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Jun 17, 2020

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dariahiddleston
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• Slide 1 / 189

This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others.

Click to go to website:www.njctl.org

New Jersey Center for Teaching and Learning

Progressive Mathematics Initiative

Slide 2 / 189

www.njctl.org

2014-06-03

Geometry

Slide 3 / 189

· 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 4 / 189

Angles of Polygons

Slide 5 / 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 6 / 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

http://www.njctl.orghttp://www.njctl.orghttp://www.njctl.orghttp://www.njctl.orghttp://www.njctl.orgnextPage();page4svgpage56svgpage75svgpage6svgpage115svgpage143svgpage163svgpage172svgpage157svgpage1svg

• Slide 7 / 189

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 8 / 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 9 / 189

1 The figure below is a polygon.

True

False

Ans

wer

1 The figure below is a polygon.

True

False

[This object is a pull tab]A

nsw

er

False

Slide 10 / 189

2 The figure below is a polygon.

True

False

Ans

wer

2 The figure below is a polygon.

True

False

[This object is a pull tab]

Ans

wer

True

• Slide 11 / 189

3 Indentify the polygon.

A Pentagon

B Octagon

D HexagonE Decagon

F Triangle Ans

wer

3 Indentify the polygon.

A Pentagon

B Octagon

D HexagonE Decagon

F Triangle

[This object is a pull tab]

Ans

wer

D

Slide 12 / 189

4 Is the polygon convex or concave?

A Convex

B Concave

Ans

wer

4 Is the polygon convex or concave?

A Convex

B Concave

[This object is a pull tab]

Ans

wer

B

Slide 13 / 189

5 Is the polygon convex or concave?

A ConvexB Concave

Ans

wer

5 Is the polygon convex or concave?

A ConvexB Concave

[This object is a pull tab]

Ans

wer

A

• Slide 14 / 189

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 15 / 189

6 Describe the polygon. (Choose all that apply)

A Pentagon

B Octagon

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular

4

60o

60o

60o

44

Ans

wer

6 Describe the polygon. (Choose all that apply)

A Pentagon

B Octagon

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular

4

60o

60o

60o

44

[This object is a pull tab]

Ans

wer

E, F, H, I, J

Slide 16 / 189

7 Describe the polygon. (Choose all that apply)

A Pentagon

B Octagon

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular

Ans

wer

7 Describe the polygon. (Choose all that apply)

A Pentagon

B Octagon

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular

[This object is a pull tab]

Ans

wer

F, H

Slide 17 / 189

8 Describe the polygon. (Choose all that apply)

A Pentagon

B Octagon

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular

Ans

wer

• Slide 17 (Answer) / 189

8 Describe the polygon. (Choose all that apply)

A Pentagon

B Octagon

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular

[This object is a pull tab]

Ans

wer

C, F, I

Slide 18 / 189

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 19 / 189

Polygon Number of SidesNumber of Triangular

RegionsSum 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 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

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

_____________

_____________

_____________

[This object is a pull tab]

Ans

wer

Heptagon

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

F

A B

C

DE

F

G

Slide 21 / 189

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 SidesSum 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 22 / 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 23 / 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 oclick to reveal

Slide 24 / 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 anglesmeasure of each

angle

triangle 3 180o 60o

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 25 / 189

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

Ans

wer

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

[This object is a pull tab]

Ans

wer

1080o

Slide 26 / 189

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

Ans

wer

• Slide 26 (Answer) / 189

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

[This object is a pull tab]

Ans

wer

135o

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

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

[This object is a pull tab]

Ans

wer

C

Slide 28 / 189

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

A 162

B 3240

C 180

D 60 Answ

er

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

A 162

B 3240

C 180

D 60

[This object is a pull tab]

Ans

wer

A

Slide 29 / 189

13 What is the measure of each interior angle of a regular 16-gon?

A 2520 B 2880 C 3240 D 157.5

Ans

wer

• Slide 29 (Answer) / 189

13 What is the measure of each interior angle of a regular 16-gon?

A 2520 B 2880 C 3240 D 157.5

[This object is a pull tab]

Ans

wer

D

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 What is the value of x?

