Quadrilaterals Geometrycontent.njctl.org/courses/math/...presentation-2014-06-03-3-slides-per-page-w-answers.pdfpentagon and hexagon. In each polygon, diagonals are drawn from o n

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www.njctl.org

2014-06-03

Quadrilaterals

Geometry

Slide 2 / 189

Table of Contents

· Angles of Polygons· Properties of Parallelograms

· Proving Quadrilaterals are Parallelograms· Constructing Parallelograms

· Rhombi, Rectangles and Squares

· Trapezoids· Kites

· Coordinate Proofs· Proofs

Click on a topic to go to that section.· Families of Quadrilaterals

Slide 3 / 189

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Angles of Polygons

Return to the Table of Contents

Slide 4 / 189

A polygon is a closed figure made of line segments connected end to end. Since it is made of line segments, there can be no curves. Also, it has only one inside regioin, so no two segments can cross each other.

A

BC

D

Can you explain why the figure below is not a polygon?

· DA is not a segment (it has a curve). · There are two inside regions.

Polygon

click to reveal

Slide 5 / 189

Types of Polygons

Polygons are named by their number of sides.

Number of Sides Type of Polygon

3 triangle4 quadrilateral5 pentagon 6 hexagon7 heptagon8 octagon 9 nonagon10 decagon 11 11-gon12 dodecagonn n-gon

Slide 6 / 189

page1svg

A polygon is convex if no line that contains a

side of the polygon contains a point in the

interior of the polygon.

interior

Convex polygons

Slide 7 / 189

A polygon is concave if a line that contains a side of the polygon

contains a point in the interior of the

polygon. interior

Concave polygons

Slide 8 / 189

1 The figure below is a polygon.

True

False

Ans

wer

Slide 9 / 189


True

False

[This object is a pull tab]

Ans

wer

False

Slide 9 (Answer) / 189


True

False

Ans

wer

Slide 10 / 189


True

False


Ans

wer

True


3 Indentify the polygon.

A Pentagon

B Octagon

C Quadrilateral

D HexagonE Decagon

F Triangle Ans

wer

Slide 11 / 189

3 Indentify the polygon.

A Pentagon

B Octagon

C Quadrilateral

D HexagonE Decagon

F Triangle


Ans

wer

D


4 Is the polygon convex or concave?

A Convex

B Concave

Ans

wer

Slide 12 / 189


A Convex

B Concave


Ans

wer

B



A ConvexB Concave

Ans

wer

Slide 13 / 189


A ConvexB Concave


Ans

wer

A


A polygon is equilateral if all its sides are congruent.

A polygon is equiangular if all its angles are congruent.

A polygon is regular if it is equilateral and equiangular.

Equilateral, Equiangular, Regular

Slide 14 / 189

6 Describe the polygon. (Choose all that apply)

A Pentagon

B Octagon

C Quadrilateral

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular

4

60o

60o

60o

44

Ans

wer

Slide 15 / 189


A Pentagon

B Octagon

C Quadrilateral

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular

4

60o

60o

60o

44


Ans

wer

E, F, H, I, J



A Pentagon

B Octagon

C Quadrilateral

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular

Ans

wer

Slide 16 / 189


A Pentagon

B Octagon

C Quadrilateral

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular


Ans

wer

F, H



A Pentagon

B Octagon

C Quadrilateral

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular

Ans

wer

Slide 17 / 189


A Pentagon

B Octagon

C Quadrilateral

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular


Ans

wer

C, F, I


Angle Measures of Polygons

Above are examples of a triangle, quadrilateral, pentagon and hexagon. In each polygon, diagonals are

drawn from one vertex.

What do you notice about the regions created by the diagonals?

They are triangularclick

Slide 18 / 189

Polygon Number of Sides

Number of Triangular Regions

Sum of the Interior Angles

triangle 3 1 1(180o) = 180o

quadrilateral 4 2 2(180o) = 360o

pentagon 5 3 3(180o) = 540o

hexagon 6 4 4(180o) = 720o

Complete the table

Slide 19 / 189

Given:Polygon ABCDEFG

Classify the polygon.

How many triangular regions can be drawn in polygon ABCDEFG?

What is the sum of the measures of the interior angles on ABCDEFG?

A B

C

DE

F

G

_____________

_____________

_____________

Ans

wer

Slide 20 / 189

Given:Polygon ABCDEFG

Classify the polygon.

How many triangular regions can be drawn in polygon ABCDEFG?

What is the sum of the measures of the interior angles on ABCDEFG?

A B

C

DE

F

G

_____________

_____________

_____________


Ans

wer

Heptagon

The sum of the interior angles is 5(180o) = 900o

F

A B

C

DE

F

G


The sum of the measures of the interior angles of a convex polygon with n sides is 180(n-2).

Complete the table.

Polygon Number of Sides

Sum of the measures of the interior angles.

hexagon 6 180(6-2) = 720o

heptagon 7 180(7-2) = 900o

octagon 8 180(8-2) = 1080o

nonagon 9 180(9-2)=1260o

decagon 10 180(10-2)=1440o

11-gon 11 180(11-2) = 1620o

dodecagon 12 180(12-2) = 1800o

Polygon Interior Angles Theorem Q1

Slide 21 / 189

Example:Find the value of each angle.

L M

N

O

xo

(3x)o

146o

(2x+3)o

(3x+4)o

P

The figure above is a pentagon.

The sum of measures of the interior angles a pentagon is 540o.

