Geometry...

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2014-06-03

Quadrilaterals

Geometry

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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

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Angles of Polygons

Return to the Table of Contents

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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

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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

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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

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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

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1 The figure below is a polygon.

True

False

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2 The figure below is a polygon.

True

False

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3 Indentify the polygon.

A Pentagon

B Octagon

C Quadrilateral

D Hexagon

E Decagon

F Triangle

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4 Is the polygon convex or concave?

A Convex

B Concave

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5 Is the polygon convex or concave?

A ConvexB Concave

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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

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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

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A Pentagon

B Octagon

C Quadrilateral

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular

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A Pentagon

B Octagon

C Quadrilateral

D Hexagon

E Triangle

F Convex

G Concave

H Equilateral

I Equiangular

J Regular

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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

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Polygon Number of S ides

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

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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

_____________

_____________

_____________

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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 S ides

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

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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.

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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

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

180(n-2)n

Complete the table.

regular polygon number of s idessum of interior

anglesmeasure 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

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

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

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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

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

A 162

B 3240

C 180

D 60

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

A 2520 B 2880 C 3240 D 157.5

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14 What is the value of x?

(5x+

15)o

(9x-6) o

(8x) o

(11x+16)o

(10x+8)o

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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

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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

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

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

A 72

B 60C 51.43

D 45

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

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

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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.

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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

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

o

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Properties of Parallelograms


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Lab - Investigating Parallelograms

Lab - Properties of Parallelograms

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

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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

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page1svg

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

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

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.

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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

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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

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Example:

ABCD is parallelogram.

Find w, x, y, and z.

A B

CD

12

2y

x-5

9

65o

5zo

wo

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A B

CD

12

2y

x-5

9

65o

5zo

wo

The opposite sides are congruent.

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A B

CD

12

2y

x-5

9

65o

5zo

wo

The opposite angles are congruent.

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A B

CD

12

2y

x-5

9

65o

5zo

wo

The consecutive angles are supplementary.

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21 DEFG is a parallelogram. Find w.

D E

FG

70o

15

3x-32w z+12

21

y2

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22 DEFG is a parallelogram. Find x.

D E

FG

70o

15

3x-32w z+12

21

y2

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23 DEFG is a parallelogram. Find y.

D E

FG

70o

15

3x-32w z+12

21

y2

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24 DEFG is a parallelogram. Find z.

D E

FG

70o

15

3x-32w z+12

21

y2

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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

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Example:

LMNP is a parallelogram. Find QN and MP.

L M

NP

Q

4

6(The diagonals bisect each other)

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

A

B E

R

S

x 4y

8 10

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

A sometimes

B always

C never

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26 MATH is a parallelogram. Find RT.

A 6

B 7

C 8

D 9 12

M A

TH

R

7

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27 MATH is a parallelogram. Find AR.

A 6

B 7

C 8

D 912

M A

TH

R

7

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28 MATH is a parallelogram. Find m H.

M A

TH98o

2x-4

14

(3y+8)o

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29 MATH is a parallelogram. Find x.

M A

TH98o

2x-4

14

(3y+8)o

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30 MATH is a parallelogram. Find y.

M A

TH98o

2x-4

14

(3y+8)o

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Proving Quadrilaterals are

Parallelograms


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page1svg

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.

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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

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Example

Tell whether PQRS is a parallelogram. Explain.

P

Q

R

S6

6

4

4

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Example

Tell whether PQRS is a parallelogram. Explain.P Q

RS

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31 Tell whether the quadrilateral is a parallelogram.

Yes

No

78o

136o

2

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Yes

No3 3

5

4.99

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Yes

No

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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

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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

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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

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Yes

No

Slide 74 / 189


Yes

No141o

39o

49o

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Yes

No

89.5

819

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Yes

No

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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

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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

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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

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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)

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Slide 82 / 189




6

63(6-4)

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Constructing Parallelograms


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page1svg

To construct a parallelogram, there are 3 steps.

Construct a Parallelogram

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

Construct a Parallelogram - Step 1

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


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Why is this a parallelogram?

Step 3 - Connect the endpoints of the segments.


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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.

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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.


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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?


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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

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41 The opposite angles of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary

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42 The consecutive angles of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary

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43 The diagonals of a parallelogram ______ each other.

A bisect

B congruent

C parallel

D supplementary

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44 The opposite sides of a parallelogram are ...

A bisect

B congruent

C parallel

D supplementary

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Rhombi, Rectanglesand Squares


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three special parallelograms

Rhombus

Rectangle

Square

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

and square.

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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

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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

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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

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

A B

CD

If ABCD is a rhombus,

then AC BD.

Theorem Q11

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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

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Example

EFGH is a rhombus.

Find x, y, and z.E F

G H

72o

z

2x-6

5y

10

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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.

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Try this ...

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

8

86o

3x+2

z

12 y2

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47 This is a rhombus. Find x.

xo

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13

x-3

9

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126ox

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

H J

KL

6 M16

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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

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

6y

12

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

If ABCD is a rectangle,

then AC BD.