(5x+

15)o

(9x-6) o

(8x) o

(11x+16)

o

(10x+8)o

[This object is a pull tab]

Ans

wer

14

Slide 31 / 189

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 32 / 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 33 / 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 (Answer) / 189

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

[This object is a pull tab]

Ans

wer

B

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

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

A 72

B 60C 51.43

D 45

[This object is a pull tab]

Ans

wer

C

Slide 35 / 189

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

Ans

wer

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

[This object is a pull tab]

Ans

wer

360o

Slide 36 / 189

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

Ans

wer

• Slide 36 (Answer) / 189

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

[This object is a pull tab]

Ans

wer

72o

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

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.

[This object is a pull tab]

Ans

wer

180(n-2)n

= 172(n) (n)

180(n-2) = 172n

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

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

Slide 38 / 189

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

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

o

[This object is a pull tab]

Ans

wer

D

Slide 39 / 189

20 The measure of each angle of a regular convex polygon is 162 . Find the number of sides of the polygon.

Ans

wer

o

• Slide 39 (Answer) / 189

20 The measure of each angle of a regular convex polygon is 162 . Find the number of sides of the polygon.

o

[This object is a pull tab]

Ans

wer

20

Slide 40 / 189

Properties of Parallelograms

Slide 41 / 189

Lab - Investigating Parallelograms

Lab - Properties of Parallelograms

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

Slide 42 / 189

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 43 / 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 44 / 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 45 / 189

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

supplementary.

yo

xo

xo

yoA B

CD

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

Theorem Q5

Slide 46 / 189

Example:

ABCD is parallelogram.

Find w, x, y, and z.

A B

CD

12

2y

x-5

9

65o

5zo

wo

Slide 47 / 189

A B

CD

12

2y

x-5

9

65o

5zo

wo

The opposite sides are congruent.

Ans

wer

A B

CD

12

2y

x-5

9

65o

5zo

wo

The opposite sides are congruent.

[This object is a pull tab]A

nsw

er AB = DC2y = 92 2y = 4.5

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

Slide 48 / 189

A B

CD

12

2y

x-5

9

65o

5zo

wo

The opposite angles are congruent.

Ans

wer

A B

CD

12

2y

x-5

9

65o

5zo

wo

The opposite angles are congruent.

[This object is a pull tab]

Ans

wer m C = m A

5z = 65 5 5 z = 13

• Slide 49 / 189

A B

CD

12

2y

x-5

9

65o

5zo

woThe consecutive angles are supplementary.

Ans

wer

A B

CD

12

2y

x-5

9

65o

5zo

woThe consecutive angles are supplementary.

[This object is a pull tab]

Ans

wer m B + m A = 180o

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

Slide 50 / 189

21 DEFG is a parallelogram. Find w.

D E

FG

70o

15

3x-32w z+12

21

y2 An

swer

21 DEFG is a parallelogram. Find w.

D E

FG

70o

15

3x-32w z+12

21

y2

[This object is a pull tab]

Ans

wer

55o

Slide 51 / 189

22 DEFG is a parallelogram. Find x.

D E

FG

70o

15

3x-32w z+12

21

y2

Ans

wer

22 DEFG is a parallelogram. Find x.

D E

FG

70o

15

3x-32w z+12

21

y2

[This object is a pull tab]

Ans

wer

8

• Slide 52 / 189

23 DEFG is a parallelogram. Find y.

D E

FG

70o

15

3x-32w z+12

21

y2

Ans

wer

23 DEFG is a parallelogram. Find y.

D E

FG

70o

15

3x-32w z+12

21

y2

[This object is a pull tab]

Ans

wer

30

Slide 53 / 189

24 DEFG is a parallelogram. Find z.

D E

FG

70o

15

3x-32w z+12

21

y2

Ans

wer

24 DEFG is a parallelogram. Find z.

D E

FG

70o

15

3x-32w z+12

21

y2

[This object is a pull tab]

Ans

wer

58

Slide 54 / 189

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 55 / 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 (Answer) / 189

Example:

LMNP is a parallelogram. Find QN and MP.