Slide 22 / 189

m L + m M + m N + m O + m P = 540o

(3x+4) + 146 + x + (3x) + (2x+3) = 540 (Combine Like Terms)

9x + 153 = 540 - 153 -153 9x = 387 9 9 x = 43

m L=3(43)+4=133 m M=146 m N=x=43

m O=3(43)=129 m P=2(43)+3=89

o

o o o

o

Check: 133 +146 +43 +129 +89 =540 o o o o o o

click to reveal

Slide 23 / 189

The measures of each interior angle of a regular polygon is:

180(n-2)n

Complete the table.

regular polygon number of sides sum of interior angles

measure of each angle

triangle 3 180o 60o

quadrilateral 4 360o 90o

pentagon 5 540o 108o

hexagon 6 720o 120o

octagon 8 1080o 135o

decagon 10 1440o 144o

15-gon 15 2340o 156o

Polygon Interior Angles Theorem Corollary

Slide 24 / 189

9 What is the sum of the measures of the interior angles of the stop sign?

Ans

wer

Slide 25 / 189

9 What is the sum of the measures of the interior angles of the stop sign?


Ans

wer

1080o


10 If the stop sign is a regular polygon. What is the measure of each interior angle?

Ans

wer

Slide 26 / 189

10 If the stop sign is a regular polygon. What is the measure of each interior angle?


Ans

wer

135o


11 What is the sum of the measures of the interior angles of a convex 20-gon?

A 2880

B 3060

C 3240

D 3420

Ans

wer

Slide 27 / 189

11 What is the sum of the measures of the interior angles of a convex 20-gon?

A 2880

B 3060

C 3240

D 3420


Ans

wer

C


12 What is the measure of each interior angle of a regular 20-gon?

A 162

B 3240

C 180

D 60 Ans

wer

Slide 28 / 189


A 162

B 3240

C 180

D 60


Ans

wer

A



A 2520 B 2880 C 3240 D 157.5

Ans

wer

Slide 29 / 189


A 2520 B 2880 C 3240 D 157.5


Ans

wer

D


14 What is the value of x?

(5x+

15)o

(9x-6) o

(8x) o

(11x+16)o

(10x+8)oA

nsw

er

Slide 30 / 189

14 What is the value of x?

(5x+

15)o

(9x-6) o

(8x) o

(11x+16)o

(10x+8)o


Ans

wer

14


The sum of the measures of the

exterior angles of a convex polygon, one at each vertex, is 360o.

x

yz

In other words, x + y + z = 360 o

Polygon Exterior Angle Theorem Q2

Slide 31 / 189

The measure of each exterior angle

of a regular polygon with n sides

is 360 n a

The polygon is a hexagon.

n=6

a=360 6

a = 60o

Polygon Exterior Angle Theorem Corollary

Slide 32 / 189

15 What is the sum of the measures of the exterior angles of a heptagon? A 180B 360C 540D 720

Ans

wer

Slide 33 / 189

15 What is the sum of the measures of the exterior angles of a heptagon? A 180B 360C 540D 720


Ans

wer

B


16 If a heptagon is regular, what is the measure of each exterior angle?

A 72

B 60C 51.43

D 45 Ans

wer

Slide 34 / 189

16 If a heptagon is regular, what is the measure of each exterior angle?

A 72

B 60C 51.43

D 45


Ans

wer

C


17 What is the sum of the measures of the exterior angles of a pentagon?

Ans

wer

Slide 35 / 189

17 What is the sum of the measures of the exterior angles of a pentagon?


Ans

wer

360o


18 If a pentagon is regular, what is the measure of each exterior angle?

Ans

wer

Slide 36 / 189

18 If a pentagon is regular, what is the measure of each exterior angle?


Ans

wer

72o


Example:The measure of each angle of a regular convex polygon is 172 . Find the number of sides of the polygon.o

180(n-2)n

We need to use to find n.

Ans

wer

Slide 37 / 189

Example:The measure of each angle of a regular convex polygon is 172 . Find the number of sides of the polygon.o

180(n-2)n

We need to use to find n.


Ans

wer

180(n-2)n

= 172(n) (n)

180(n-2) = 172n

180n-360 = 172n-180n -180n

-360 = -8n-8 -845 = n


19 The measure of each angle of a regular convex polygon is 174 . Find the number of sides of the polygon.

A 64

B 62 C 58

D 60 Ans

wer

o

Slide 38 / 189


A 64

B 62 C 58

D 60

o


Ans

wer

D



Ans

wer

o

Slide 39 / 189


o


Ans

wer

20


Properties of Parallelograms


Slide 40 / 189

Lab - Investigating Parallelograms

Lab - Properties of Parallelograms

Click on the links below and complete the two labs before the Parallelogram lesson.

Slide 41 / 189

page1svg

https://njctl.org/courses/math/geometry/quadrilaterals/investigating-parallelograms/

https://njctl.org/courses/math/geometry/quadrilaterals/properties-of-parallelograms/

A Parallelogram is a quadrilateral whose both pairs of opposite sides are parallel.

D E

G F

In parallelogram DEFG,

DG EF and DE GF

Parallelograms

Slide 42 / 189

Theorem Q3

A B

CD

If ABCD is a parallelogram,

then AB = DC and DA = CB

If a quadrilateral is a parallelogram, then

its opposite sides are congruent.

Slide 43 / 189

A B

CD

If ABCD is a parallelogram,then m A = m C and m B = m D

If a quadrilateral is a parallelogram, then

its opposite angles are congruent.

Theorem Q4

Slide 44 / 189

If a quadrilateral is a parallelogram, then the consecutive angles are

supplementary.

yo

xo

xo

yo

A B

CD

If ABCD is a parallelogram, then xo + yo = 180o

Theorem Q5

Slide 45 / 189

Example:

ABCD is parallelogram.

Find w, x, y, and z.

A B

CD

12

2y

x-5

9

65o

5zo

wo

Slide 46 / 189

A B

CD

12

2y

x-5

9

65o

5zo

wo

The opposite sides are congruent.

Ans

wer

Slide 47 / 189

A B

CD

12

2y

x-5

9

65o

5zo

wo

The opposite sides are congruent.


Ans

wer AB = DC

2y = 92 2y = 4.5

BC = ADx-5 = 12 +5 +5 x = 17


A B

CD

12

2y

x-5

9

65o

5zo

wo

The opposite angles are congruent.

