
Æ
Æ
opposite
are
Æ
Æ
Æ
Æ
Æ
Æ
PQ
RS
andÆ
SP £ Æ
QR
THEOREM 6.2 –PQ
IfÆ£a£RS
quadrilateral
is a parallelogram,
Æ
Æ
Æ
Æ
Æ
Æ
Æ
and
SP
£ QR
PQ
S
PQ££RS
RSand
andSP
SP££QR
QR £ RS and SP £PQR
PQ
then both THEOREM
pairs of opposite
sides are congruent.
6.3
P
P
PP
THEOREM 6.3
THEOREM 6.3
THEOREM
6.3
THEOREM
6.3
quadrilateral
is
a parallelogram,
S
S
SS
qq
If a
thenthen
its its
If a quadrilateral is a THEOREM
parallelogram,
6.3 its
aIf quadrilateral
isisa aparallelogram,
quadrilateral
parallelogram,then
thenits
its
a aquadrilateral
is a congruent.
parallelogram,
then
oppositeIfIfopposite
angles
are
congruent.
angles
are
its
RR
R
RR
q
qq
angles
anglesare
arecongruent.
congruent.
opposite
angles
are
congruent.
THEOREM 6.3 –opposite
Ifopposite
a£ ™R
quadrilateral
isa aquadrilateral
parallelogram,is a parallelogram, then its
™P
and
™Q £ If
™S
™P £ ™R
and
™Qand
£ ™Q
™S
™P
£
™R
£
™S
™P££™R
™Rand
and
™Q££™S
™S congruent.
™P
™Q
then both pairs of opposite
angles
are
opposite
angles are congruent.
P
S
P PP
SS
SS
P
™P £ ™R and ™Q £ ™S
THEOREM 6.4
THEOREM
6.4
THEOREM
6.4
THEOREM
6.4
THEOREM
6.4
If a quadrilateral is a parallelogram, then its
is
aquadrilateral
aparallelogram,
parallelogram,
then
its
If
isisaaare
parallelogram,
then
its
If a6.4
quadrilateral
is aangles
parallelogram,
thenthen
its its
consecutive
THEOREM
–Ifconsecutive
IfaIfaquadrilateral
aquadrilateral
quadrilateral
issupplementary.
a parallelogram,
angles
are
consecutive
angles
aresupplementary.
supplementary.
consecutive
angles
are
supplementary.
consecutive
angles
are
supplementary.
m™Pangles
+ m™Q are
= 180°,
m™Q + m™R = 180°,
then its’ consecutive
supplementary.
m™P
+
m™Q
=
m™Q
=
m™P
m™Q
180°,
m™Q+
m™R
180°,
THEOREM
6.4
m™P
++
m™Q
==180°,
180°,
m™Q
++m™R
m™R
==180°,
180°,
m™R +
180°, m™S
+ m™P
= 180°
m™P + m™Q
=m™S
180°,=m™Q
+ m™R
= 180°,
m™R
+
m™S
=
m™R
m™S
180°,m™S
m™S+
m™P=
180°
m™R
++
m™S
==180°,
180°,
m™S
++m™P
m™P
==180°
180°
q
R
RRR R
qqqq
P
PP
S
S
P
S
quadrilateral
is aP parallelogram,
then its
S
m™R + m™S
= 180°, m™SIf+am™P
= 180°
THEOREM 6.5
consecutive
angles
are
supplementary.
THEOREM
6.5
THEOREM
6.5
q
R
6.5
THEOREM 6.5 – THEOREM
If a quadrilateral
is a parallelogram,
qq
q
If a quadrilateral is a parallelogram, then its
RR
THEOREM
Ifaquadrilateral
aquadrilateral
quadrilateral
isaaaparallelogram,
parallelogram,
thenits
its
IfIf
a6.5
is
then
M
is
parallelogram,
then
its
diagonals
bisecteach
each
other.
then its’ diagonals
will
bisect
other.
q
M m™R =
m™P
+
M
R 180°,
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THEOREM 6.5
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Chapter 6 Quadrilaterals
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Theorem
6.4, and Theorem
330
Chapter
Quadrilaterals
330
66 6Quadrilaterals
PA R A L L E L O G R A M S
Chapter 6 Quadrilaterals
6
PARALLELOGRAMS
 PROVING QUADRILATERALS ARE PARALLELOGRAMS
To prove that a quadrilateral is a parallelogram, you need to do the following…
(1) establish that the shape has four sides (quadrilateral)
(2) establish that the shape meets any one of the four properties
Each possible way to prove this is represented by its’ own theorem
(Theorems 6.6 – 6.9)
SEE TEXTBOOK PAGE 340
These theorems are nothing more than the converses of the ones listed earlier in
the notes.
< end of page > CHAPTER 6 NOTES
SPECIAL PARALLELOGRAMS
6.4
Rhombuse
and Square
7
By definition, all we need to have for a parallelogram is both pairs of opposite
sides parallel.
BUT, if we add something extra to that, then the parallelogram
can turn
1 intoPone
GOAL
ROPERTI
you should
learn
of three special parallelograms (which
are also
quadrilaterals).
What
GOAL 1 Use properties of
In this lesson you will stud
sides and angles of
rectangles, and squares.
