Chapter 6 Notes

6.1 – Polygons

Poly Many

A polygon is a plane figure that meets the following conditions:

1) It is formed by three or more segments called sides such that no two sides with a common endpoint are collinear.

2) Each side intersects exactly two other sides, one at each endpoint.

Names of polygons

# of sides Name

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

8 Octagon

10 decagon

12 dodecagon

n n-gon

When naming polygons, you list the vertices in order!

P R

T

BL

E

Hexagon PRTBLE

Or

Hexagon TBLEPR

Or other names

Convex A polygon such that no line containing a side of a polygon contains a point in the interior of the polygon.

Equilateral Sides are the same.

Equiangular Angles are the same.

Regular Both

Is it a polygon? If so, name it and say if it is convex or concave.

Diagonal – Segment that joins two nonconsecutive vertices

Interior angles of a Quadrilateral

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

3604321 mmmm

2xo

xo

100o

x+20o2xo

xo

100o

x+20o

Solve for x

(3x + 2)o

(2x – 10 )o (x + 5)o

(2x – 7)o

6.2 – Properties of Parallelograms

Definition of a parallelogram Both pairs of opposite sides are parallel.

A

B

D

C

M

DCAB,BCAD

congruent are sides opposite its

then -gram,|| is ABCD quad If

6.2 Theorem

DBC,A

congruent are angles opposite

then-gram,|| is ABCD quad If

6.3 Theorem

180AmDm;180DmCm

180CmBm;180BmAm

arysupplement are angles econsecutiv

then-gram,|| is ABCD quad If

6.4 Theorem

DMBM,CMAM

othereach bisect diagonals its

then -gram,|| is ABCD quad If

6.5 Theorem

A

B

D

C

M

Find all information

BC

10 AD

BD

8BM

AM

10AC

Dm

Cm

Bm

Am

4xBm

10xAm 10x 4x

x + 24

4x – 12

65 y

z

7y

3y + 28

30oxo

zo

wo

30o

9q + 4

write

Proving Theorem 6.2

A

B

D

C

DCAB,BCAD:Prove

ramparallelog a is ABCD :Given

A

B

D

C

E

F

isosceles is DEF:Prove

FCFDE,A

-gram,|| is ABCD :Given

A

B

D

C

E

DB :Prove

-gram,|| is EYDZBXEF, :Given

F

X

Y

Z

12

6.3 – Proving Quadrilaterals are Parallelograms

T

Q

S

R1 4

23

THRM 6.6 If both pairs of opposite sides of a quad are congruent, then the quadrilateral is a parallelogram.

THRM 6.8, If an angle of a quad is supplementary to both its consecutive angles,

then the quadrilateral is a ||- gram

THRM 6.7 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a ||-gram

THRM 6.9 If the diagonals of a quad bisect each other, then the quadrilateral is a ||-gram

Opposite sides are parallel, then it’s a ||-gram by def.

T

Q

S

R1 4

23

M

THRM 6.6 If both pairs of opposite sides of a quad are congruent, then the quadrilateral is a parallelogram.

THRM 6.9 If the diagonals of a quad bisect each other, then the quadrilateral is a ||-gram

Let’s discuss how to prove two of these theorems

T

Q

S

R1 4

23

M

THRM 6.10, If one pair of opposite sides are both CONGRUENT and PARALLEL, then the quadrilateral is a ||-gram

Yes parallelogram, not a parallelogram, and why?

Prove that the following coordinates make a parallelogram by the given theorems\definitions using slope formula and distance formula.

The diagonals of a quad bisect each other

Opposite sides are parallel, then it’s a ||-gram by def.

One pair of opposite sides are both CONGRUENT and PARALLEL.

(-1, 3)(3, 2)

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

AB

DC

12

12

xx

yySlope

212

212Distance yyxx

12

12

xx

yySlope

212

212Distance yyxx

Do these points make a parallelogram?

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

Do these points make a parallelogram?

(-1,-3) (-2, 1) (2, 2) (1, -3)

6.4 – Rhombuses, Rectangles, and Squares

Rectangle – Quad with 4 rt angles.

Rhombus – Quad with 4 congruent sides

Square – 4 rt angles AND 4 congruent sides, It’s a rectangle AND a rhombus!!

Why are these parallelograms?

World O’ Parallelograms

Answer with always, sometimes, or never

A Rhombus is a Square Always Sometimes Never

A Square is a Parallelogram Always Sometimes Never

Rhombus Corollary A quadrilateral is a rhombus iff it has four congruent sides.

Rectangle Corollary A quadrilateral is a rectangle iff it has four right angles.

Square Corollary A quadrilateral is a square iff it is a rhombus and a rectangle.

Corollaries

Thrm 6.11 A ||-gram is a rhombus iff its diagonals are perpendicular.

A

B

D

C

M

Thrm 6.12 A ||-gram is a rhombus iff each diagonal bisects a pair of opposite angles.

