Download Objectives

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
Definition of a Parallelogram (
)
If the opposite sides of a quadrilateral are parallel,
then it is a parallelogram.
ABCD
A
D
B
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
Theorem
Opposite sides of a parallelogram are congruent.
A
D
B
C
Theorem
Opposite angles of a parallelogram are congruent.
A
D
B
C
Theorem
The diagonals of a parallelogram bisect each other.
A
D
B
C
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
White Board Practice
Given
AB || DC
BC || AD
ABCD
White Board Practice
Given
ABCD
Name all pairs of congruent angles
White Board Practice
Given
ABCD
 BAD   DCB
 ABC   CDA
 BEA   DEC
 BEC   DEA
 CBD   ADB
 ABD   CDB
 BCA   DAC
 BAC   DCA
White Board Practice
Given
ABCD
Name all pairs of congruent segments
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
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.
x = 80
y = 45
a=6
b=9
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
White Board Groups
• Quadrilateral RSTU is a parallelogram.
Find the values of x, y, a, and b.
x = 100
y = 45
a = 12
b=9
White Board Groups
• Given this parallelogram with the diagonals
drawn.
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
Use the Definition of a
Parallelogram
• Show that both pairs of opposite sides of a
quadrilateral are parallel
• Then the quadrilateral is a parallelogram
A
D
B
C
Theorem
• Show that both pairs of opposite sides are
congruent.
• If both pairs of opposite sides of a quadrilateral are
congruent, then it is a parallelogram.
A
D
B
C
Theorem
• Show that one pair of opposite sides are both
congruent and parallel.
• If one pair of opposite sides of a quadrilateral are
both congruent and parallel, then it is a
parallelogram.
A
D
B
C
Theorem
• Show that both pairs of opposite angles are
congruent.
• If both pairs of opposite angles of a quadrilateral
are congruent, then it is a parallelogram.
A
D
B
C
Theorem
• Show that the diagonals bisect each other.
• If the diagonals of a quadrilateral bisect each other,
then it is a parallelogram.
A
B
X
D
C
Five ways to prove a
Quadrilateral is a Parallelogram
• Show that both pairs of opposite sides parallel
• Show that both pairs of opposite sides congruent
• Show that one pair of opposite sides are both
congruent and parallel
• Show that both pairs of opposite angles
congruent
• 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
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.
m
n
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 of a
triangle and is parallel to a another side passes
through the midpoint of the third side.
A
X
B
Y
C
Theorem
A segment that joins the midpoints of two sides of a
triangle is parallel to the third side and its length is
half the length of the third side.
A
X
B
Y
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
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
5.4 Special Parallelograms
Objectives
• Apply the
definitions and
identify the special
properties of a
rectangle, rhombus
and 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
A
C
D
Theorem
The diagonals of a rectangle are congruent.
WY  XZ
W
Z
P
X
Y
Theorem
The diagonals of a rhombus are perpendicular.
K
X
J
L
M
Theorem
Each diagonal of a rhombus bisects the
opposite angles.
K
X
J
L
M
Theorem
The midpoint of the hypotenuse of a right
triangle is equidistant from the three
vertices.
A
X
B
C
Theorem
If an angle of a parallelogram is a right angle,
then the parallelogram is a rectangle.
R
S
V
T
Theorem
If two consecutive sides of a parallelogram
are congruent, then the parallelogram is a
rhombus.
B
C
A
D
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
eah 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
A
B
C
D
Anatomy Of a Trapezoid
• The bases are the parallel sides
Base
R
S
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
K
L
Theorem 5-18
The base angles of an isosceles trapezoid
are congruent.
F
E
G
H
The Median of a Trapezoid
A segment that joins the midpoints of the
legs.
B
X
A
C
Y
D
Theorem
The median of a trapezoid is parallel to
the bases and its length is the average
of the bases.
B
X
A
A
C
Y
D
White Board Practice
• Complete
1. AD = 25, BC = 13, XY = ______
B
X
A
C
Y
D
White Board Practice
• Complete
1. AD = 25, BC = 13, XY = 19
B
X
A
C
Y
D
White Board Practice
• Complete
2. AX = 11, YD = 8, AB = _____, DC = ____
B
X
A
C
Y
D
White Board Practice
• Complete
2. AX = 11, YD = 8, AB = 22, DC = 16
B
X
A
C
Y
D
White Board Practice
• Complete
3. AD = 29, XY = 24, BC =______
B
X
A
C
Y
D
White Board Practice
• Complete
3. AD = 29, XY = 24, BC =19
B
X
A
C
Y
D
White Board Practice
• Complete
4. AD = 7y + 6, XY = 5y -3, BC = y – 5, y =____
B
X
A
C
Y
D
White Board Practice
• Complete
4. AD = 7y + 6, XY = 5y -3, BC = y – 5, y = 3.5
B
X
A
C
Y
D
Homework Set 5.5
• WS PM 28
• 5-5 #1-27 odd
• Quiz next class
day