Download Quadrilaterals

Warm – up 11/07
• MAKE SURE YOU ARE SITTING IN
YOUR NEW SEAT!!
• Start a new sheet for Ch. 5 warm-ups
• Answer the following
– Come up with your own definition of a
quadrilateral (Remember a quad is a polygon)
– What is the internal angle sum of a
quadrilateral?
Chapter 5
Quadrilaterals
• Apply the definition of
a parallelogram
• Prove that certain
quadrilaterals are
parallelograms
• Apply the theorems
and definitions about
the special
quadrilaterals
5-1 Properties of Parallelograms
Objectives
• Apply the definition of a
parallelogram
• List the other properties
of a parallelogram
through new theorems
Quadrilaterals
• Any 4 sided figure
He’s Back….
• Parallelograms are special types of
quadrilaterals with unique properties
• If you know you have a parallelogram, then you can prove
that these unique properties exist…
• With each property we learn, say the following to yourself..
– “If a quadrilateral is a parallelogram, then _____________.”
Definition of a Parallelogram (
)
What
do you thinkisthe
definition is based
thepairs
If a quadrilateral
a parallelogram,
then on
both
diagram?
of opposite sides are parallel.
ABCD
A
B
Partners: What do we
know about the angles
of a parallelogram b/c
it has parallel sides?
D
C
Naming a Parallelogram
Use the symbol for parallelogram
and name using the 4
vertices in order either clockwise or counter clockwise.
ABCD
A
D
B
C
A
D
B
C
• The fact that we know opposite sides are
parallel, we can deduce addition properties
through theorems…
Theorem
Opposite sides of a parallelogram are congruent.
A
D
B
C
What would be our
plan for solving
this theorem?
Theorem
Opposite
angles
of a at
parallelogram
What
did we
discuss
the beginningareofcongruent.
the lesson
about s-s int. angles?
A
D
B
C
Theorem
The diagonals of a parallelogram bisect each other.
A
D
B
C
What is another
name for AC and
BD?
Parallelograms: What we now
know…
If a quad is a parallelogram, then…
• From the definition..
1. Both pairs of opposite sides are parallel
• From theorems…
1. Both pairs of opposite sides are congruent
2. Both pairs of opposite angles are congruent
3. The diagonals of a parallelogram bisect each
other
Remote Time
• True or False
True or False
• Every parallelogram is a quadrilateral
True or False
• Every quadrilateral is a parallelogram
True or False
• All angles of a parallelogram are congruent
True or False
• All sides of a parallelogram are congruent
True or False
• In
RSTU, RS | |TU.
 Hint draw a picture
True or False
• In ABCD, if m  A = 50, then m  C =
130.
 Hint draw a picture
True or False
• In
XWYZ, XY WZ
 Hint draw a picture
True or False
• In
ABCD, AC and BD bisect each other
 Hint draw a picture
White Board Practice
Given
ABCD
Name all pairs of parallel sides
AB || DC
BC || AD
White Board Practice
Given
Given
ABCD
ABCD
Name
all pairs
of congruent
angles
 BAD
 DCB
 CBD
  ADB
•Draw
ABCthe
 parallelogram
CDA
 with
ABDthe
 diagonals
CDB
intersecting at E
 BEA   DEC
 BCA   DAC
 BEC   DEA
 BAC   DCA
White Board Practice
Given
ABCD
Name all pairs of congruent segments
• Draw the parallelogram with the diagonals
intersecting at E
White Board Practice
Given
AB  CD
BC  DA
BE  ED
AE  EC
ABCD
White Board Groups
• Quadrilateral RSTU is a parallelogram.
Find the values of x, y, a, and b.
6
R
x = 80
y = 100
a=6
b=9
xº S
yº
9
b
80º
U
a
T
White Board Groups
• Quadrilateral RSTU is a parallelogram.
Find the values of x, y, a, and b.