L M

NP

Q

4

6(The diagonals bisect each other)

[This object is a pull tab]

Ans

wer

QN = LQLQ = 4

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

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

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

A

B E

R

S

x 4y

8 10

[This object is a pull tab]

Ans

wer x = SA = 104y = RS = 8

y = 2ER = RS + SE = 16

Slide 57 / 189

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

A sometimesB always

C never Answ

er

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

A sometimesB always

C never

[This object is a pull tab]

Ans

wer

B

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

• Slide 58 (Answer) / 189

26 MATH is a parallelogram. Find RT.

A 6

B 7

C 8

D 9 12

M A

TH

R

7

[This object is a pull tab]

Ans

wer

B

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

27 MATH is a parallelogram. Find AR.

A 6

B 7

C 8

D 912

M A

TH

R

7

[This object is a pull tab]

Ans

wer

A

Slide 60 / 189

28 MATH is a parallelogram. Find m H.

M A

TH98o

2x-4

14

(3y+8)o

Ans

wer

28 MATH is a parallelogram. Find m H.

M A

TH98o

2x-4

14

(3y+8)o

[This object is a pull tab]

Ans

wer

82

Slide 61 / 189

29 MATH is a parallelogram. Find x.

M A

TH98o

2x-4

14

(3y+8)o

Ans

wer

• Slide 61 (Answer) / 189

29 MATH is a parallelogram. Find x.

M A

TH98o

2x-4

14

(3y+8)o

[This object is a pull tab]

Ans

wer

9

Slide 62 / 189

30 MATH is a parallelogram. Find y.

M A

TH98o

2x-4

14

(3y+8)o

Ans

wer

30 MATH is a parallelogram. Find y.

M A

TH98o

2x-4

14

(3y+8)o

[This object is a pull tab]

Ans

wer

30

Slide 63 / 189

Parallelograms

Slide 64 / 189

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 65 / 189

A D and B C,

A B

CD

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

Theorem Q7

page1svg

• Slide 66 / 189

Example

Tell whether PQRS is a parallelogram. Explain.

P

Q

R

S6

6

4

4 Ans

wer

Example

Tell whether PQRS is a parallelogram. Explain.

P

Q

R

S6

6

4

4

[This object is a pull tab]

Ans

wer

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

Slide 67 / 189

Example

Tell whether PQRS is a parallelogram. Explain.P Q

RS

Ans

wer

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.

Slide 68 / 189

31 Tell whether the quadrilateral is a parallelogram.

Yes

No

78o

136o 2

Ans

wer

31 Tell whether the quadrilateral is a parallelogram.

Yes

No

78o

136o 2

[This object is a pull tab]

Ans

wer

No

• Slide 69 / 189

32 Tell whether the quadrilateral is a parallelogram.

Yes

No3 3

5

4.99

Ans

wer

32 Tell whether the quadrilateral is a parallelogram.

Yes

No3 3

5

4.99

[This object is a pull tab]

Ans

wer

No

Slide 70 / 189

33 Tell whether the quadrilateral is a parallelogram.

Yes

No

Ans

wer

33 Tell whether the quadrilateral is a parallelogram.

Yes

No

[This object is a pull tab]

Ans

wer

No

Slide 71 / 189

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 72 / 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 73 / 189

If one pair of sides of a quadrilateral is

parallel and congruent, then the

A B

CD

Theorem Q10

Slide 74 / 189

34 Tell whether the quadrilateral is a parallelogram.

Yes

No

Ans

wer

34 Tell whether the quadrilateral is a parallelogram.

Yes

No

[This object is a pull tab]

Ans

wer

No

Slide 75 / 189

35 Tell whether the quadrilateral is a parallelogram.

Yes

No141o

39o

49o

Ans

wer

35 Tell whether the quadrilateral is a parallelogram.

Yes

No141o

39o

49o

[This object is a pull tab]

Ans

wer

No

Slide 76 / 189

36 Tell whether the quadrilateral is a parallelogram.

Yes

No

89.5

819

Ans

wer

• Slide 76 (Answer) / 189

36 Tell whether the quadrilateral is a parallelogram.

Yes

No

89.5

819

[This object is a pull tab]

Ans

wer

Yes

Slide 77 / 189

37 Tell whether the quadrilateral is a parallelogram.

Yes

No

Ans

wer

37 Tell whether the quadrilateral is a parallelogram.

Yes

No

[This object is a pull tab]

Ans

wer

Yes

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

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

[This object is a pull tab]

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

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

• Slide 79 (Answer) / 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

[This object is a pull tab]

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.

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 _____________,

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

[This object is a pull tab]

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.

Slide 81 / 189

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

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)

[This object is a pull tab]

Ans

wer

E

Slide 82 / 189

39 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.