Ans

wer

Slide 48 / 189

A B

CD

12

2y

x-5

9

65o

5zo

wo

The opposite angles are congruent.


Ans

wer m C = m A

5z = 65 5 5 z = 13


A B

CD

12

2y

x-5

9

65o

5zo

wo

The consecutive angles are supplementary.

Ans

wer

Slide 49 / 189

A B

CD

12

2y

x-5

9

65o

5zo

wo

The consecutive angles are supplementary.


Ans

wer m B + m A = 180o

w + 65 = 180 - 65 -65 w = 115o


21 DEFG is a parallelogram. Find w.

D E

FG

70o

15

3x-32w z+12

21

y2 A

nsw

er

Slide 50 / 189

21 DEFG is a parallelogram. Find w.

D E

FG

70o

15

3x-32w z+12

21

y2


Ans

wer

55o


22 DEFG is a parallelogram. Find x.

D E

FG

70o

15

3x-32w z+12

21

y2

Ans

wer

Slide 51 / 189

22 DEFG is a parallelogram. Find x.

D E

FG

70o

15

3x-32w z+12

21

y2


Ans

wer

8


23 DEFG is a parallelogram. Find y.

D E

FG

70o

15

3x-32w z+12

21

y2

Ans

wer

Slide 52 / 189

23 DEFG is a parallelogram. Find y.

D E

FG

70o

15

3x-32w z+12

21

y2


Ans

wer

30


24 DEFG is a parallelogram. Find z.

D E

FG

70o

15

3x-32w z+12

21

y2

Ans

wer

Slide 53 / 189

24 DEFG is a parallelogram. Find z.

D E

FG

70o

15

3x-32w z+12

21

y2


Ans

wer

58


If a quadrilateral is a parallelogram,

then the diagonals bisect each other.

A B

CD

E

If ABCD is a parallelogram,

then AE EC and BE ED

Theorem Q5

Slide 54 / 189

Example:

LMNP is a parallelogram. Find QN and MP.

L M

NP

Q

4

6(The diagonals bisect each other)

Ans

wer

Slide 55 / 189

Example:

LMNP is a parallelogram. Find QN and MP.

L M

NP

Q

4

6(The diagonals bisect each other)


Ans

wer

QN = LQLQ = 4

MQ + QP = MP (Segment Addition)MQ = QPMQ = 4 4 + 4 = MP 8 = MP


Try this...BEAR is a parallelogram. Find x, y, and ER.

A

B E

R

S

x 4y

8 10

Ans

wer

Slide 56 / 189

Try this...BEAR is a parallelogram. Find x, y, and ER.

A

B E

R

S

x 4y

8 10


Ans

wer x = SA = 10

4y = RS = 8y = 2

ER = RS + SE = 16


25 In a parallelogram, the opposite sides are ________ parallel.

A sometimesB always

C never Ans

wer

Slide 57 / 189

25 In a parallelogram, the opposite sides are ________ parallel.

A sometimesB always

C never


Ans

wer

B


26 MATH is a parallelogram. Find RT.

A 6

B 7

C 8

D 9 12

M A

TH

R

7

Ans

wer

Slide 58 / 189

26 MATH is a parallelogram. Find RT.

A 6

B 7

C 8

D 9 12

M A

TH

R

7


Ans

wer

B


27 MATH is a parallelogram. Find AR.

A 6

B 7

C 8

D 912

M A

TH

R

7

Ans

wer

Slide 59 / 189

27 MATH is a parallelogram. Find AR.

A 6

B 7

C 8

D 912

M A

TH

R

7


Ans

wer

A


28 MATH is a parallelogram. Find m H.

M A

TH98o

2x-4

14

(3y+8)o

Ans

wer

Slide 60 / 189

28 MATH is a parallelogram. Find m H.

M A

TH98o

2x-4

14

(3y+8)o


Ans

wer

82


29 MATH is a parallelogram. Find x.

M A

TH98o

2x-4

14

(3y+8)o

Ans

wer

Slide 61 / 189

29 MATH is a parallelogram. Find x.

M A

TH98o

2x-4

14

(3y+8)o


Ans

wer

9


30 MATH is a parallelogram. Find y.

M A

TH98o

2x-4

14

(3y+8)o

Ans

wer

Slide 62 / 189

30 MATH is a parallelogram. Find y.

M A

TH98o

2x-4

14

(3y+8)o


Ans

wer

30


Proving Quadrilaterals are

Parallelograms


Slide 63 / 189

In quadrilateral ABCD,

AB DC and AD BC,

so ABCD is a parallelogram.

A B

CD

Theorem Q6

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Slide 64 / 189

In quadrilateral ABCD,

A D and B C,

so ABCD is a quadrilateral.

A B

CD

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Theorem Q7

Slide 65 / 189

page1svg

Example

Tell whether PQRS is a parallelogram. Explain.

P

Q

R

S6

6

4

4 Ans

wer

Slide 66 / 189

Example

Tell whether PQRS is a parallelogram. Explain.

P

Q

R

S6

6

4

4


Ans

wer

Yes, PQRS is a quadrilateral. The opposite sides are congruent.


Example

Tell whether PQRS is a parallelogram. Explain.P Q

RS

Ans

wer

Slide 67 / 189

Example

Tell whether PQRS is a parallelogram. Explain.P Q

RS[This object is a pull tab]

Ans

wer

Because PQRS is a quadrilateral, m Q + m R = 180o. But, we can't assume that Q and R are right angles. We can't prove PQRS is a

parallelogram.


31 Tell whether the quadrilateral is a parallelogram.

Yes

No

78o

136o

2

Ans

wer

Slide 68 / 189


Yes

No

78o

136o

2


Ans

wer

No



Yes

No3 3

5

4.99

Ans

wer

Slide 69 / 189


Yes

No3 3

5

4.99


Ans

wer

No



Yes

No

Ans

wer

Slide 70 / 189


Yes

No


Ans

wer

No


If an angle of a quadrilateral is

supplementary to both of its consecutive

angles, then the quadrilateral is a

parallelogram.

