Chapter 5 Quadrilaterals Apply the definition of a parallelogram Prove that certain quadrilaterals...
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Transcript of Chapter 5 Quadrilaterals Apply the definition of a parallelogram Prove that certain quadrilaterals...
Chapter 5Quadrilaterals
• 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.
ABCDA B
CD
Naming a Parallelogram
Use the symbol for parallelogram and name using the 4 vertices in order either clockwise or counter clockwise.
ABCDA B
CD
Opposite sides of a parallelogram are congruent.
Theorem
A B
CD
Opposite angles of a parallelogram are congruent.
Theorem
A B
CD
The diagonals of a parallelogram bisect each other.
Theorem
A B
CD
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 ABCD
AB || DC
BC || AD
White Board Practice
Given ABCD
Name all pairs of congruent angles
White Board Practice
Given ABCD
BAD DCB CBD ADB
ABC CDA ABD CDB
BEA DEC BCA DAC
BEC DEA BAC DCA
White Board Practice
Given ABCD
Name all pairs of congruent segments
White Board Practice
Given ABCD
AB CD
BC DA
BE ED
AE EC
• Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b.
White Board Groups
a
R
U T
S
9 b
6
yº
80º
xº
• 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.
White Board Groups
a
R
U T
S
9
b
12
yº
35º
xº
45º
• 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
18
2x + 84y - 2
22
• Given this parallelogram with the diagonals drawn.
x = 5y = 6
White Board Groups
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 B
CD
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 B
CD
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 B
CD
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 B
CD
Theorem• Show that the diagonals bisect each other.• If the diagonals of a quadrilateral bisect each other, then
it is a parallelogram.
A B
CD
X
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. Sometimes
B. Always
C. Never
D. I don’t know
If the measure of two angles of a quadrilateral are equal, then the quadrilateral is ____________ a
parallelogram
A) Sometimes
B) Always
C) Never
D) I don’t know
If one pair of opposite sides of a quadrilateral is congruent and
parallel, then the quadrilateral is ___________ a parallelogram
A. Sometimes
B. Always
C. Never
D. I don’t know
If both pairs of opposite sides of a quadrilateral are congruent, then the
quadrilateral is __________ a parallelogram
A.) Sometimes
B.) Always
C.) Never
D.) 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.) Sometimes
B.) Always
C.) Never
D.) 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
B
C
D
E
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
B C
X Y
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
B C
X Y
White Board Practice• Given: R, S, and T are midpoint of the sides of ABC
CA
B
T
R S
AB BC AC ST RT RS
12 14 18
15 22 10
10 9 7.8
White Board Practice• Given: R, S, and T are midpoint of the sides of ABC
CA
B
T
R S
AB BC AC ST RT RS
12 14 18 6 7 9
20 15 22 10 7.5 11
10 18 15.6 5 9 7.8
White Board Practice
• Given that AR | | BS | | CT;
RS ST
A
B
C
T
SR
White Board Practice
• Given that AR | | BS | | CT;
RS ST
If RS = 12, then ST = ____
A
B
C
TS
R
White Board Practice
• Given that AR | | BS | | CT;
RS ST
If RS = 12, then ST = 12
A
B
C
TS
R
White Board Practice
• Given that AR | | BS | | CT;
RS ST
If AB = 8, then BC = ___
A
B
C
TS
R
White Board Practice
• Given that AR | | BS | | CT;
RS ST
If AB = 8, then BC = 8
A
B
C
TS
R
White Board Practice
• Given that AR | | BS | | CT;
RS ST
If AC = 20, then AB = ___
A
B
C
TS
R
White Board Practice
• Given that AR | | BS | | CT;
RS ST
If AC = 20, then AB = 10
A
B
C
TS
R
White Board Practice
• Given that AR | | BS | | CT;
RS ST
If AC = 10x, then BC =____
A
B
C
TS
R
White Board Practice
• Given that AR | | BS | | CT;
RS ST
If AC = 10x, then BC = 5x
A
B
C
TS
R
5.4 Special Parallelograms
Objectives
• Apply the definitions and identify the special properties of a rectangle, rhombus and square.
RectangleBy definition, it is a quadrilateral with four
right angles.
R
S T
V
RhombusBy definition, it is a quadrilateral with four
congruent sides.
A
B C
D
SquareBy definition, it is a quadrilateral with four
right angles and four congruent sides.
A
B C
D
TheoremThe diagonals of a rectangle are congruent.WY XZ
W
X Y
Z
P
TheoremThe diagonals of a rhombus are
perpendicular.