 RHOMBUS
rhombuses,
rectangles,
U
SING
D
IAGONALS
OF
S
PECIAL
P
ARALLELOGRAMS
GOAL 2
parallelogram
has four congruent
then P
it ARALLELOGRAMS
is
SING DIAGONALS
OF Ssides,
PECIAL
GOAL 2If a U
and squares.
called a rhombus.
2 Use properties of
The following theorems are aboutGOAL
diagonals
of rhombuses and rectangles. You
The
following
theorems
are
about
diagonals
ofExercises
rhombuses
diagonals
rhombuses,
are asked to prove Theorems 6.12 and 6.13of in
51,and
52,rectangles.
59, and 60.You
re asked to prove Theorems 6.12rectangles,
and 6.13 and
in Exercises
squares. 51, 52, 59, and 60.
THEOREMS
THEOREMS
Why you should learn it
RE
ABCD is a rhombus if and only if Æ
AC fi BD
Æ.
ABCD is a rhombus if and only if AC fi BD .
A rhombus is a
parallelogram with
four congruent sides.
B
B
C
C
FE
PROPERTIES OF RHOMBUSES
! To simplify real-life tasks,
THEOREM 6.11
such as checking whether a
THEOREM 6.11
A parallelogram is a rhombus
if and
if its
theater
flatonly
is rectangular
in
A parallelogram
is a rhombus
if and only if its
THEOREM
6.11
– A parallelogram
IFF
diagonals
are perpendicular.
Example 6.is a rhombus
AL LI
diagonals
are perpendicular.
its diagonals
are perpendicular. Æ Æ
AThe
A
Venn diagram
at the ri
D
D
shows the relationships am
THEOREM 6.12
THEOREM 6.12
parallelograms, rhombuses
B
C
A parallelogram is a rhombus if and only if each
B
C
rectangles,
and squares.
Ea
A parallelogram is a rhombus if and only if each
diagonal
bisects
a
pair
of
opposite
angles.
THEOREM
6.12
–
A
parallelogram
is
a
rhombus
IFF
diagonal bisects a pair of opposite angles.
has the properties of every
eachisdiagonal
bisects
a pair
of opposite
angles.
ABCD
a rhombus
if and
only
if
that it belongs to. For insta
Æ is a rhombus if and only if
ABCD
A
D
Æ
AC bisects ™DAB and ™BCD and
A
square is a Drectangle, a rho
AC
Æ bisects ™DAB and ™BCD and
Æ
BD bisects ™ADC and ™CBA.
and a parallelogram, so it h
BD bisects ™ADC and ™CBA.
of the properties of each of
THEOREM 6.13
THEOREM 6.13
shapes.
A
B
Notice that each of these Theorems say what a parallelogram
must have in order
A
B
A parallelogram is a rectangle if and only if its
A parallelogram
a rectangle if and only if its
to be called aisrhombus.
diagonals are congruent.
diagonals are congruent.
Æ
Æ
Æ
Æ.
ABCD
a rectangle
if and
only
AC
£ BD
Thisisismeans
that you
mustonly
meetifif AC
the £
requirements
of
D theE parallelogram
X A M P L E 1 CCfirstDescrib
ABCD
a rectangle
if and
BD .
D
before you can try to classify it as a rhombus.
Decide whether the statem
< end of page >
a.
A rhombus is a rectang
Youcan
canrewrite
rewriteTheorem
Theorem 6.11
6.11 as
as aa conditional
conditional statement
statement and
and its
its converse.
converse.
You
CHAPTER 6 NOTES
b. A parallelogram
is a re
Conditional statement: If the diagonals of a parallelogram are perpendicular,
then
onditional statement: If the diagonals of a parallelogram are perpendicular, then
the parallelogram
parallelogram is
is aa rhombus.
rhombus.
the
SOLUTION
Converse: If a parallelogram is a rhombus, then its diagonals are perpendicular.
6.4
8
What you should learn
and Squares
GOAL 1
PROPERTIES OF SPECIAL PARALLEL
SPECIAL PARALLELORGRAMS
GOAL 1 Use properties of
In this lesson you will study three special types of parallelog
sides
and angles of
 RECTANGLE
rectangles, and squares.
rhombuses, rectangles,
If aand
parallelogram
has
four
right
angles, then it is
2 USING DIAGONALS OF SPECIAL PARALLELOGRAMS
GOALsquares.
called a rectangle.
GOAL 2
Use properties of
of rhombuses,
Thediagonals
following
theorems are about diagonals of rhombuses and rectangles. You
and squares.
are rectangles,
asked to prove
Theorems 6.12 and 6.13 in Exercises 51, 52, 59, and 60.
learn
rties of
es,
rties of
ses,
res.
learn it
fe tasks,
hether a
gular in
FE
LI
A rhombus is a
such as checking whether a
four congruent sides.
A rectangle is a
parallelogram with
four right angles.
Rhombuses, Rectangles,
THEOREM
– A parallelogram is a rectangle IFF its
and6.13Squares
diagonals are congruent.
THEOREM
theater flat is6.11
rectangular in
RE
Example
6.
A
parallelogram
LI a rhombus if and only if its
ALis
diagonals are perpendicular.