A B

CD

75o

75o

105o

In quadrilateral ABCD, m A + m B=180

and m B + m C=180, so ABCD is a parallelogram.

o o

Theorem Q8

Slide 71 / 189

If the diagonals of a quadrilateral bisect each

other, then the quadrilateral is a parallelogram.

In quadrilateral ABCD,AE EC and DE EB, so ABCD is a quadrilateral.

A B

CD

E

Theorem Q9

Slide 72 / 189

If one pair of sides of a quadrilateral is

parallel and congruent, then the

quadrilateral is a parallelogram.

In quadrilateral ABCD,AD BC and AD BC, so ABCD is a parallelogram.

A B

CD

Theorem Q10

Slide 73 / 189


Yes

No

Ans

wer

Slide 74 / 189


Yes

No


Ans

wer

No



Yes

No141o

39o

49o

Ans

wer

Slide 75 / 189


Yes

No141o

39o

49o


Ans

wer

No



Yes

No

89.5

819

Ans

wer

Slide 76 / 189


Yes

No

89.5

819


Ans

wer

Yes



Yes

No

Ans

wer

Slide 77 / 189


Yes

No


Ans

wer

Yes


Example:

Three interior angles of a quadrilateral measure 67 , 67 and 113 . Is this enough information to tell whether the quadrilateral is a parallelogram? Explain.

o o o

Ans

wer

Slide 78 / 189

Example:

Three interior angles of a quadrilateral measure 67 , 67 and 113 . Is this enough information to tell whether the quadrilateral is a parallelogram? Explain.

o o o


Ans

wer

NO, the question did not state the position of the measurements in the quadrilateral. We cannot assume their position.

67 67

113

This is not a parallelogram

o o o


In a parallelogram...

the opposite sides are _________________ and ____________,

the opposite angles are _____________, the consecutive angles are _____________

and the diagonals ____________ each other.

parallel perpendicularbisect congruent supplementary

Fill in the blank

Ans

wer

Slide 79 / 189

In a parallelogram...

the opposite sides are _________________ and ____________,

the opposite angles are _____________, the consecutive angles are _____________

and the diagonals ____________ each other.

parallel perpendicularbisect congruent supplementary

Fill in the blank


Ans

wer

In a parallelogram...the opposite sides are parallel and congruent,the opposite angles are congruent, the consecutive angles are supplementary and the diagonals bisect each other.


To prove a quadrilateral is a parallelogram...

both pairs of opposite sides of a quadrilateral must be _____________,

both pairs of opposite angles of a quadrilateral must be ____________,

an angle of the quadrilateral must be _____________ to its consecutive

angles, the diagonals of the quadrilateral __________ each other, or one pair of opposite sides of a quadrilateral are ___________ and _________.

bisect congruent parallel perpendicular supplementary

Fill in the blankA

nsw

er

Slide 80 / 189

To prove a quadrilateral is a parallelogram...

both pairs of opposite sides of a quadrilateral must be _____________,

both pairs of opposite angles of a quadrilateral must be ____________,

an angle of the quadrilateral must be _____________ to its consecutive

angles, the diagonals of the quadrilateral __________ each other, or one pair of opposite sides of a quadrilateral are ___________ and _________.

bisect congruent parallel perpendicular supplementary

Fill in the blank


Ans

wer

To prove a quadrilateral is a parallelogram...both pairs of opposite sides of a quadrilateral must be congruent,both pairs of opposite angles of a quadrilateral must be congruent,an angle of the quadrilateral must be supplementary to its consecutive angles, the diagonals of the quadrilateral bisect each other, or one pair of opposite sides of a quadrilateral are parallel and congruent.


38 Which theorem proves the quadrilateral is a parallelogram?

A The opposite angle are congruent.

B The opposite sides are congruent. C An angle in the quadrilateral is supplementary to its consecutive angles. D The diagonals bisect each other. E One pair of opposite sides are congruent and parallel. F Not enough information.

3(2)3

6(7-3)

Ans

wer

Slide 81 / 189




3(2)3

6(7-3)


Ans

wer

E





Ans

wer

Slide 82 / 189



B The opposite sides are congruent. C An angle in the quadrilateral is supplementary to its consecutive angles. D The diagonals bisect each other. E One pair of opposite sides are congruent and parallel. F Not enough information. [This object is a pull tab]

Ans

wer

F





6

63(6-4)

Ans

wer

Slide 83 / 189




6

63(6-4)


Ans

wer

F


Constructing Parallelograms


Slide 84 / 189

To construct a parallelogram, there are 3 steps.

Construct a Parallelogram

Slide 85 / 189

Step 1 - Use a ruler to draw a segment and its midpoint.

Construct a Parallelogram - Step 1

Slide 86 / 189

page1svg

Step 2 - Draw another segment such that the midpoints coincide.


Slide 87 / 189

Why is this a parallelogram?

Step 3 - Connect the endpoints of the segments.

Construct a Parallelogram - Step 3 A

nsw

er

Slide 88 / 189


Step 3 - Connect the endpoints of the segments.



Ans

wer The diagonals

bisect each other


3 steps to draw a parallelogram in a coordinate plane

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

12 units

Step 1 - Draw a horizontal segment in the plane. Find the length of the segment.

Slide 89 / 189

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

12 units

12 units

Step 2 - Draw another horizontal line of the same length, anywhere in the plane.


Slide 90 / 189

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

12 units

12 units

Step 3 - Connect the endpoints



Ans

wer

Slide 91 / 189

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

12 units

12 units

Step 3 - Connect the endpoints




Ans

wer

Remember all horizontal lines have

a slope of zero.One pair of opposite

sides are parallel and congruent.


Note: this method also works with vertical lines.