R
S
xº
yº
9
12
a
b
U
45º
35º
T
x = 100
y = 45
a = 12
b=9
White Board Groups
• Given this parallelogram with the diagonals
drawn.
x=5
y=6
5-2:Ways to Prove that
Quadrilaterals are Parallelograms
Objectives
• Learn about ways
to prove a
quadrilateral is a
parallelogram
What we already know…
• If a quad is a parallelogram, then…
– 4 properties
• What we are going to learn..
– What if we don’t know if a quad is a
parallelogram, how can we prove that it is one?
In the courtroom…
“What
we have ladies and
gentlemen is a quadrilateral, and I
believe this particular
quadrilateral happens to be a
parallelogram, and I have the
evidence to prove it!”
Use the Definition of a
Parallelogram
• If both pairs of opposite sides of a quadrilateral are
parallel then…
• the quadrilateral is a parallelogram.
A
B
The definition is
a biconditional,
so it can be used
either way.
D
C
Theorem
• If both pairs of opposite sides of a quadrilateral are
congruent, then it is a parallelogram.
• Show that both pairs of opposite sides are
congruent.
A
D
B
C
Theorem
• If one pair of opposite sides of a quadrilateral are
both congruent and parallel, then it is a
parallelogram.
• Show that one pair of opposite sides are both
congruent and parallel.
A
D
B
C
Theorem
• If both pairs of opposite angles of a quadrilateral
are congruent, then it is a parallelogram.
• Show that both pairs of opposite angles are
congruent.
A
D
B
C
Theorem
• If the diagonals of a quadrilateral bisect each other,
then it is a parallelogram.
• Show that the diagonals bisect each other.
A
B
X
D
C
Five ways to prove a
Quadrilateral is a Parallelogram
1. Show that both pairs of opposite sides parallel
2. Show that both pairs of opposite sides
congruent
3. Show that one pair of opposite sides are both
congruent and parallel
4. Show that both pairs of opposite angles
congruent
5. Show that diagonals that bisect each other
The diagonals of a quadrilateral
_____________ bisect each other
A.
B.
C.
D.
Sometimes
Always
Never
I don’t know
If the measure of two angles of a
quadrilateral are equal, then the
quadrilateral is ____________ a
parallelogram
A)
B)
C)
D)
Sometimes
Always
Never
I don’t know
If one pair of opposite sides of a
quadrilateral is congruent and
parallel, then the quadrilateral is
___________ a parallelogram
A.
B.
C.
D.
Sometimes
Always
Never
I don’t know
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is __________ a
parallelogram
A.)
B.)
C.)
D.)
Sometimes
Always
Never
I don’t know
To prove a quadrilateral is a
parallelogram, it is ________
enough to show that one pair of
opposite sides is parallel.
A.)
B.)
C.)
D.)
Sometimes
Always
Never
I don’t know
Whiteboards
• Open book to page 173
–Answer the following…
• #2
• #3
• #6
• #9
5-3 Theorems Involving Parallel
Lines
Objectives
• Apply the theorems about parallel lines and
triangles
Theorem
If two lines are parallel, then all points on one line
are equidistant from the other line.
Demo: 6 volunteers
How do we measure the
distance from a point to a
line?
m
What does equidistant mean?
n
Demo: Popsicle Sticks and lined paper
Theorem
If three parallel lines cut off congruent segments on
one transversal, then they do so on any transversal.
A
D
B
E
C
F
Theorem
A line that contains the midpoint of one side 1 of a
triangle and is parallel to a another side 2 passes
through the midpoint of the third side 3 .
A
3
X
Y
1
B
2
C
Theorem
A segment that joins the midpoints of two sides 1 2
of a triangle is parallel to the third side and its
length is half the length of the third side. 3
A
If BC is 12 then XY =?
2
X
Y
1
3
B
C
White Board Practice
• Given: R, S, and T are midpoint of the sides of  ABC
AB BC AC ST RT RS
12
14
B
18
R
15
10
22
S
10
9
7.8
A
T
C
White Board Practice
• Given: R, S, and T are midpoint of the sides of  ABC
AB BC AC ST RT RS
12
14
18
6
7
B
9
R
20
15
22
10
10
18
15.6 5
S
7.5 11
9
7.8
A
T
C
B
• ST is parallel to
what side?