J
K
L
M
X
TheoremEach diagonal of a rhombus bisects the
opposite angles.
J
K
L
M
X
TheoremThe midpoint of the hypotenuse of a right
triangle is equidistant from the three vertices.
A
B C
X
TheoremIf an angle of a parallelogram is a right angle,
then the parallelogram is a rectangle.
R
S T
V
TheoremIf two consecutive sides of a parallelogram are
congruent, then the parallelogram is a rhombus.
A
B C
D
White Board Practice
• Quadrilateral ABCD is a rhombus
Find the measure of each angle
1. ACD
2. DEC
3. EDC
4. ABC
D
A B
C
E
62º
White Board Practice
• Quadrilateral ABCD is a rhombus
Find the measure of each angle
1. ACD = 62
2. DEC = 90
3. EDC = 28
4. ABC = 56
D
A B
C
E
62º
White Board Practice
• Quadrilateral MNOP is a rectangle
Find the measure of each angle
1. m PON =
2. m PMO =
3. PL =
4. MO =
P
M N
O
L
29º
12
White Board Practice
• Quadrilateral MNOP is a rectangle
Find the measure of each angle
1. m PON = 90
2. m PMO = 61
3. PL = 12
4. MO = 24
P
M N
O
L
29º
12
White Board Practice ABC is a right ; M is the midpoint of
AB
1. If AM = 7, then MB = ____, AB = ____,
and CM = _____ .
C
A
B
M
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 .
C
A
B
M
White Board Practice ABC is a right ; M is the midpoint of
AB
1. If AB = x, then AM = ____, MB = _____,
and MC = _____ .
C
A
B
M
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 .
C
A
B
M
Remote Time
A. Always
B. Sometimes
C. Never
D. I don’t know
A. AlwaysB. SometimesC. NeverD. I don’t know
• A square is ____________ a rhombus
A. AlwaysB. SometimesC. NeverD. I don’t know
• The diagonals of a parallelogram ____________ bisect the angles of the parallelogram.
A. AlwaysB. SometimesC. NeverD. I don’t know
• A quadrilateral with one pairs of sides congruent and one pair parallel is _________________ a parallelogram.
A. AlwaysB. SometimesC. NeverD. I don’t know
• The diagonals of a rhombus are ___________ congruent.
A. AlwaysB. SometimesC. NeverD. I don’t know
• A rectangle ______________ has consecutive sides congruent.
A. AlwaysB. SometimesC. NeverD. I don’t know
• A rectangle ____________ has perpendicular diagonals.
A. AlwaysB. SometimesC. NeverD. I don’t know
• The diagonals of a rhombus ___________ bisect each other.
A. AlwaysB. SometimesC. NeverD. 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.
A
B C
D
Trap. ABCD
Anatomy Of a Trapezoid
R S
TV
Base
Base
• The bases are the parallel sides
Anatomy Of a Trapezoid
R S
TV
LegLeg
• The legs are the non-parallel sides
Isosceles TrapezoidA trapezoid with congruent legs.
J
K L
M
Theorem 5-18The base angles of an isosceles trapezoid
are congruent.
E
F G
H
The Median of a TrapezoidA segment that joins the midpoints of the
legs.
A
B C
D
X Y
TheoremThe median of a trapezoid is parallel to the
bases and its length is the average of the bases.
B C
D
X Y
AA
B C
D
X Y
White Board Practice
B C
D
X Y
A
• Complete
1. AD = 25, BC = 13, XY = ______
White Board Practice
B C
D
X Y
A
• Complete
1. AD = 25, BC = 13, XY = 19
White Board Practice
B C
D
X Y
A
• Complete
2. AX = 11, YD = 8, AB = _____, DC = ____
White Board Practice
B C
D
X Y
A
• Complete
2. AX = 11, YD = 8, AB = 22, DC = 16
White Board Practice
B C
D
X Y
A
• Complete
3. AD = 29, XY = 24, BC =______
White Board Practice
B C
D
X Y
A
• Complete
3. AD = 29, XY = 24, BC =19
White Board Practice
B C
D
X Y
A
• Complete
4. AD = 7y + 6, XY = 5y -3, BC = y – 5, y =____
White Board Practice
B C
D
X Y
A
• Complete
4. AD = 7y + 6, XY = 5y -3, BC = y – 5, y = 3.5
Homework Set 5.5
• WS PM 28
• 5-5 #1-27 odd
• Quiz next class day