Lesson 6.1 Polygons

Lesson 6.1Polygons

Today, we will learn to…> identify, name, and describe polygons > use the sum of the interior angles of a quadrilateral

# of Sides Name3

4

5

6

7

8

9

10

12

trianglequadrilateral

pentagonhexagonheptagon

octagonnonagon

decagon

dodecagon

Theorem 6.1Interior Angles of a

Quadrilateral

The sum of the measures of the interior angles of a quadrilateral is ______360°

Section 6.1 Vocabulary

ConvexConcave

EquilateralEquiangular

RegularDiagonal

Sides:

Vertices:

Diagonals:

S

T

U

DY

ST TU UD DY YS

S, T, U, D, Y

SU SD TD TY UY

S

T

U

DY

There are 10 possible names of this pentagon.

STUDYSYDUTTUDYSTSYDUUDYSTUTSYD

DYSTUDUTSYYSTUDYDUTS

How many diagonals can be drawn from N?

N M

O

PQ

R

Starting with N, give 2 names for the hexagon.

N M

O

PQ

R

NMOPQR NRQPOM

Is this a polygon? If not, explain. If so, is it

convex or concave?

Yes, it’s a convex

pentagon


convex or concave?

No, polygons must be made of

segments


convex or concave?

Yes, it’s a concave

dodecagon


convex or concave?

No, polygons must be closed

figures

Find x.

90 + 87 + 93 + x = 360x = 90

Find x.

3x + 3x + 2x + 2x = 360x = 36

Lesson 6.2Properties of Parallelograms

RULERS AND PROTRACTORS

Today, we will learn to…> use properties of parallelograms

A quad is a parallelogram if and only if two pairs of opposite sides are parallel

parallelogram

Draw a Parallelogram.

Measure each angle.Measure each side in centimeters.

Theorems 6.2-6.5If a quadrilateral is a parallelogram, then…

1) 6.22) 6.33) 6.44) 6.5

… opposite sides are __________congruent

… opposite angles are__________.congruent

… consecutive angles are__________.supplementary

1 2

34

m m m m

m m m m

1 2 180 1 4 180

3 2 180 3 4 180

… diagonals __________each other.

bisect

ABCD is a parallelogram. Find the missing angle and side measures.

1.A B

CD

105˚10

66

10

75˚

75˚

105˚

ABCD is a parallelogram. Find AC and DB.

2. A

CD

8

85

B

5

AC = 10 DB = 16

3. In ABCD, m C = 115˚. Find mA and mD.

4. Find x in JKLM.J K

LM(4x-9)˚

(3x+18)˚

mA = 115˚ mD = 65˚

x = 27

ABCD is a parallelogram.

EC =

m BCD =

m ADC =

AD =

5

8

70° 110°

The figure is a parallelogram.

x = y = 5 4

2x – 6 = 4 2y = 8


x = y = 30 6 4x + 2x = 180 2y + 3 = y + 9


x = y = 3 6

y

y

3x + 1 = 10 2y – 1 = y + 5


x = y = 40 8 3x – 9 = 2x + 31 4y + 5 = 2y + 21

Lesson 6.3Proving that Quadrilaterals

are Parallelograms

What is a converse?

Today, we will learn to…> prove that a quadrilateral is a

parallelogram

Theorem 6.6

If both pairs of opposite sides are __________,

then it is a parallelogram.congruent

Theorem 6.7If both pairs of opposite angles are __________,

then it is a parallelogram.congruent

Is ABCD a parallelogram? Explain.

1. 2.A B

CD

10

6

10

6

A B

CDyes

no

Theorem 6.8If an angle is

_______________ to both of its consecutive angles, then it is a parallelogram.

supplementary

1

2

3 m1 + m3 = 180˚m1 + m2 = 180˚

Theorem 6.9If the diagonals

__________________, then it is a parallelogram.

bisect each other

AE = ECand

DE = EB

A

D

B

C

E

Is ABCD a parallelogram? Explain.

3. 4. A B

CD

A B

CD

104˚

86˚ 104˚

no yes

Theorem 6.10If one pair of opposite sides are ___________

and __________, then it is a parallelogram.

congruentparallel

5.

8.

7.

6.

No Yes

Yes No

9. List 3 ways to prove that a quadrilateral is a parallelogram

1) prove that both pairs of opposite sides are __________

2) prove that both pairs of opposite sides are __________3) prove that one pair of opposite sides are both ________ and ________

parallel

congruent

parallel congruent

A ( , ) B ( , ) C ( , ) D ( , )

Prove that this is a parallelogram…

slope of AB isslope of BC isslope of CD isslope of AD is

0

4-2/5

-2/5

AB =BC =CD = AD =

4.15.44.15.4

2 3 4 -2 6 -3 2

4

Lesson 6.4Special

Parallelograms

Today, we will learn to…> use properties of a rectangle,

a rhombus, and a square

A square is a parallelogram with four congruent sides and four right angles.