R
S
• BC is parallel to
what side?
A
T
C
White Board Practice
• Given that AR | | BS | | CT;
RS  ST
R
S
A
T
B
C
White Board Practice
• Given that AR | | BS | | CT;
RS  ST
If RS = 12, then ST = ____
R
A
S
T
B
C
White Board Practice
• Given that AR | | BS | | CT;
RS  ST
If RS = 12, then ST = 12
R
A
S
T
B
C
White Board Practice
• Given that AR | | BS | | CT;
RS  ST
If AB = 8, then BC = ___
R
A
S
T
B
C
White Board Practice
• Given that AR | | BS | | CT;
RS  ST
If AB = 8, then BC = 8
R
A
S
T
B
C
White Board Practice
• Given that AR | | BS | | CT;
RS  ST
If AC = 20, then AB = ___
R
A
S
T
B
C
White Board Practice
• Given that AR | | BS | | CT;
RS  ST
If AC = 20, then AB = 10
R
A
S
T
B
C
White Board Practice
• Given that AR | | BS | | CT;
RS  ST
If AC = 10x, then BC =____
R
A
S
T
B
C
White Board Practice
• Given that AR | | BS | | CT;
RS  ST
If AC = 10x, then BC = 5x
R
A
S
T
B
C
QUIZ REVIEW
1. Know the properties of a parallelogram
2. Know the 5 ways to prove a quad is a
parallelogram
1. “Show that…..
3. Solve problems using the theorems from
section 5-3
5.4 Special Parallelograms
Objectives
• Apply the
definitions and
identify the special
properties of a
rectangle, rhombus
and square.
Parallelograms: What we now
know…
• From the definition..
1. Both pairs of opposite sides are parallel
• From theorems…
1. Both pairs of opposite sides are congruent
2. Both pairs of opposite angles are congruent
3. The diagonals of a parallelogram bisect each
other
QUADRILATERALS
parallelogram
Rhombus
Rectangle
Square
Rectangle
By definition, it is a quadrilateral with four
right angles.
R
S
V
T
Rhombus
By definition, it is a quadrilateral with four
congruent sides.
B
C
A
D
Square
By definition, it is a quadrilateral with four
right angles and four congruent sides.
B
C
What do you notice
about the definition
compared to the
previous two?
The square is the
most specific type
of quadrilateral.
A
D
Theorem
The diagonals of a rectangle are congruent.
WY  XZ What can we conclude about the
W
smaller segments that make up the
diagonals?
Z
P
X
Y
Finding the special properties of a
Rhombus
Apply the properties of a
parallelogram to find 2 special
properties that apply to the
Rhombus. Hint: both properties
involve angles.
Theorem
The diagonals of a rhombus are perpendicular.
K
X
J
L
M
What does the
definition of
perpendicular lines
tell us?
Theorem
Each diagonal of a rhombus bisects the
opposite angles.
K
X
J
L
M
Theorem
If an angle of a parallelogram is a right angle,
then the parallelogram is a rectangle.
R
V
WHY?????????
S
T
Theorem
If two consecutive sides of a parallelogram
are congruent, then the parallelogram is a
rhombus.
B
C
WHY?
A
D
X
M
Z
Y
Theorem
The midpoint of the hypotenuse of a right
triangle is equidistant from the three
vertices.
X
M
Z
Y
White Board Practice
• Quadrilateral ABCD is a rhombus
Find the measure of each angle
A
1.  ACD
2.  DEC
E
3.  EDC
4.  ABC
B
62º
D
C
White Board Practice
• Quadrilateral ABCD is a rhombus
Find the measure of each angle
A
1.  ACD = 62
2.  DEC = 90
E
3.  EDC = 28
4.  ABC = 56
B
62º
D
C
White Board Practice
• Quadrilateral MNOP is a rectangle
Find the measure of each angle
1. m  PON =
M
29º
2. m  PMO =
L
3. PL =
4. MO =
P
N
12
O
White Board Practice
• Quadrilateral MNOP is a rectangle
Find the measure of each angle
1. m  PON = 90 M
29º
2. m  PMO = 61
L
3. PL = 12
4. MO = 24
P
N
12
O
White Board Practice
•  ABC is a right ; M is the midpoint of
AB
1. If AM = 7, then MB = ____, AB = ____,
and CM = _____ .