B
A
pa
fo
an
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4
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T
EO
R E Mreal-life
SOF RECTANGLES
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parallelogram with
simplify
tasks,
!HTo
The Venn Æ
diagram
Æ at the right
para
ABCD is a rhombus if and only if AC fi BD .
shows the relationships amongA
D
GOAL 1 PROPERTIESparallelograms,
OF SPECIAL
PARALLELOGRAMS
rhombuses,
THEOREM 6.12
rectangles, and squares. Each shape
C
parallelogram
a rhombus
and
onlytypes
if each
hasifthe
properties
ofofevery
group B rhombuses,
InA this
lesson you is
will
study three
special
parallelograms:
rhombuses
diagonal
bisects
a
pair
of
opposite
angles.
that it belongs to. For instance, a
rectangles,
SQUARE and squares.
square
rectangle,
rhombus,
ABCD is a rhombus
only is
if asides
If a parallelogram
has fourif and
congruent
ANDa four
Æ
D
and a and
parallelogram, so it has Aall
AC bisects
™DAB
anda™BCD
right angles,
then
it
is
called
square.
Æ
of the properties of each of those
BD bisects ™ADC and ™CBA.
shapes.
THEOREM 6.13
A
B
A parallelogram is a rectangle if and only if its
Adiagonals
rhombus
a
AE Xrectangle
is a
areiscongruent.
A M P L E 1is a DescribingAa square
Special Parallelogram
PROPERTIES
parallelogram OF
withSQUARESparallelogram
Æ with
Æ
parallelogram with
D
C
four right angles.
four congruent sides
Decide
whether
the
statement
is
always,
sometimes,
or never
Notice that squares require properties of rhombuses AND rectangles…
and four right angles.
(four congruent
sides….four
angles)
a. A
rhombus isright
a rectangle.
ABCD is a rectangle if and only if AC £ BD .
four congruent sides.
para
b. A parallelogram is a rectangle.
Thecan
Venn
diagram
at the
right
You
rewrite
Theorem
6.11
as a conditional
statement
andfor
itsrhombuses
converse. and
Therefore,
the
properties
of
square
are the same
as those
parallelograms
shows the relationships among
rectangles.
SOLUTION
Conditional statement: If the diagonals
of a parallelogram are perpendicular, then
parallelograms, rhombuses,
rhombuses
the
parallelogram
a rhombus.is sometimes true.
a.
The
rectangles, and squares. Each shapeis statement
s
In
the
Venn
diagram,
the
regions
has the If
properties
of every
Converse:
a parallelogram
is group
a rhombus, then its diagonals
are perpendicular.
rhombuses
rectangles
that it belongs to. For instance, a for rhombuses and rectangles
squares
To prove the theorem, you must prove
both statements.
square is a rectangle, a rhombus, overlap. If the rhombus is a
and a parallelogram, so it has all square, it is a rectangle.
of the properties of each of those
b. The statement is sometimes true. Some parallelograms a
X A M P L E 4 Proving Theorem 6.11
< Eend
of page >
shapes.
Venn diagram, you can see that some of the shapes in th
are in the region for rectangles, but many aren’t.
Write a paragraph proof of the converse above.
B
CHAPTER 6 NOTES
A
E
X
A
M
P
L
E
1
6.4
Rhombuses, Rectangles
Describing a Special Parallelogram
GIVEN ! ABCD is a rhombus.
Æ
Æ
PROVE ! AC fi BD
X
Decide whether the statement is always, sometimes, or never
true.
D
C
6.5Trapezoids and Kites
6.5
6.5 Trapezoids and Kites
TRAPEZOIDS
9
Trapezoids and Kites
There are some quadrilaterals that do not meet the requirements of being
parallelograms. These
the last two
we will examine.
1 shapes
GOALtwo
USING are
PROPERTIES
OF that
TRAPEZOIDS
What you should learn
GOAL 1
A
Use properties of
t you should
learn

TRAPEZOID
trapezoids.
GOAL 1
USING PROPERTIES OF TRAPEZOIDS
base
trapezoid is a quadrilateral with exactly one pair
A
of parallel sides. The parallel sides are the bases.
leg
1
GOAL
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U
SING
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A
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For
2
GOAL
of properties
Use
A oftrapezoid is a quadrilateral with exactly one pair
A
should learn
instance, in trapezoid ABCD, ™D and ™C are one
kites.sides is called a trapezoid.
oids.
of parallel sides. The parallel sides are the bases.
D
pair of base angles. The other pair is ™A and ™B.
leg
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angles.
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two pairs
of are
For
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should
it The
trapezoid
A learn
ishas
a quadrilateral
with
one
pair
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the exactly
trapezoid.
A
instance,
in
trapezoid
ABCD,
™D
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™C
are
one
solve two
real-life
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ofsides
parallel
sides.
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parallel
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the
! ToThe
that Ifare
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are called
the bases.
the legs
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are congruent,
then the
Dleg
pair of base
angles.
Theof
other
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is ™A and
™B.
problems, such as
base
Aplanning
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Forlegs.
trapezoid
isparallel
an isosceles
se properties
of two
The
sides
thathas
aretwo
notpairs
aretrapezoid.
called the
6.5
leg
base
Trapezoids and Kites
Why
B
B
C
leg
base
base
B
C
leg
RE
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base
the layers
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Thein nonparallel
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of™C
the are
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instance,
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ABCD,
™D
and
one
L
Example 3.
