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