A rhombus is a parallelogram with

four congruent sides.

A rectangle is a parallelogram with four right angles.four congruent sides. four right angles.

four congruent sides four right angles

parallelograms

rhombuses rectangles

squares

Sometimes, always, or never true?

1. A rectangle is a parallelogram.

2. A parallelogram is a rhombus.

3. A square is a rectangle.

4. A rectangle is a rhombus.

5. A rhombus is a square.

always true

sometimes true

always true

sometimes true

sometimes true

Geometer’s Sketchpad

mAEB = 90CD = 4.48 cmBC = 4.48 cmAD = 4.48 cmAB = 4.48 cm

E

C

A B

DWhat do we know about the diagonals in a

rhombus?

The diagonals of a rhombus are _____________.perpendicular

Theorem 6.11

What do we know about the diagonals in a rhombus?

mECD = 40

mEDA = 50 mEDC = 50

mEAD = 40 mEAB = 40

mECB = 40mEBC = 50 mEBA = 50

E

C

A B

D

The diagonals of a rhombus _____________________.bisect opposite angles

Theorem 6.12

What do we know about the diagonals in a rectangle?

ED = 4.51 cmEB = 4.51 cm

EC = 4.51 cmEA = 4.51 cm

E

C

A B

D

The diagonals of a rectangle are _____________. congruent

Theorem 6.13

6. In the diagram, PQRS is a rhombus. What is the value of y?

2y + 3

5y – 6

P Q

RS

y = 3

Find x. 7. rhombus

A

B

C

Dxº

52º

x = 38º

Find m CDB. 8. rhombus

A

B

C

D32º

mCDB =32º

Find AB.9. rectangle

A B

CD

10 12

AB = 16

?

202 = x2 + 122

10

Find x.10. square

A B

CD

xº xº

x = 45˚

Find EA & AB.11. square

EA =

A B

CD

4

EAB = 5.7

x2 = 42 + 42

x2 = 16 + 16x2 = 32x = 5.7

4

4

Lesson 6.5Trapezoids

& Kites

Today, we will learn to…> use properties of trapezoids

and kites

A trapezoid is a quadrilateral with only

one pair of parallel sides.

A B

D C

base

base

leg leg

B A

D

C

Compare leg angles.


mC = 65mD = 115mA = 90mB = 90

In ALL trapezoids, leg angles are

_______________supplementary

A trapezoid is an

isosceles trapezoid

if its legs are congruent.


Compare base angles.Compare leg angles.How do you know it is isosceles?

mA = 67 mD = 67 mC = 113 mB = 113 CD = 3.7 cmAB = 3.7 cm

A D

B C

Theorem 6.14 & 6.15A trapezoid is isosceles if and

only if base angles are ___________.congruent

Base angles are congruent.

A B

CDAC BD

The trapezoid is isosceles.

The triangles share CD.ADC BCD by SAS

CPCTC

Theorem 6.16A trapezoid is isosceles if

and only if its diagonals are __________.congruent

AC BD

A B

CD

ABCD is an isosceles trapezoid. Find the missing angle measures.

1. A B

CD100°

80° 80°

100°

2. The vertices of ABCD are A(-1,2), B(-4,1), C(4,-3), and D(3,0). Show that ABCD is an isosceles trapezoid.

Figure is graphed on next slide.

3

2

1

-1

-2

-3

-4

-6 -4 -2 2 4 6

D(3, 0)

C(4, -3)

B(-4, 1)

A(-1, 2)

AD || BC ?

AB =CD =

- ½ - ½

Legs are ? Diagonals are ? AC=BD =

50 10 10 50

OR?

Slope of AD isSlope of BC is

x = 118 Find x.

The midsegment is a segment that connects the midpoints of

the 2 legs of a trapezoid.

Geometer’s SketchPad

EF = 8 cmCD = 12 cm

AB = 4 cm

EF = 7 cmCD = 11 cm

AB = 3 cm

A

EF = 5 cmCD = 6 cm

AB = 4 cm

EF = 7 cmCD = 9 cm

AB = 5 cm

FE

A B

D C

Theorem 6.17Midsegment Theorem for

TrapezoidsThe midsegment of a

trapezoid is _________ to each base and its length is ______________ of the

bases.

parallel

the average

Find x.

3. 4.

7

11

x

x

17

20

x = 9 x = 23

KITE

A kite has two pairs of consecutive congruent

sides but opposite sides are not congruent and no sides

are parallel.

Kite

What do we know if these points are equidistant from the endpoint of the segment?