A

B D

C

BDAC iffrhombusisABCD

ADC and ABC bisects BD

and BCD and BAD bisects

iffrhombusisABCD

AC

Thrm 6.13 A ||-gram is a rectangle iff its diagonals are congruent.A

B

D

C

BDAC rectangleisABCD

Prove this theorem.

Stuwork

• Which shape could it be?

• What can be true about it?

70o

xo

Solve for x and y.

4y – 10 2y + 16

Given the figure is a rectangle,

Solve for x and y.

(2x + 5)o

5y – 10

2y + 35

Stuwork

55o

xo

Solve for x and y.

y

Given the figure is a square,

Solve for x and y.

AC = 4x – 10

Stuwork

A B

D C

BD = 2x + 2

yo105

A ||-gram is a rectangle iff its diagonals are congruent. We’ll now prove this using coordinate proofA

B

D

C

BDAC :Prove

rectangleisABCD :Given

Stuwork

( , )

( , )

( , )

( , )

Well do some coordinate proof stuff with rhombus, rectangles, and squares added on.

6.5 – Trapezoids and Kites

A quadrilateral with EXACTLY one pair of parallel sides is called a TRAPEZOID. The parallel sides are called BASES. The other sides are LEGS

ISOSCELES TRAPEZOID – LEGS are CONGRUENT!

Trapezoids have two pairs of base angles.

KITE – A quadrilateral with two pairs of consecutive sides, BUT opposite sides aren’t congruent.

Theorem 6.14 Base angles of an isosceles trapezoid are congruent.

BA Y;X :Prove

AXBY with ABYX Trapezoid :Given

X Y

A B

Z

Congruent, opp sides ||-gram congruent.

Transitive to work all sides congruent.

Corresponding, base angles thrm, transitive.

Same side interior angles, measure, supp, subtraction.

ONLY TRUE FOR ISOS TRAPEZOID, NOT REGULAR TRAPEZOID!

A B

C D

A B

C D

trapezoidisosisABCDthen

DCtrapezoidaABCD ,

BCADifftrapisosisABCD

Theorem 6.15 If a trapezoid has one pair of congruent base angles, then it’s an isosceles trapezoid.

Theorem 6.16 A trapezoid is isosceles iff its diagonals are congruent.

basesbothofaverage

theisandbasesboth

toismidsegmentThe

THEOREMMIDSEGMENTThrm

||

17.6

221 bb

midsegment

Just like in a triangle, the midsegment will go through the midpoint of the legs.

A B

C D

b1

b2

midsegmentE F

A B

C D

10

18

xE F

A B

C D

12

y

15E F

A B

C D

z

14

11E F

A B

C D

142x+2

5y+10 7y-10

x

13

y

19

z

A B

C D

R

Write in

AD = 15 AR = 6

BC = _____ BR = __

RC = ____ RD = __

___

___

___

110

BDCm

ABDm

ACDm

BACm

45

x

55

y22

44

z

A

B D

C

A

B D

C

Theorem 6.18

If a quadrilateral is a kite, then its diagonals are perpendicular

Theorem 6.18

If a quadrilateral is a kite, then EXACTLY one pair of opposite angles are congruent.

BDAC

DBCA ;

A

B D

C

A

B D

C

12

5

x

60o

140o

yo

zo

A

B D

C

A

B D

C

24

x

25

zo

100o

yo

70o

• Go Over HW

• Verify Quadrilateral while HW checked

• Properties of shapes of Parallelogram, Rhombus, Rectangle, Square

• Discuss what’s on quiz

6.6 – Special Quadrilaterals

Quad-Tree-LateralsQuadrilateral

Kite Parallelogram Trapezoid

Rhombus Rectangle

Square

Isosceles Trapezoid

For which shape(s) is/are the following always true?

Make a table with the following shapes as the heading:

Parallelogram

Rectangle

Rhombus

Square

Trapezoid

Isosceles Trapezoid

Kite

6.7 – Areas of Triangles and Quadrilaterals

Postulate: The area of a square is the square of the length of its side A = s2

Area congruence Postulate If two figures are congruent, they have the same area.

s

Area Addition Postulate The area of a region is the sum of the areas of the non overlapping part. Just means you can cut up the parts and add them together.

In a parallelogram, a side can be considered the base. The line perpendicular to the base and going to the other side is the altitude. The length of the altitude is called the height. (Altitude segment, height number)

Area of a rectangle equals the product of the base and height. (A = bh)

b

h

b

a

Area of Parallelogram

A = bh

b

h

bh2

1A

trianglea of Area

b1

b2

h

2

)b(bA

: trapezoida of Area

21 h

21dd2

1A

rhombus a of Area

d1

d2

21 d

2

1d

2

1A

triangles theof One

21dd4

1A

2121 dd2

1dd

4

122A

Kite works the same way.

4

5

1010

2222

54 8

15

10

Write bottom two

10

The sides of a rectangle are x and x – 14. The area is 120. Find the value of x.

x

x – 14

h

10

Area is 40 ft2. Find h

x

x

16

Area is 80 ft2. Find x

812 17

6

8

12

14

6

4

5 11

4

14

Find Area

Write bottom two

Find Area

Write bottom two