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

Slide 92 / 189

41 The opposite angles of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary

Ans

wer

Slide 93 / 189

41 The opposite angles of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary


Ans

wer

B


42 The consecutive angles of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary

Ans

wer

Slide 94 / 189

42 The consecutive angles of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary


Ans

wer

D


43 The diagonals of a parallelogram ______ each other.

A bisect

B congruent

C parallel

D supplementary Ans

wer

Slide 95 / 189

43 The diagonals of a parallelogram ______ each other.

A bisect

B congruent

C parallel

D supplementary


Ans

wer

A


44 The opposite sides of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary

Ans

wer

Slide 96 / 189

44 The opposite sides of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary


Ans

wer

B & C


Rhombi, Rectanglesand Squares


Slide 97 / 189

three special parallelograms

Rhombus

Rectangle

Square

All the same properties of a parallelogram apply to the rhombus, rectangle,

and square.

Slide 98 / 189

page1svg

A quadrilateral is a rhombus if and only if it has four congruent sides.

A B

CD

AB BC CD DAIf ABCD is a quadrilateral with four congruent sides,

then it is a rhombus.

Rhombus Corollary

Slide 99 / 189

45 What is the value of y that will make the quadrilateral a rhombus?

A 7.25

B 12

C 20

D 25

35

y

12

Ans

wer

Slide 100 / 189


A 7.25

B 12

C 20

D 25

35

y

12[This object is a pull tab]

Ans

wer

C



A 7.25

B 12

C 20

D 25

2y+29

6y

Ans

wer

Slide 101 / 189


A 7.25

B 12

C 20

D 25

2y+29

6y[This object is a pull tab]

Ans

wer

A


If a parallelogram is a rhombus, then its diagonals are perpendicular.

A B

CD

If ABCD is a rhombus,

then AC BD.

Theorem Q11

Slide 102 / 189

A B

CD

If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

If ABCD is a rhombus, then

DAC BAC BCA DCA

and

ADB CDB ABD CBD

Theorem Q12

Slide 103 / 189

Example

EFGH is a rhombus.

Find x, y, and z.E F

G H

72o

z

2x-6

5y

10

Slide 104 / 189

All sides of a rhombus are congruent.

EF = HG2x-6 = 10 +6 +6 2x = 16 2 2 x = 8

EG = HG5y = 105 5 y = 2

Because the consecutive angles of parallelogram are supplementary, the consecutive angles of a rhombus are supplementary.

m E + m F = 180 72 + m F = 180-72 -72 m F = 108 z = m F

z = (108 )

z = 54

12

12

o

o

o

o

o The diagonals of a rhombus bisect the opposite angles.

Slide 105 / 189

Try this ...

The quadrilateral is a rhombus. Find x, y, and z.

8

86o

3x+2

z

12 y2

Ans

wer

Slide 106 / 189

Try this ...

The quadrilateral is a rhombus. Find x, y, and z.

8

86o

3x+2

z

12 y2 [This object is a pull tab]

Ans

wer x = 2

y = 4z = 470


47 This is a rhombus. Find x.

xo

Ans

wer

Slide 107 / 189


xo


Ans

wer

90o



13

x-3

9 Ans

wer

Slide 108 / 189


13

x-3

9


Ans

wer

36



126ox A

nsw

er

Slide 109 / 189


126ox


Ans

wer

27o


50 HJKL is a rhombus. Find the length of HJ.

H J

KL

6 M16

Ans

wer

Slide 110 / 189

50 HJKL is a rhombus. Find the length of HJ.

H J

KL

6 M16


Ans

wer

10


A quadrilateral is a rectangle if and only if it has four right angles.

A, B, C and D are right angles.

If a quadrilateral is a rectangle, then

it has four right angles.

Rectangle Corollary

Slide 111 / 189

51 What value of y will make the quadrilateral a rectangle?

6y

12

Ans

wer

Slide 112 / 189

51 What value of y will make the quadrilateral a rectangle?

6y

12


Ans

wer

15


If a quadrilateral is a rectangle, then its diagonals are congruent.

If ABCD is a rectangle,

then AC BD.

A B

CD

Theorem Q13

Slide 113 / 189

Example

RECT is a rectangle. Find x and y.

2x-5 13

63o9yo

R E

CT

Ans

wer

Slide 114 / 189

Example

RECT is a rectangle. Find x and y.

2x-5 13

63o9yo

R E

CT[This object is a pull tab]

Ans

wer

Option A2x - 5 = 13

2x = 18x = 9

Option B2x - 5 + 13 = 26

2x + 8 = 262x = 18

x = 9The measure of each angle in a

rectangle is 90o.

9y + 63 = 909y = 27y = 3


52 RSTU is a rectangle. Find z.R S

TU8z

Ans

wer

Slide 115 / 189


TU8z


Ans

wer

11.25



TU

4z-9

7

Ans

wer

Slide 116 / 189


TU

4z-9

7


Ans

wer

4


A quadrilateral is a square if and only if it is a rhombus and a rectangle.

A square has all the properties of a

rectangle and rhombus.

Square Corollary

Slide 117 / 189

Example

The quadrilateral is a square. Find x, y, and z.

z - 4

(5x)o

6

3y

Ans

wer

Slide 118 / 189

Example


z - 4

(5x)o

6

3y


Ans

wer

In a rhombus the diagonals are perpendicular.

In a rhombus the diagonal

bisect the opposite angles.

5x = (90)5x = 45

x = 9

12

3y = 90y = 30

z - 4 = 6z = 10


Try this ...


3y

12z

8y - 1

0

(x2 + 9)o

Ans

wer

Slide 119 / 189

Try this ...