Ans

wer

• Slide 82 (Answer) / 189

39 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. [This object is a pull tab]

Ans

wer

F

Slide 83 / 189

40 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.

6

63(6-4)

Ans

wer

40 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.

6

63(6-4)

[This object is a pull tab]

Ans

wer

F

Slide 84 / 189

Constructing Parallelograms

Slide 85 / 189

To construct a parallelogram, there are 3 steps.

Construct a Parallelogram

Slide 86 / 189

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

Construct a Parallelogram - Step 1

page1svg

• Slide 87 / 189

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

Construct a Parallelogram - Step 2

Slide 88 / 189

Why is this a parallelogram?

Step 3 - Connect the endpoints of the segments.

Construct a Parallelogram - Step 3

Ans

wer

Why is this a parallelogram?

Step 3 - Connect the endpoints of the segments.

Construct a Parallelogram - Step 3

[This object is a pull tab]

Ans

wer The diagonals

bisect each other

Slide 89 / 189

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

3 steps to draw a parallelogram in a coordinate plane

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

Why is this a parallelogram?

3 steps to draw a parallelogram in a coordinate plane

Ans

wer

• Slide 91 (Answer) / 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

Why is this a parallelogram?

3 steps to draw a parallelogram in a coordinate plane

[This object is a pull tab]

Ans

wer

Remember all horizontal lines have

a slope of zero.One pair of opposite

sides are parallel and congruent.

Slide 92 / 189

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 93 / 189

41 The opposite angles of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary

Ans

wer

41 The opposite angles of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary

[This object is a pull tab]

Ans

wer

B

Slide 94 / 189

42 The consecutive angles of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary

Ans

wer

42 The consecutive angles of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary

[This object is a pull tab]

Ans

wer

D

• Slide 95 / 189

43 The diagonals of a parallelogram ______ each other.

A bisect

B congruent

C parallel

D supplementary Answ

er

43 The diagonals of a parallelogram ______ each other.

A bisect

B congruent

C parallel

D supplementary

[This object is a pull tab]

Ans

wer

A

Slide 96 / 189

44 The opposite sides of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary

Ans

wer

44 The opposite sides of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary

[This object is a pull tab]A

nsw

er

B & C

Slide 97 / 189

Rhombi, Rectanglesand Squares

Slide 98 / 189

three special parallelograms

Rhombus

Rectangle

Square

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

and square.

page1svg

• Slide 99 / 189

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 100 / 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

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[This object is a pull tab]

Ans

wer

C

Slide 101 / 189

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

A 7.25

B 12

C 20

D 25

2y+29

6y

Ans

wer

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

A 7.25

B 12

C 20

D 25

2y+29

6y[This object is a pull tab]

Ans

wer

A

Slide 102 / 189

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 103 / 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

Theorem Q12

Slide 104 / 189

Example

EFGH is a rhombus.

Find x, y, and z.E F

G H

72oz

2x-6

5y

10

Slide 105 / 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 106 / 189

Try this ...

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

8

86o

3x+2

z

12 y

2

Ans

wer

Try this ...

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

8

86o

3x+2

z

12 y

2 [This object is a pull tab]

Ans

wer x = 2

y = 4z = 470

Slide 107 / 189

47 This is a rhombus. Find x.

xoA

nsw

er

• Slide 107 (Answer) / 189

47 This is a rhombus. Find x.

xo

[This object is a pull tab]

Ans

wer

90o

Slide 108 / 189

48 This is a rhombus. Find x.

13

x-3

9 Ans

wer

48 This is a rhombus. Find x.

13

x-3

9

[This object is a pull tab]

Ans

wer

36

Slide 109 / 189

49 This is a rhombus. Find x.

126ox An

swer

49 This is a rhombus. Find x.

126ox

[This object is a pull tab]

Ans

wer

27o

Slide 110 / 189

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

H J

KL

6 M16

Ans

wer

• Slide 110 (Answer) / 189

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

H J

KL

6 M16

[This object is a pull tab]

Ans

wer

10

Slide 111 / 189

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 112 / 189

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

6y

12

Ans

wer

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

6y

12

[This object is a pull tab]A

nsw

er

15

Slide 113 / 189

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 114 / 189

Example

RECT is a rectangle. Find x and y.

2x-513

63o9yo

R E

CT

Ans

wer

• Slide 114 (Answer) / 189

Example

RECT is a rectangle. Find x and y.