A B

CD

Theorem Q13

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Example

RECT is a rectangle. Find x and y.

2x-5 13

63o9yo

R E

CT

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52 RSTU is a rectangle. Find z.R S

TU8z

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53 RSTU is a rectangle. Find z.R S

TU

4z-9

7

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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

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Example

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

z - 4

(5x)o

6

3y

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Try this ...

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

3y

12z

8y - 1

0

(x2 + 9)o

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54 The quadrilateral is a square. Find y.

A 2

B 3

C 4

D 5

18y

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55 The quadrilateral is a rhombus. Find x.

A 2

B 3

C 4

D 5

2x + 6

4x

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112o

(4x)o

56 The quadrilateral is parallelogram. Find x.

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57 The quadrilateral is a rectangle. Find x.

10x

3x + 7

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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

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Lab - Quadrilaterals in the Coordinate Plane

Click on the link below and complete the lab.

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Trapezoids

Return to theTable of Contents

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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

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An isosceles trapezoid is a trapezoid with congruent legs.

isosceles trapezoid

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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

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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

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59 The quadrilateral is an isosceles trapezoid. Find x.

A 3

B 5

C 7

D 9 64o (9x + 1)o

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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

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Example

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

112o 147o

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

P

R

Q

S

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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

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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 = 11

OR

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

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

Option B

o

o o

o

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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

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60 The trapezoid is isosceles. Find x.

9

4

6x + 3

2x + 2

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61 The trapeziod is isosceles. Find x.

137o

xo

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

H I

JK

YesNo

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

midsegment of a trapezoid

Lab - Midsegments of a Trapezoid


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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

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P

Q R

S

L M

15

7

Example

PQRS is a trapezoid. Find LM.

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P

Q R

S

L M

20

14.5

Example

PQRS is a trapezoid. Find PS.

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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.

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

H I

JK

E F

6

x

15

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

HI

J K

EF

x

19

10

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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.

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Kites


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

kites

Lab - Properties of Kites


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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

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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

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Example

LMNP is a kite. Find x.

72

(x2-1)

48

M

N

P

o

o

oL

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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

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

A EA

B ADC DR R

E

A

D

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

A EB D

C RR

E

A

D

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68 Find the value of z in the kite.

z 5z-8

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69 Find the value of x in the kite.

68o

(8x+4)o

44o

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70 Find the value of x.

36

(3x 2 + 3)

24

o

o

o

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Theorem Q19

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

In kite ABCDAC BD

A

B

C

D

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71 Find the value of x in the kite.

x

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72 Find the value of y in the kite.

12y

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Families of Quadrilaterals


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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.

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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)

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QUADRILATERALS

Kite

Trapezoid

IsoscelesTrapezoid

Parallelogram

Rhombus Rectangle

Squa

re

Rhombus

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73 A rhombus is a square.

A alwaysB sometimes

C never

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74 A square is a rhombus.

A alwaysB sometimes

C never

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75 A rectangle is a rhombus.

A alwaysB sometimes

C never

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76 A trapezoid is isosceles.

A alwaysB sometimes

C never

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77 A kite is a quadrilateral.

A alwaysB sometimes

C never

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78 A parallelogram is a kite.

A alwaysB sometimes

C never

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Coordinate Proofs


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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

page1svg

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

= 3 2 + 3 2 = (-3) 2 + 3 2 = 9 + 9 = 9 + 9 = 18 = 18 = 4.24 = 4.24

#

##

##

##

#

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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

= 5 2 + (-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.

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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

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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

= 3 2 + 4 2 = (-4) 2 + 3 2

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

# #### #

##

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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

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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

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Proofs


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

T E

AM

1

2

≅ ≅

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T E

AM

1

2

Option A

s tatements reasons

1) TE ≅ MA, <1 ≅ <2 1) Given

2) EM ≅ EM 2) Reflexive Property

3) Triangle MTE ≅ Triangle EAM 3) S ide Angle S ide

4) TM ≅ AE 4) CPCTC

5) TEAM is a paralle logram5) The oppos ite s ides of a paralle logram are congruent

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page1svg

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

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

F G

HJ

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F G

HJ

s tatements reasons

1) FGHJ is a paralle logramand F is a right angle 1) Given

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

3) H is a right angle 3) The oppos ite angles of a paralle logram are congruent

4) TEAM is a rectangle 4) Rectangle Corollary

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

Prove: COLD is a trapezoid

C O

LD

140o

40o60o

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C O

LD

140o

40o60o

s tatements 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 paralle l to LD 5) Consecutive Interior Angles Converse

6) CL is not paralle l to OD 6) Consecutive Interior Angles Converse

7) COLD is a trapezoid 7) Definition of a Trapezoid(A trapezoid has one pair of paralle l s ides)

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Try this ...

Given: FCD FEDProve: FD CE

F

C

D

E

≅

Slide 189 / 189

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