3y

12z

8y - 1

0

(x2 + 9)o


Ans

wer x = 6

y = 2 z = 7.5


54 The quadrilateral is a square. Find y.

A 2

B 3

C 4

D 5

18y

Ans

wer

Slide 120 / 189

54 The quadrilateral is a square. Find y.

A 2

B 3

C 4

D 5

18y


Ans

wer

5


55 The quadrilateral is a rhombus. Find x.

A 2

B 3

C 4

D 5

2x + 6

4x

Ans

wer

Slide 121 / 189

55 The quadrilateral is a rhombus. Find x.

A 2

B 3

C 4

D 5

2x + 6

4x


Ans

wer

3


112o

(4x)o

56 The quadrilateral is parallelogram. Find x.

Ans

wer

Slide 122 / 189

112o

(4x)o

56 The quadrilateral is parallelogram. Find x.


Ans

wer

17


57 The quadrilateral is a rectangle. Find x.

10x

3x + 7

Ans

wer

Slide 123 / 189

57 The quadrilateral is a rectangle. Find x.

10x

3x + 7


Ans

wer

1


Opposite sidesare

Diagonals bisectopposite <'s

Has 4 sides

Has 4 right <'s

Diagonals are

Slide the description under the correct special parallelogram.

rhombus rectangle square

Diagonals are

Has 4 right <'s

Has 4 sides

Diagonals are

Opposite sidesare

Diagonals are

Slide 124 / 189

Lab - Quadrilaterals in the Coordinate Plane

Click on the link below and complete the lab.

Slide 125 / 189

Trapezoids

Return to theTable of Contents

Slide 126 / 189

https://njctl.org/courses/math/geometry/quadrilaterals/quadrilaterals-in-the-coordinate-plane/

page1svg

A trapezoid is a quadrilateral with one pair of parallel sides. base

legbase angles

base

leg

The parallel sides are called bases.

The nonparallel sides are called legs.

A trapezoid also has two pairs of base angles.

trapezoid

Slide 127 / 189

An isosceles trapezoid is a trapezoid with congruent legs.

isosceles trapezoid

Slide 128 / 189

If a trapezoid is isosceles, then each pair of base angles are congruent.

ABCD is an isosceles trapezoid. <A <B

and <C <D.

A B

CD

Theorem Q14

Slide 129 / 189

If a trapezoid has at least one pair of congruent base angles, then the trapezoid is isosceles.

A B

CD

In trapezoid ABCD, A B. ABCD is an isosceles trapezoid.

Theorem Q15

Slide 130 / 189

Slide 131 / 189


59 The quadrilateral is an isosceles trapezoid. Find x.

A 3

B 5

C 7

D 9 64o (9x + 1)o

Ans

wer

Slide 132 / 189

59 The quadrilateral is an isosceles trapezoid. Find x.

A 3

B 5

C 7

D 9 64o (9x + 1)o


Ans

wer

C


A trapezoid is isosceles if and only if its diagonals are congruent.

In trapeziod ABCD,

AC BD. ABCD is isosceles.

A B

CD

Theorem Q16

Slide 133 / 189

Example

PQRS is a trapeziod. Find the m S and m R.

112o 147o

(6w+2)o (3w)o

P

R

Q

S

Slide 134 / 189

Option A

(6w+2) + (3w) + 147 + 112 = 3609w + 261 = 360

9w = 99w = 11

m S = 6w+2 = 6(11)+2 = 68

m R = 3w = 3(11) = 33

o o

The sum of the interior angles of a quadrilateral is 360 .o

Slide 135 / 189

The parallel lines in a trapezoid create pairs of consecutive interior angles.

m P + m S = 180 and m Q + m R = 180

(6w+2) + 112 = 1806w + 114 = 180

w = 11

(3w) + 147 = 1803w = 33w = 11OR

m S = 6w+2 = 6(11)+2 = 68

m R = 3w = 3(11) = 33

Option B

o

o o

o

Slide 136 / 189

Try this ...

PQRS is an isosceles trapezoid. Find the m Q, m R and m S.

123o

(4w+1)o

(9w-3)oP Q

RS

Ans

wer

Slide 137 / 189

Try this ...

PQRS is an isosceles trapezoid. Find the m Q, m R and m S.

123o

(4w+1)o

(9w-3)oP Q

RS


Ans

wer

123 + 4w + 1 = 180124 + 4w = 180

4w = 56w = 14

m S = 4(14) + 1 = 57m Q = 9(14) - 3 = 123m R = 180 - 123 = 57

o o o


60 The trapezoid is isosceles. Find x.

9

4

6x + 3

2x + 2

Ans

wer

Slide 138 / 189

60 The trapezoid is isosceles. Find x.

9

4

6x + 3

2x + 2


Ans

wer

6x + 3 = 9or 4 = 2x + 2

or 6x + 3 + 2x + 2 = 4 + 9

x = 1


61 The trapeziod is isosceles. Find x.

137o

xo

Ans

wer

Slide 139 / 189

61 The trapeziod is isosceles. Find x.

137o

xo


Ans

wer

143


62 In trapezoid HIJK, can HI and KJ have slopes that are opposite reciprocals?

H I

JK

Ans

wer

YesNo

Slide 140 / 189

62 In trapezoid HIJK, can HI and KJ have slopes that are opposite reciprocals?

H I

JK

YesNo


Ans

wer

No, the bases are parallel, and have the same slope


The midsegment of a trapezoid is a segment that joins the midpoints of the legs.

midsegment of a trapezoid

Lab - Midsegments of a Trapezoid


Slide 141 / 189

https://njctl.org/courses/math/geometry/quadrilaterals/midsegments-of-trapezoids/

The midsegment is parallel to both the bases, and the length of the midsegment is half the sum of the

bases.

AB EF DCEF = (AB+DC)1

2

A B

CD

E F

Theorem Q17

Slide 142 / 189

P

Q R

S

L M

15

7

Example

PQRS is a trapezoid. Find LM.A

nsw

er

Slide 143 / 189

P

Q R

S

L M

15

7

Example

PQRS is a trapezoid. Find LM.


Ans

wer


P

Q R

S

L M

20

14.5

Example

PQRS is a trapezoid. Find PS.