2x-513

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

Slide 115 / 189

52 RSTU is a rectangle. Find z.R S

TU8z

Ans

wer

52 RSTU is a rectangle. Find z.R S

TU8z

[This object is a pull tab]

Ans

wer

11.25

Slide 116 / 189

53 RSTU is a rectangle. Find z.R S

TU

4z-9

7

Ans

wer

53 RSTU is a rectangle. Find z.R S

TU

4z-9

7

[This object is a pull tab]

Ans

wer

4

Slide 117 / 189

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 118 / 189

Example

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

z - 4

(5x)o

6

3y

Ans

wer

Example

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

z - 4

(5x)o

6

3y

[This object is a pull tab]

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

Slide 119 / 189

Try this ...

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

3y

12z

8y - 1

0

(x2 + 9)o

Ans

wer

Try this ...

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

3y

12z

8y - 1

0

(x2 + 9)o

[This object is a pull tab]

Ans

wer x = 6

y = 2 z = 7.5

Slide 120 / 189

54 The quadrilateral is a square. Find y.

A 2

B 3

C 4

D 5

18y

Ans

wer

54 The quadrilateral is a square. Find y.

A 2

B 3

C 4

D 5

18y

[This object is a pull tab]

Ans

wer

5

• Slide 121 / 189

55 The quadrilateral is a rhombus. Find x.

A 2

B 3

C 4

D 5

2x + 6

4x

Ans

wer

55 The quadrilateral is a rhombus. Find x.

A 2

B 3

C 4

D 5

2x + 6

4x

[This object is a pull tab]

Ans

wer

3

Slide 122 / 189

112o

(4x)o

56 The quadrilateral is parallelogram. Find x.A

nsw

er

112o

(4x)o

56 The quadrilateral is parallelogram. Find x.

[This object is a pull tab]

Ans

wer

17

Slide 123 / 189

57 The quadrilateral is a rectangle. Find x.

10x

3x + 7

Ans

wer

57 The quadrilateral is a rectangle. Find x.

10x

3x + 7

[This object is a pull tab]

Ans

wer

1

• Slide 124 / 189

Opposite sidesare

Diagonals bisectopposite

• Slide 130 / 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 131 / 189

Slide 131 (Answer) / 189 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

59 The quadrilateral is an isosceles trapezoid. Find x.

A 3

B 5

C 7

D 9 64o (9x + 1)o

[This object is a pull tab]

Ans

wer

C

Slide 133 / 189

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 134 / 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 135 / 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 136 / 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 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

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

[This object is a pull tab]

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

Slide 138 / 189

60 The trapezoid is isosceles. Find x.

9

4

6x + 3

2x + 2 An

swer

• Slide 138 (Answer) / 189

60 The trapezoid is isosceles. Find x.

9

4

6x + 3

2x + 2

[This object is a pull tab]

Ans

wer

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

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

x = 1

Slide 139 / 189

61 The trapeziod is isosceles. Find x.

137o

xo

Ans

wer

61 The trapeziod is isosceles. Find x.

137o

xo

[This object is a pull tab]

Ans

wer

143

Slide 140 / 189

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

H I

JK

Ans

wer

YesNo

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

H I

JK

YesNo

[This object is a pull tab]

Ans

wer

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

Slide 141 / 189

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

midsegment of a trapezoid

Lab - Midsegments of a Trapezoid

Click on the link below and complete the lab.

• Slide 142 / 189

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 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

15

7

Example

PQRS is a trapezoid. Find LM.

[This object is a pull tab]

Ans

wer

Slide 144 / 189

P

Q R

S

L M

20

14.5

Example

PQRS is a trapezoid. Find PS.

Ans

wer

P

Q R

S

L M

20

14.5

Example

PQRS is a trapezoid. Find PS.

[This object is a pull tab]

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

• Slide 145 (Answer) / 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.

[This object is a pull tab]

Ans

wer x = 18

y = 7z = 5

Slide 146 / 189

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

H I

JK

E F

6

x

15

Ans

wer

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

H I

JK

E F

6

x

15

[This object is a pull tab]

Ans

wer

10.5

Slide 147 / 189

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

HI

J K

EF

x

19

10

Ans

wer

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

HI

J K

EF

x

19

10

[This object is a pull tab]

Ans

wer

1

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

• Slide 148 (Answer) / 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.