L
You
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the
following
theorems
in
the
1
I
GOAL
USING PROPERTIES OF TRAPEZOIDS
A
solve you
real-life
D
pairIfofthe
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What
should learn
isosceles trapezoid
legsangles.
of a trapezoid
arepair
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then™B.
the
exercises.
base
ms,
suchlearn
as planning
The
that areison
each
oflegs
theofbases
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hould
it angles
isosceles
an
The trapezoid
nonparallel
sides
areend
thetrapezoid.
the trapezoid.
OAL 1
yers
of a Use
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A trapezoid is a quadrilateral with exactly one pair
eal-life
apezoids.
H
E Oprove
R The
EM
S
ple
3.
You
to
the
following
theorems
in the
bases.
If the
legs
ofasked
a Ttrapezoid
are
congruent,
then
ofare
parallel
sides.
parallel
sides are
thethe
AL LI
ch as planning
exercises.
A trapezoid
has two trapezoid.
pairs of base angles. For
OAL 2 Use properties of
isosceles
trapezoid
is an THEOREM
6.14
a layer cake Ifin the two legsinstance,
are congruent,
the™D
trapezoid
in trapezoidthen
ABCD,
and ™C is
arecalled
one
RE
B
B
A
If aprove
trapezoid
is
isosceles, theorems
then each pairthe
of
You arepair
asked
to
the
following
of base
angles.
The
other pair is ™A andin™B.
an isosceles
trapezoid.
angles is congruent.
H Enonparallel
O Rbase
EMS
Why you should
learnexercises.
it TThe
sides are the legs of the trapezoid.
AL LI
D
FE
RE
base
A
legisosceles trapezoid leg
FE
es.
C
angles.
™A £ ™B, ™C £ ™D
C
base
isosceles
trapezoid
C
D
6
RE
FE
! To solve real-life
If the legsTHEOREM
of
a trapezoid
are congruent, then the
B
A
THEOREM
6.14
6.15
oblems, such as planning
B
A
trapezoid
is
an isosceles trapezoid.
T
H
E
O
R
E
M
S
If a trapezoid
has a pair
of each
congruent
If a trapezoid
is isosceles,
then
pair base
of
e layers of a layer cake in
angles,
then
it is
anfollowing
isosceles trapezoid.
xample 3.
You are
asked
prove
the
theorems in the
base
angles
istocongruent.
AL LI
 PROPERTIES
OF
TRAPEZOIDS
BC C
THEOREM
6.14 ABCD is an isosceles trapezoid.
isosceles trapezoid
DA
exercises.
D
™A £ ™B, ™C £ ™D
If a trapezoid
is isosceles,
then each pair of
B
A
THEOREM
6.16
THEOREM 6.15
B
A
base anglesAistrapezoid
congruent.
is
isosceles
if
and
only
if
its
HEOREMS
If aTtrapezoid
has are
a pair
of isosceles,
congruent base
C is
D
THEOREM 6.14
–
If
a
trapezoid
is
then
each
pair
of
base
angles
diagonals
congruent.
™A £ ™B, ™C £ ™D
angles, then it is an isosceles trapezoid.Æ Æ
C
D
THEOREMABCD
6.14 is isosceles if and only if AC £ BD .
congruentTHEOREM
6.15
ABCD
is an isosceles
trapezoid.
HEOREM
If a T
trapezoid
isSisosceles, then each pair of
If a trapezoid
has
a
pair
of
congruent
base
base angles
THEOREM
6.16 is congruent.
A
DA
AD £ BC
B
A
angles, then it is an isosceles trapezoid.
D
™A
™C £ ™D
trapezoid
if and
onlyaifofits
THEOREM A6.15
– AIf£Mis™B,
aisosceles
has
pair
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P
L Etrapezoid
1 Using
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ABCD EisX an
isosceles
trapezoid.
D
diagonals
are
congruent.
THEOREM
6.15
congruent base
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then it is an isosceles
Æ
Æ
PQRS
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THEOREM
6.16
If
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trapezoid
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base
ABCD is isosceles
iftrapezoid.
and
if AC S£
BD
50! .
trapezoid.
Find m™P, m™Q, and m™R.
angles, then it is an isosceles trapezoid.
ATtrapezoid
H E O R E MisSisosceles if and only if its
is an isosceles trapezoid.
diagonalsABCD
are congruent.
D
A
A
B
BC
C
C
R
AD £ BC
B
C
B
q
P
C
D
D m™S = 50°. Because C
Æ
Æ
SOLUTION
PQRS
an isosceles
so.m™R =
B
A
THEOREM
6.16
ABCD
is isosceles
if isand
only if trapezoid,
AC £ BD
THEOREM 6.16 ™S
– A trapezoid
is isosceles
IFF its’by parallel lines,
BC are
are consecutive
interior
angles formed
E X A M P L Eand
1 ™P
Using
Properties
of Isosceles
Trapezoids AD £they
A
trapezoid
is
isosceles
if
and
only
if
its
T
H
E
O
R
E
M
S
supplementary.
So,
m™P
=
180°
º
50°
=
130°, and m™Q = m™P = 130°.
diagonals are congruent.
diagonals are congruent.
356
R
S Æ
PQRS is an isosceles trapezoid.
D
Æ
Chapter 6 Quadrilaterals
ABCD is isosceles if and only if AC50!£ BD .
AD £ BC
Find m™P,
m™Q, and m™R.