Ans

wer

Slide 144 / 189

P

Q R

S

L M

20

14.5

Example

PQRS is a trapezoid. Find PS.


Ans

wer


P

QR

S

LM

y

5

10

14

xz

7

Try this ...

PQRS is an trapezoid. ML is the midsegment. Find x, y, and z.

Ans

wer

Slide 145 / 189

P

QR

S

LM

y

5

10

14

xz

7

Try this ...

PQRS is an trapezoid. ML is the midsegment. Find x, y, and z.


Ans

wer x = 18

y = 7z = 5


63 EF is the midsegment of trapezoid HIJK. Find x.

H I

JK

E F

6

x

15

Ans

wer

Slide 146 / 189


H I

JK

E F

6

x

15


Ans

wer

10.5



HI

J K

EF

x

19

10

Ans

wer

Slide 147 / 189


HI

J K

EF

x

19

10


Ans

wer

1


65 Which of the following is true of every trapezoid? Choose all that apply.

A Exactly 2 sides are congruent.

B Exactly one pair of sides are parallel.C The diagonals are perpendicular.D There are 2 pairs of base angles.

Ans

wer

Slide 148 / 189

65 Which of the following is true of every trapezoid? Choose all that apply.

A Exactly 2 sides are congruent.

B Exactly one pair of sides are parallel.C The diagonals are perpendicular.D There are 2 pairs of base angles.


Ans

wer

B and D


Kites


Slide 149 / 189

A kite is a quadrilateral with two pairs of adjacent congruent sides. The opposite sides are not congruent.

kites

Lab - Properties of Kites


Slide 150 / 189

page1svg

https://njctl.org/courses/math/geometry/quadrilaterals/properties-of-kites/

In kite ABCD, <B <D

and <A <D

If a quadrilateral is a kite, then it has one pair of congruent opposite angles.

A

B

C

D

Theorem Q18

Slide 151 / 189

Theorem Q18

If a quadrilateral is a kite, then it has one pair of congruent opposite angles.

In kite ABCD, B D and A D

Slide 152 / 189

Example

LMNP is a kite. Find x.

72

(x2-1)

48

M

N

P

o

o

oL

Slide 153 / 189

m L + m M +m N +m P = 360 (Remember M ≅ P)

72 + (x2-1) + (x2-1) + 48 = 3602x2 + 118 = 360

2x2 = 242x2 = 121x = ±11

o

Slide 154 / 189

66 READ is a kite. RE is congruent to ____.

A EA

B ADC DR R

E

A

D

Ans

wer

Slide 155 / 189

66 READ is a kite. RE is congruent to ____.

A EA

B ADC DR R

E

A

D


Ans

wer

A


67 READ is a kite. A is congruent to ____.

A EB D

C RR

E

A

D

Ans

wer

Slide 156 / 189

67 READ is a kite. A is congruent to ____.

A EB D

C RR

E

A

D


Ans

wer

C


68 Find the value of z in the kite.

z 5z-8

Ans

wer

Slide 157 / 189

68 Find the value of z in the kite.

z 5z-8


Ans

wer

2


69 Find the value of x in the kite.

68o

(8x+4)o

44o

Ans

wer

Slide 158 / 189


68o

(8x+4)o

44o


Ans

wer

15


70 Find the value of x.

36

(3x 2 + 3)

24 Ans

wer

o

o

o

Slide 159 / 189

70 Find the value of x.

36

(3x 2 + 3)

24

o

o

o


Ans

wer

7


Theorem Q19

If a quadrilateral is a kite then the diagonals are perpendicular.

In kite ABCDAC BD

A

B

C

D

Slide 160 / 189


x

Ans

wer

Slide 161 / 189


x


Ans

wer

90o


72 Find the value of y in the kite.

12y

Ans

wer

Slide 162 / 189

72 Find the value of y in the kite.

12y


Ans

wer

7.5


Families of Quadrilaterals


Slide 163 / 189

In this unit, you have learned about several special quadrilaterals. Now you will study what

links these figures.

quadrilateral

kite trapezoidparallelogram

rhombus

square

rectangle isosceles trapezoid

Every rhombus is a special kite

Each quadrilateral shares the properties with the quadrilateral above it.

Slide 164 / 189

page1svg

Complete the chart by sliding the special quadrilateral next to its description. (There can be more than one answer).

squarerectanglerhombusparallelogram kite

trapezoid isosceles trapezoid

Description Answer(s)

An equilateral quadrilateral

An equiangular quadrilateral

The diagonals are perpendicular

The diagonals are congruent

Has at least 1 pair of parallel sides

rectangle & square

rhombus & square

rhombus, square & isosceles trapezoid

rectangle, square & kite

All except kite

Special Quadrilateral(s)

Slide 165 / 189

QUADRILATERALS

Kite

Trapezoid

IsoscelesTrapezoid

Parallelogram

Rhombus Rectangle

Squa

re

Rhombus

Slide 166 / 189

73 A rhombus is a square.

A alwaysB sometimes

C never

Ans

wer

Slide 167 / 189

73 A rhombus is a square.

A alwaysB sometimes

C never


Ans

wer

B


74 A square is a rhombus.

A alwaysB sometimes

C never

Ans

wer

Slide 168 / 189

74 A square is a rhombus.

A alwaysB sometimes

C never


Ans

wer

A


75 A rectangle is a rhombus.

A alwaysB sometimes

C never

Ans

wer

Slide 169 / 189

75 A rectangle is a rhombus.

A alwaysB sometimes

C never


Ans

wer

C


76 A trapezoid is isosceles.

A alwaysB sometimes

C never

Ans

wer

Slide 170 / 189

76 A trapezoid is isosceles.

A alwaysB sometimes

C never


Ans

wer

B


77 A kite is a quadrilateral.

A alwaysB sometimes

C never

Ans

wer

Slide 171 / 189

77 A kite is a quadrilateral.

A alwaysB sometimes

C never


Ans

wer

A


78 A parallelogram is a kite.

A alwaysB sometimes

C never

Ans

wer

Slide 172 / 189

78 A parallelogram is a kite.

A alwaysB sometimes

C never


Ans

wer

C


Coordinate Proofs


Slide 173 / 189

page1svg

Given: PQRS is a quadrilateralProve: PQRS is a kite

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

P

Q

R

(-1,6)

(-4,3) (2,3)

(-1,-2)

S

Slide 174 / 189

P

Q

R

(-1,6)

(-4,3) (2,3)

(-1,-2)

S

A kite has one unique property.The adjacent sides are congruent.