Theorem 6.18

In a kite, the longer

diagonal is the _________________

of the shorter diagonal.perpendicular bisector

Kite

What do we know about congruent triangles?

How do we know the triangles are congruent?

Kite

Theorem 6.19In a kite, exactly one pair of opposite angles

are ________.congruent

The congruent angles are formed by the noncongruent sides.

Find x and y.

5. 6.

5

x yx˚ 125˚

y˚

(y+30)˚29

x = 2 y = 2

x = 125

y = 40

Theorem 6.19*

In a kite, the longer diagonal

________________.bisects opposite angles

mJ =70°

mL = 70°

Find the missing angles.

x =35

Find x.

Find x.

x = 110

Find x.

x = 5

Based on our theorems, list all of the properties that must be true for the quadrilateral.

1. Parallelogram (definition plus 4 facts)

2. Rhombus (plus 3 facts)

3. Rectangle (plus 2 facts)

4. Square (plus 5 facts)

Parallelogram

1) opposite sides are parallel

2) opposite sides are congruent

3) opposite angles are congruent

4) consecutive angles are supplementary

5) diagonals bisect each other

Rhombus1) equilateral2) diagonals are perpendicular3) diagonals bisect opposite angles

Rectangle1) equiangular2) diagonals are congruent

Square1) equilateral2) equiangular3) diagonals are perpendicular4) diagonals bisect opposite angles5) diagonals are congruent

Lesson 6.6Identifying Special

Quadrilaterals

Complete the chart of characteristics of special quadrilaterals.

Today, we will learn to…> identify special quadrilaterals

with limited information

Given the following coordinates, identify the quadrilateral.

(-2, 1)(-2, 3)(3, 6) (0, 1)

kite


(0, 0)(4, 0)(3, 7) (1, 7)

trapezoid


rectangle

(-1, -3)(4, -3)(4, 3) (-1, 3)


rhombus

(-2, 0)(3, 0)(6, 4) (1, 4)

In quadrilateral WXYZ, WX = 15, YZ = 20, XY = 15,

ZW = 20. What is it?

It is a kite!

Lesson 6.7Areas of Triangles and

Quadrilaterals

Today, we will learn to…> find the area of triangles and

quadrilaterals

Postulate 22Area of a Square

Area = side2

A=s2

Postulate 23Area Congruence Postulate

If two polygons are congruent, then they have the same area.

Theorem 6.20Area of a Rectangle

Area = base ( height )

A = bh

1. Find the area of the polygon made up of rectangles.

4 m

10 m

2 m

9 m

11 m

7 m11(2) = 22 m2

8(4) =

32 m2

5(4)= 20 m2

74 m2

?

??

Postulate 24

Area Addition Postulate

The area of a region is the sum of

the areas of its nonoverlapping

parts.

Theorem 6.21Area of a Parallelogram

Area = base ( height)

A=bh

Do experiment.

Theorem 6.22Area of a Triangle

A=½ bh

Area of a Trapezoid

hh

b2

A = ½ h b1 + ½ h b2

b1

A = ½ h (b1 + b2)

A = ½ h b1 + ½ h b2

Theorem 6.23Area of a Trapezoid

A = ½ height (sum of bases)

A=½ h (b1+b2)

2. parallelogram 3. trapezoid

6

4 55 5

3

4

9

A = 6(4)

A = 24 units2A = ½ 4(9+3)

A = 24 units2

Area of a Kite

b

b

x

y

A = ½ bx + ½ by

A = ½ b (x + y)What is b? a diagonal

What is x + y? a diagonal

A = ½ d1 d2

Theorem 6.24Area of a Kite

Area = ½ (diag.)(diag.)

A=½ d1 d2

Area of a RhombusA = ½ bx + ½ by

A = ½ b(x + y)What is b? a diagonal

What is x + y? a diagonal

A = ½ d1 d2

b

b

x

y

Theorem 6.25Area of a Rhombus

Area = ½ (diag.)(diag.)

A=½ d1 d2

4. Rhombus 5. Kite

4

35

34

A = ½ 6(8)

A = 24 units2

A = ½ 6(9)

A = 27 units2

6. Rhombus 7. Trapezoid

8

x

A = 80 units2

x = 5

A = 55 units2

h = 5

h

13

9

8. Find the total area.

15

8 A = ½(10)(8+20)

A = 440 units2

20

25A = 140

A = 20(15)

A = 300

?10

A = 12(11)

blue A = ½ (12)(5)

11

12

A = 132

132 = 122 + x2

x = 513

just blue?

blue A = 30

pink A = 132 – 60

pink A = 72

2 blue regions A = 60

?5

9. Find the areas of the blue and pink regions.

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Lesson 6.1 Polygons