EXAM
PEL O
E R1E MUsing
TH
S
Properties of Isosceles Trapezoids
P
C
q
S
PQRS
is an isosceles
SOLUTION
PQRS trapezoid.
is an isosceles trapezoid,
so m™R = m™S =R50°. Because
50!
E
X A™P
M Pare
L E and
1 m™R.
Using Properties
of Isosceles
Find™S
m™P,
m™Q,
and
consecutive
interior angles
formed byTrapezoids
parallel lines, they are
< end of page
>
supplementary.
So, m™P = 180° º 50° = 130°, and m™Q = m™P = 130°.
PQRS is an isosceles trapezoid.
Find m™P, m™Q, and m™R.
SOLUTION PQRS
Chapter 6 Quadrilaterals
S
P
50!
q
R
CHAPTER 6 NOTES
is an isosceles trapezoid, so m™R = m™S = 50°. Because
q lines, they are
™S and ™P are consecutive interior angles formedP by parallel
supplementary. So, m™P = 180° º 50° = 130°, and m™Q = m™P = 130°.
SOLUTION PQRS is an isosceles trapezoid, so m™R = m™S = 50°. Because
HOMEWORK HELP
Æ
Æ
Æ Æ
Visit our Web site
The slopes
and CD
equal, so AB ∞ CD.
Show of
thatAB
ABCD
is a are
trapezoid.
ww.mcdougallittell.com
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r extra examples.
7º5
2
1
1
x
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slope of BC = !! = !! = !!.
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10
A(5, 0)
1
4º0
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C(4, 7)
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The slope of AD = !! = !! = 2.
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7
º3 that connected
Æ
Æ
Æ
!
!
!
!
=
=
=
º1.
slope
of
CD
So, because AB ∞ CD and BC is not parallel to AD, ABCD is a trapezoid. D(7, 4)
In a
two of !its’ sides.
7º4
3
Æ
Æ
Æ Æ
. . . . .The
. . . slopes
..
of AB and CD are equal, so AB ∞ CD.
7º5
2 segment
1
The midsegment
a trapezoid
!! =
!!.
The slope of BC
= !!is=the
4º0
4
2
Æ
1
x
A(5, 0)
1
B
For a that
trapezoid,
islegs.
a swegment
connectsthe
the midsegment
midpoints of its
Theorem
Æ
4 º 0two
4
6.17 is similar
to
the
Midsegment
Theorem
that connects
the
midpoints
of
the
= 2.for
The slope of AD = !! = !!legs.
midsegment
º5
2
triangles. You will justify part7 of
this theorem
in
Æ
Æ
Æ
Æ
A parallel to AD .
The
slopes
of
BC
and
AD
are
not
equal, so BC is not
Exercise 42. A proof appears on page 839.
Æ
Æ
C
Æ
D
Æ
So, because AB ∞ CD and BC is not parallel to AD, ABCD is a trapezoid.
.
PROPERTIES OF TRAPEZOID
MIDSEGMENTS
!
T H. E
. .O. R
. .E. M
..
TrapezoidTHEOREM
midsegments
have
main
properties
are
B similar to the ones for C
6.17 Midsegment
Theorem
for that
Trapezoids
The midsegment
of two
a trapezoid
is the segment
triangles.The
They
are
represented
by
a
Midsegment
Theorem.
B
that
connects theof
midpoints
of itsislegs.
Theorem
midsegment C
midsegment
a trapezoid
parallel
6.17 isbase
similar
thelength
Midsegment
to each
andtoits
is oneTheorem
half the for
triangles.
You
will
justify
part
of
this
theorem in
sum
of
the
lengths
of
the
bases.
midsegment of a trapezoid…
The
Exercise 42. A proof appears on page 839.
Æ to
Æeach
Æ ofÆ
 …is parallel
the two bases
AND
1
MN ∞ AD , MN ∞ BC , MN = !!(AD + BC)
 …is half the sum of the two bases2
N
M
A
D
D
A
T H E O RTEHM
EOREM
THEOREM 6.17
Midsegment Theorem for Trapezoids
Finding Midsegment Lengths of Trapezoids
B
EXAMPLE 3
M
FE
RE
THEOREM 6.17
– The midsegment
of ais trapezoid
is
The midsegment
of a trapezoid
parallel
to
each
base
and
its
length
is
one
half
the
L
L I the bases AND is half the sum of the lengths
parallel Ato
Lsum
AYERofCthe
AKE
A baker
lengths
of is
themaking
bases.a cake like the
of the bases.one
at the right. The top layer has a diameter of
Æ
Æ Æ
Æ
1
8 inches andMN
the bottom
layer
has, MN
a diameter
∞ AD , MN
∞ BC
= !!(ADof+20
BC)
2
inches.
How
big
should the middle layer be?
TH
EOR
EM
C
A
C
E
F
D
N
G
D
H
Shown: MN is parallel to BC and AD. Also, MN = ½(BC + AD)
SOLUTION
EXAMPLE 3
Finding Midsegment Lengths of Trapezoids
Use the Midsegment Theorem for Trapezoids.
L
RE
E
FE
AL I 1 LAYER CAKE A
1 baker is making a cake like the
+ CH) = !!(8
+ 20) = 14 inches
DG = !!(EF
2 one at the right.2The top layer has a diameter of
8 inches and the bottom layer has a diameter of 20
inches. How big should the middle layer be?