SP = (6-3)2 + (-1-(-4))2 PQ = (3-6)2 + (2-(-1))2 = 32 + 32 = (-3)2 + 32 = 9 + 9 = 9 + 9 = 18 = 18 = 4.24 = 4.24

√

√√

√√

√√

√

Slide 175 / 189

P

Q

R

(-1,6)

(-4,3) (2,3)

(-1,-2)

S

SR = (3-(-2))2 +(-4-(-1))2 RQ = (-2-3)2 + (-1-2)2 = 52 + (-3)2 = (-5)2 + (-3)2

= 25 + 9 = 25 + 9 = 34 = 34 = 5.83 = 5.83

√

√√

√√

√√

√

So, because SP=PQ and SR=RQ, PQRS is a kite.

Slide 176 / 189

Given: JKLM is a parallelogramProve: JKLM is a square

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

J (1,3)

K (4,-1)

L (0,-4)

(-3,0) M

Slide 177 / 189

J (1,3)

K (4,-1)

L (0,-4)

(-3,0) M

Since JKLM is a parallelogram, we know the opposite sides are parallel and congruent.

We also know that a square is a rectangle and a rhombus.We need to prove the sides are congruent and perpendicular.

MJ = (3-0)2 + (1-(-3))2 JK = (-1-3)2 + (4-1)2 = 32 + 42 = (-4)2 + 32

= 9 + 16 = 9 + 16 = 25 = 25 = 5 = 5

√ √√√√ √

√√

Slide 178 / 189

J (1,3)

K (4,-1)

L (0,-4)

(-3,0) M

mMJ = = mJK = =

3 - 0 31-(-3) 4

-1-3 -4 4-1 3

MJ JK and MJ JKWhat else do you know?

MJ LK and JK LM (Opposite sides are congruent)MJ LM and JK LK (Perpendicular Transversal Theorem)

JKLM is a square

Slide 179 / 189

Try this ...

Given: PQRS is a trapezoidProve: LM is the midsegment

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

P (2,2)

(1,0) LQ (5,1)

M (7,-2)

R (9,-5)

(0,-2) S

Hin

t

Slide 180 / 189

Try this ...

Given: PQRS is a trapezoidProve: LM is the midsegment

2

4

6

8

10

-2

-4

-6

-8

-10

2 4 6 8 10-2-4-6-8-10 0

P (2,2)

(1,0) LQ (5,1)

M (7,-2)

R (9,-5)

(0,-2) S


Hin

t

Remember, the midsegment is the segment that joins the

midpoints of the legs and is parallel to the bases.

You need to show that:1. SL = LP2. QM = MR

3. slope of LM = slope of SR


Proofs


Slide 181 / 189

page1svg

Given: TE MA, <1 <2Prove: TEAM is a parallelogram.

T E

AM

1

2

≅ ≅

Slide 182 / 189

T E

AM

1

2

Option A

statements reasons

1) TE ≅ MA, <1 ≅ <2 1) Given

2) EM ≅ EM 2) Reflexive Property

3) Triangle MTE ≅ Triangle EAM 3) Side Angle Side

4) TM ≅ AE 4) CPCTC

5) TEAM is a parallelogram 5) The opposite sides of a parallelogram are congruent

Slide 183 / 189

T E

AM

1

2

Option B

We are given that TE MA and 2 3. TE AM, by the alternate interior angles converse.

So, TEAM is a parallelogram because each pair of opposite sides is parallel and congruent.

≅≅

click

click to reveal

Slide 184 / 189

Given: FGHJ is a parallelogram, F is a right angleProve: FGHJ is a rectangle

F G

HJ

Slide 185 / 189

F G

HJ

statements reasons

1) FGHJ is a parallelogramand F is a right angle 1) Given

2) J and G are right angles 2) The consecutive angles of a parallelogram are supplementary

3) H is a right angle 3) The opposite angles of a parallelogram are congruent

4) TEAM is a rectangle 4) Rectangle Corollary

Slide 186 / 189

Given: COLD is a quadrilateral, m O=140o, m D =40o, m L=60o

Prove: COLD is a trapezoid

C O

LD

140o

40o60o

Slide 187 / 189

C O

LD

140o

40o60o

statements reasons

1) COLD is a quadrilateral,m O=140,m L=40,m D=60 1) Given

2) m O + m L = 180m L + m D = 100 2) Angle Addition

3) O and D are supplementary 3) Definition of Supplementary Angles

4) L and D are not supplementary 4) Definition of Supplementary Angles

5) CO is parallel to LD 5) Consecutive Interior Angles Converse

6) CL is not parallel to OD 6) Consecutive Interior Angles Converse

7) COLD is a trapezoid 7) Definition of a Trapezoid(A trapezoid has one pair of parallel sides)

Slide 188 / 189

Try this ...

Given: FCD FEDProve: FD CE

F

C

D

E

≅H

int

Slide 189 / 189

Try this ...

Given: FCD FEDProve: FD CE

F

C

D

E

≅


Hin

t

If you prove CDEF is kite, then the diagonals must be perpendicular.


Quadrilaterals Geometrycontent.njctl.org/courses/math/...presentation-2014-06-03-3-slides-per-page-w-answers.pdfpentagon and hexagon. In each polygon, diagonals are drawn from o n

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