[This object is a pull tab]

Ans

wer

B and D

Slide 149 / 189

Kites

Slide 150 / 189

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

kites

Lab - Properties of Kites

Click on the link below and complete the lab.

Slide 151 / 189

In kite ABCD,

• Slide 154 / 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 155 / 189

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

A EA

E

A

D

Ans

wer

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

A EA

E

A

D

[This object is a pull tab]

Ans

wer

A

Slide 156 / 189

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

A EB D

C RR

E

A

D

Ans

wer

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

A EB D

C RR

E

A

D

[This object is a pull tab]

Ans

wer

C

Slide 157 / 189

68 Find the value of z in the kite.

z 5z-8

Ans

wer

• Slide 157 (Answer) / 189

68 Find the value of z in the kite.

z 5z-8

[This object is a pull tab]

Ans

wer

2

Slide 158 / 189

69 Find the value of x in the kite.

68o

(8x+4)o

44o

Ans

wer

69 Find the value of x in the kite.

68o

(8x+4)o

44o

[This object is a pull tab]

Ans

wer

15

Slide 159 / 189

70 Find the value of x.

36

(3x 2 + 3)

24 Ans

wer

o

o

o

70 Find the value of x.

36

(3x 2 + 3)

24

o

o

o

[This object is a pull tab]

Ans

wer

7

Slide 160 / 189

Theorem Q19

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

In kite ABCDAC BD

A

B

C

D

• Slide 161 / 189

71 Find the value of x in the kite.

x

Ans

wer

71 Find the value of x in the kite.

x

[This object is a pull tab]

Ans

wer

90o

Slide 162 / 189

72 Find the value of y in the kite.

12y

Ans

wer

72 Find the value of y in the kite.

12y

[This object is a pull tab]A

nsw

er

7.5

Slide 163 / 189

Slide 164 / 189

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

kite trapezoidparallelogram

rhombus

square

rectangle isosceles trapezoid

Every rhombus is a special kite

page1svg

• Slide 165 / 189

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

squarerectanglerhombusparallelogram kite

trapezoid isosceles trapezoid

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

Slide 166 / 189

Kite

Trapezoid

IsoscelesTrapezoid

Parallelogram

Rhombus Rectangle

Squa

re

Rhombus

Slide 167 / 189

73 A rhombus is a square.

A alwaysB sometimes

C never

Ans

wer

73 A rhombus is a square.

A alwaysB sometimes

C never

[This object is a pull tab]A

nsw

er

B

Slide 168 / 189

74 A square is a rhombus.

A alwaysB sometimes

C never

Ans

wer

74 A square is a rhombus.

A alwaysB sometimes

C never

[This object is a pull tab]

Ans

wer

A

• Slide 169 / 189

75 A rectangle is a rhombus.

A alwaysB sometimes

C never

Ans

wer

75 A rectangle is a rhombus.

A alwaysB sometimes

C never

[This object is a pull tab]

Ans

wer

C

Slide 170 / 189

76 A trapezoid is isosceles.

A alwaysB sometimes

C never

Ans

wer

76 A trapezoid is isosceles.

A alwaysB sometimes

C never

[This object is a pull tab]A

nsw

er

B

Slide 171 / 189

77 A kite is a quadrilateral.

A alwaysB sometimes

C never

Ans

wer

77 A kite is a quadrilateral.

A alwaysB sometimes

C never

[This object is a pull tab]

Ans

wer

A

• Slide 172 / 189

78 A parallelogram is a kite.

A alwaysB sometimes

C never

Ans

wer

78 A parallelogram is a kite.

A alwaysB sometimes

C never

[This object is a pull tab]

Ans

wer

C

Slide 173 / 189

Coordinate Proofs

Slide 174 / 189

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 175 / 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 176 / 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.

page1svg

• Slide 177 / 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 178 / 189J (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 179 / 189J (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 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

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

[This object is a pull tab]

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

Slide 181 / 189

Proofs

page1svg

• Slide 182 / 189

Given: TE MA,

• Slide 188 / 189C 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 189 / 189

Try this ...

Given: FCD FEDProve: FD CE

F

C

D

E

Hin

t

Try this ...

Given: FCD FEDProve: FD CE

F

C

D

E

[This object is a pull tab]

Hin

t

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

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