D
C
6.5 Trapezoids and Kites
F
G
357
H
SOLUTION
Use the Midsegment Theorem for Trapezoids.
1
2
1
2
DG = !!(EF + CH) = !!(8 + 20) = 14 inches
< end of page >
6.5 Trapezoids and Kites
357
CHAPTER 6 NOTES
KITES
11
The last quadrilateral we will look at is easy to recognize and work with, since it
shares very few properties with the others.
 KITE GOAL 2 USING PROPERTIES OF KITES
A quadrilateral that has two pairs of consecutive sides
kite isa akite.
quadrilateral
that has
two pairs ofOF
consecutive
2 USING
PROPERTIES
KITES
GOAL
congruent isA called
congruent sides, but opposite sides are not congruent.
You are asked
to prove
Theorem that
6.18has
and
6.19
kite
A
is a quadrilateral
twoTheorem
pairs
consecutive
2
USING PROPERTIES
OF of
KITES
GOAL
NOTE! In ainkite,
opposite
sides
congruent!
Exercises
46 and
47.arebutNOT
congruent
sides,
opposite
sides are not congruent.
You are asked to prove Theorem 6.18 and Theorem 6.19
Ainkite
is a quadrilateral
Exercises
46 and 47. that has two pairs of consecutive
mplest of flying kites
congruent sides, but opposite sides are not congruent.
use the geometric
THEOREMS ABOUT KITES
The simplest
of flying kites OF
You
are asked to prove Theorem 6.18 and Theorem 6.19
PROPERTIES
KITES
hape.
often use the geometric
T H E O R46
EM
S A
BOUT KITES
in Exercises
and
47.
C
kite shape.
THEOREM 6.18
THEOREM 6.18 – If a quadrilateral is a kite, then itsB
diagonals are perpendicular.
The simplest of flying kites
THEOREM 6.18
If a quadrilateral
is a kite, then
often use the geometric
THEOREMS ABOUT KITES
If a quadrilateral
is a kite, then
its diagonals
are perpendicular.
kite shape.
its diagonals are perpendicular.
B
THEOREM THEOREM
6.19
6.19
AC
A
Æ Æ
Æ
Æ
ACAC
fi fiBD
BD
D
C CA
If a quadrilateral
a kite, then
If a quadrilateral
is a kite,isthen
exactly
one
pair of opposite
exactly one
pair
of
opposite
anglesangles
THEOREM 6.19
are congruent.
are congruent.
Æ
Æ
AC fi BD
B B
THEOREM 6.19 – If aIfquadrilateral
kite,then
then exactly
a quadrilateral is
is aakite,
exactly one pair of opposite angles
one pair of opposite angles
are
congruent.
(across
from
are congruent.
the shorter diagonal) T H E O R E M S A B O U T K I T E S
A
DD
C
A
B ™A £ ™C, ™B ‹ ™D
™A £ ™C, ™B ‹ ™D
THEOREMS ABOUT KITES
EXAMPLE 4
D
D
B
THEOREM 6.18
If a quadrilateral is a kite, then
its diagonals are perpendicular.
C
A
™A £ ™C, ™B ‹ ™D
Using the Diagonals of a Kite
THEOREMS ABOUT KITES
xy
E X A M P L E 4 Using the Diagonals of a Kite
xy
Using
Algebra
Using
Algebra
< end of
D
X
WXYZ is a kite so the diagonals are perpendicular. You can
E
X Athe
MP
L E 4 Using
the Diagonals
a Kite
use
Pythagorean
Theorem
to find theof
side
lengths.
xWXYZ
y
is a kite so the diagonals are perpendicular. You can
20
12 X
U
2
2
X
!
"0"so"
2
"
+
"12"diagonals
"to≈find
23.32
WXYZWX
is a=kite
the
are perpendicular.
12
Pythagorean
Theorem
the
side lengths.You can W
12
12
use the Pythagorean
Theorem
to find the side lengths.
page >
2
2
12 U
20
!
1
"
2
"
"
+
"
"
1
2
"
"
≈
16.97
XY
=
2
W
20
"WX
""
0
"
+
"12"22""0"2≈
23.32
WX = !2
U 12
W
Z 12
+"
"
1"
22" ≈ 23.32
= !"
Because WXYZ is a kite, WZ = WX ≈ 23.32 and ZY = XY ≈ 16.97. 12
12
2
"XY
""
2
"
+
12
""12""2"2≈"
16.97
XY = !1
!
+"
1"
22" ≈ 16.97
="
ZZ
Because
a Angles
kite,
≈ 23.32
= XY
16.97.
Because WXYZ
aP Lkite,
= WZ
WX
23.32
and and
ZY ZY
= XY
≈≈16.97.
E X Ais
MWXYZ
E 5isWZ
of=≈
a WX
Kite
Using
use the
Algebra
Find m™G and m™J in the diagram at the right.
EXAMPLE 5
EXAMPLE 5
Y
Y
J
Angles of a Kite
Angles of a Kite
Find
m™G and m™J in the diagram at the right.
SOLUTION
Find m™GGHJK
and m™J
in the diagram at the right.
is a kite, so ™G £ ™J and m™G = m™J.
SOLUTION
2(m™G) + 132° + 60° = 360°
H 132!
J
60!
K
J
G
H 132!
60! K
Sum of measures of int. √ of a quad. is 360°.