A
M
C
B
White Board Practice
•  ABC is a right ; M is the midpoint of
AB
1. If AM = 7, then MB = 7, AB = 14,
and CM = 7 .
A
M
C
B
White Board Practice
•  ABC is a right ; M is the midpoint of
AB
1. If AB = x, then AM = ____, MB = _____,
and MC = _____ .
A
M
C
B
White Board Practice
•  ABC is a right ; M is the midpoint of
AB
1. If AB = x, then AM = ½ x, MB = ½ x,
and MC = ½ x .
A
M
C
B
Remote Time
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• A square is ____________ a rhombus
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• The diagonals of a parallelogram
____________ bisect the angles of the
parallelogram.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• A quadrilateral with one pairs of sides
congruent and one pair parallel is
_________________ a parallelogram.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• The diagonals of a rhombus are
___________ congruent.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• A rectangle ______________ has
consecutive sides congruent.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• A rectangle ____________ has
perpendicular diagonals.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• The diagonals of a rhombus ___________
bisect each other.
A.
B.
C.
D.
Always
Sometimes
Never
I don’t know
• The diagonals of a parallelogram are
____________ perpendicular bisectors of
each other.
5.5 Trapezoids
Objectives
• Apply the definitions and learn the
properties of a trapezoid and an isosceles
trapezoid.
Trapezoid
A quadrilateral with exactly one pair of
parallel sides.
Trap. ABCD
B
C
How does this definition
differ from that of a
parallelogram?
A
D
Anatomy Of a Trapezoid
• The bases are the parallel sides
Base
R
S
1 pair of base angles
2nd pair of base angles
V
T
Base
Anatomy Of a Trapezoid
• The legs are the non-parallel sides
R
S
Leg
Leg
V
T
Isosceles Trapezoid
A trapezoid with congruent legs.
J
M
What do you think would
the
definition
happen
if Iisfolded
based this
on the
figure
diagram?
in
half?
K
L
Theorem
The base angles of an isosceles trapezoid
are congruent.
F
Supplementary
E
G
Supplementary
What is something I can conclude
about 2 of the angles (other than
congruency) based on the markings
of the diagram?
H
The Median of a Trapezoid
A segment that joins the midpoints of the
legs.
B
C
X
Y
Note: this applies to any trapezoid
A
D
Theorem
The median of a trapezoid is parallel to
the bases and its length is the average
of the bases. Note: this applies to any trapezoid
B
X
C
Y
How do we find an average of the bases ?
A
A
D
White Board Practice
• Complete
19
1. AD = 25, BC = 13, XY = ______
B
X
A
C
Y
D
White Board Practice
• Complete
22 DC = ____
16
2. AX = 11, YD = 8, AB = _____,
B
X
A
C
Y
D
White Board Practice
• Complete
19
3. AD = 29, XY = 24, BC =______
B
X
A
C
Y
D
White Board Practice
• Complete
3.5
4. AD = 7y + 6, XY = 5y -3, BC = y – 5, y =____
B
X
A
C
Y
D
Test Review
• Know what properties each type of quadrilateral
has
– i.e. – all sides are congruent: square and rhombus
– #1 – 10 on pg 187
• Solving algebraic problems with a
parallelogram
– i.e. – finding the length of a side, angle, or
diagonal
• Proving a quad is a parallelogram
– Do you have enough information to say that the
quad is a parallelogram
– Knowing the 5 ways to prove
• Solving problems using TH. 5-9
– I.e. #10 – 15 on pg. 180
• Solving problems using Th. 5-11 and 5-15
– I.e. #1 – 4 on pg. 180
– I.ed # 14 , 17 on page 187
• Trapezoid Theorems
– I.e. # 1 – 9 on pg. 192
– #11 pg 193