GHJK is a kite, so ™G
£ ™J=and
m™G Simplify.
= m™J.
SOLUTION
2(m™G)
168°
H 132!
60!
G
CHAPTER 6 NOTES
GHJK is a kite,
so ™G
™J+m™G
and
=360°
84° = m™J.
Divide
each side by
2. √ of a quad. is 360°.
Sum
of measures
of int.
2(m™G)
+£
132°
60° m™G
=
G
So, m™J = m™G
= 84°.= 168°
2(m™G)
Simplify.
Sum of measures of int. √ of a quad. is 360°.
m™G = 84°
Divide each side by 2.
Simplify.
Chapter 6 Quadrilaterals2(m™G) = 168°
! + 132° + 60° = 360°
2(m™G)
358
K
Y
Quadrilaterals
12
mbuses,
Rectangles, SUMMARY OF QUADRILATERALS
RIZING
ROPERTIES
QUADRILATERALS
ThePdiagram
shown hereOF
shows
how the six quadrilaterals are organized.
Squares
quadrilateral
e studied the
PROPERTIES OF SPECIAL PARALLELOGRAMS
uadrilaterals
each
has
kitetypes parallelogram
on youshape
will study
three special
of parallelograms: trapezoid
rhombuses,
shapes
linked
and squares.
quares have the
rhombus
rectangle
isosceles
trapezoid
, rectangles,
square
drilaterals.
us is a
A rectangle is a
A square is a
am with
parallelogram with
parallelogram with
ifying
uent
sides.Quadrilaterals
four
right
angles.
four congruent
The figure below (called a Venn Diagram)
shows sides
how the special parallelograms
and
four
right angles.
are categorized.
at least one pair of opposite sides congruent. What kinds
iagram
at the right
is condition?
elationships among
ams, rhombuses,
and squares. Each shape
perties of every group
ities.
ngs to. For instance, a
rectangle, a rhombus,
lelogram, so it has all
RECTANGLE
HOMBUS
erties
of each of those
B
C
B
C
parallelograms
rhombuses
rectangles
squares
SQUARE
B
C
ISOSCELES
TRAPEZOID
C
B
Describing
a Special
Parallelogram
Notice how
a square
is a type of rhombus or rectangle. Also notice how
rectangles
and
are special types of parallelograms
D rhombuses,
A
D squares
A
D
A
D
ether the statement is always, sometimes, or never true.
E 1
mbus
is a rectangle.
l sides
are
Opposite sides
ongruent.
are congruent.
llelogram
is a rectangle.
All sides are
parallelograms
congruent.
Legs are
congruent.
rhombuses
rectangles
tement is sometimes true.
squares
< end
page >
Venn diagram,
theofregions
ecting Midpoints of Sides
mbuses and rectangles
p. If the rhombus is a
it is a rectangle.
oints
of the sides of any quadrilateral, what special
tement
Why?is sometimes true. Some parallelograms are rectangles. In the
iagram, you can see that some of the shapes in the parallelogram box
he region for rectangles, but many aren’t.
CHAPTER 6 NOTES
6.7
6.7
Areas of Triangles and
Quadrilaterals
AREAS OF QUADRILATERALS
GOAL 1
13
USING AREA FORMULAS
Areas of Triangles and
Quadrilaterals
What you should learn
geometry
wecan
are
interested
theseveral
numeric
measurements
FindIn
the areas
of
You
useoften
the postulates
below toinprove
area theorems.
squares, rectangles,
such as perimeter, area, surface area and volume.
parallelograms, and triangles.
GOAL 1
of shapes
A R E A P O S T U L AT E S
GOAL 2 Find the areas of
andIMPORTANT
AREA
CONCEPTS
trapezoids, kites,
1 22
GOAL
USING
AREA
FORMULAS
POSTULATE
Area of
a Square
Postulate
you
should
rhombuses, as
applied
in learn
Area
Congruence
Postulate
two
polygons
are of
congruent,
The area of a square–isIfthe
square
of the length
its side, or Athen
= s 2.their areas
Example
6.
GOAL 1 Find the areas of
You can use the
below to prove
several area theorems.
are
the
same.
POSTULATE
23 postulates
Area Congruence
Postulate
squares, rectangles,
you should learn it
If two polygons are congruent, then they have the same area.
parallelograms, and triangles.
A R E A P O S T U L AT E S
! To find areas of real-life
POSTULATE 24 Area Addition Postulate
GOAL 2 Area
theroof
areas
Addition
Postulate – If a figure is made from several smaller figures, then
surfaces, such Find
as the
of of
trapezoids, kites, and
area of a region
is the
thePostulate
areas of its nonoverlapping parts.
POSTULATE
of asum
the covered
bridge
the
areainof the The
whole
thing22is Area
equal
to Square
theofsum
of the individual pieces.
rhombuses,
as applied
in Exs. 48 and 49. AL LI
The area of a square is the square of the length of its side, or A = s 2.
Example 6.
What
FE
RE
Why
POSTULATE 23
Area Congruence Postulate
QUADRILATERAL
AREA
Why you should learnGOAL
it A2
RIfEtwo
AA
TREAS
H E O R OF
EM
S
polygons
are
then ,they
have,the
sameRarea.
Tcongruent,
RAPEZOIDS
KFORMULAS
ITES
AND
HOMBUSES
RE
FE
! To find areas of real-life
Area
Postulate
Aof
R E ATHEOREM
TPOSTULATE
H E O R6.20
E M24
S
Area
ofAddition
a Rectangle
surfaces, such as the roof
=
b
•
h
The area of a region isAthe
sum
of the areas of its nonoverlapping parts.h
the covered bridge Rectangle
of a rectangle is the
T H EThe
O Rarea
EMS
in Exs. 48 and 49. AL LI
=L•W
product of its base and A
height.
bb
1
THEOREM
6.23
A=
bh Area of a Trapezoid
AREA THEOREMS
Area is
ofone
a Parallelogram
The THEOREM
area of a 6.21
trapezoid
half the product
of the
height
and
the
sum
of
the
bases.
The
area
of
a
parallelogram
the
product of
THEOREM 6.20 Area of a is
Rectangle
h
h
Squarea base
A = height.
s2
its corresponding
1 and
The
area of a rectangle is the
h
b2
A = !!h(b1 + b2)
b
A2 = bh of its base and height.
product
b
THEOREM
Area
a Triangle
THEOREM
6.24
Area
of of
a Kite
A =6.22
bh
area
a6.21
triangle
isof
one
half
the product
THEOREM
Areahalf
athe
Parallelogram
The The
area
of aofkite
is one
product
of lengths
aThe
base
and
its
corresponding
of the
of
diagonals. is height.
area
ofits
a parallelogram
the product of
d1
h
h
Parallelograma base1 and its corresponding
A = b •height.
h
A1 = !! bh
2 A = bh
b
d2 b
22
A = !!d1d
THEOREM
6.22 Area
a Triangle
THEOREM
6.25 Area
of a of
Rhombus
AAREAS
OF
TTRAPEZOIDS
, ,K
ITES
, ,AND
22justify
GOAL
The
of
triangle
isRAPEZOIDS
one
half
the product
REAS
OF
K
ITES
ANDRRHOMBUSES
HOMBUSES
Youarea
can
theaarea
formulas
for
triangles
and
parallelograms
as
h follows.
GOAL
d1
The
ofarea
a rhombus
is equal
to
one
half
the
of
a
base
and
its
corresponding
height.
b
A
R E A Tof
HE
O Rlengths
E M S ofAthe
Rhombus
=
½(d
)(d
)
1
2
product
the
diagonals.
AREA THEOREMS
b
1
!! bh
AM
=S
THEO1
RE
2
TA
H=
EO
!h!R
dEdM S
2 1 2
d2
THEOREM 6.23 Area of a Trapezoid
THEOREM 6.23 Area of a Trapezoid
h
b1
b1
b
TheYou
area
a trapezoid
one half
product
canofjustify
the area is
formulas
for the
triangles
and parallelograms
as follows.
The area of a trapezoid is one half the product
The height
area of and
a parallelogram
isthe
the
Trapezoid
Asum
=in½(b
b2)h58 The area of a triangle is half the
of the
the
sum
ofExercises
b 6.23
1 +bases.
You will
justify
Theorem
of the
height
and the
of the bases.
UDENT HELP
h
h
area of1a rectangle with the same
area of a parallelogram with the
b
and 59. You
may
find
to remember the
!!1h (b
A =and
+ itb easier
)
base
same base and height. 2b2
A =2!h!hheight.
(b1 + b2 )
k Back
h
1
2
theorem this way.
2
h
1
ember that the length
2 b1 " b2
THEOREM 6.24 Area of a Kite
e midsegment of a
THEOREM
6.24
Area
of
a
Kite
372
Chapter 6 Quadrilaterals Length of
ezoid
is the average
Height
Areaarea
b
= ofMidsegment
d1
The
a kite is one •half
the product
areaarea
of a kiteparallelogram
is one
the
e lengths of the
KiteofThe
A =half
½(d1)(d2)
is theproduct The area of a triangle is half the d1
theThe
lengthsofofa its
diagonals.
of the
lengths
of its diagonals.
es. (p. 357)
area
of a rectangle
with the same
area of a parallelogram with the
1
d2
A base
= !!1d
andd height.
same base and height.
d2
A =2!!d1 1d2 2
!
E X A M P L E 42
372
"
Finding the Area of a Trapezoid
THEOREM 6.25 Area of a Rhombus
Chapter 6 Quadrilaterals
THEOREM 6.25 Area of a Rhombus
areaof
oftrapezoid
a rhombus
is equal to one half they
Find The
the area
WXYZ.
The area of a rhombus is equal to one half the
< end of pageproduct
> of the lengths of the diagonals.
Y(2, 5)
product of the lengths of the diagonals.
1
SOLUTION
A = !!1d1d2
A =2!!d1d2
2
The height of WXYZ is h = 5 º 1 = 4.
Find the lengths of the bases.
1
You
justify
Theorem
in Exercises 58
X(1, 1)
b1 will
=
YZ
=5º
2 = 3 6.23
You
will
justify
Theorem
6.23 in Exercises 58
1
and
59.
You
may
find
it
easier
to
remember
the
ook Back
and
59.XW
You=may
Look
Back
b2 =
8 ºfind
1 =it7easier to remember the
theorem
this way.
emember
that the length
d1
d1
Z(5, 5)
d2
d2
CHAPTER 6 NOTES
W (8, 1)
STUDENT HELP
STUDENT HELP
x
1
h