Name: _________________________________ Date: ________________________

Quadrilaterals Polygons Basics

Objectives: SWBAT identify, name and describe polygons.

SWBAT use the sum of the measures of the interior angles of a quadrilateral.

A. The basics on POLYGONS

3 or more sides formed by non-collinear segments

Each side intersects with two other sides,

one at each endpoint

B. Identifying Polygons

State whether the following figures are polygons are not.

1. 2. 3. 4. 5.

Yes yes yes No No

(not a segment) (can’t have

a two collinear

sides)

C. Naming Polygons (named by the number of sides they have)

# of Sides Type of polygon 3 Triangle 4 Quadrilateral

5 Pentagon 6 Hexagon Need to Memorize these!!

7 Heptagon 8 Octagon 9 Nonagon

10 Decagon 11 Hendecagon 12 Dodecagon

N n-gon

Convex vs. Concave

Convex –

Has no dips

Concave-

Has dips in the sides

Examples: Identify the polygon and state whether it is convex or concave.

1. 2.

Concave Octagon Convex Pentagon

D. Regular Polygons

If it is equiangular and equilateral

Examples:

Decide whether the polygon is regular.

1. 2. 3.

No Yes No

Convex Concave

Interior Angles of Polygon Theorem

Number of degrees in the interior angles of a polygon = 180( n-2)

N is the number of sides

Quadrilaterals A quad will always have 360 degrees.

Examples

2. Given the following diagram, find m<JKL

P

S

Q

R

80

x

70

2x

80 70 2 180( 2)

150 3 180(4 2)

150 3 360

3 210

70

x x n

x

x

x

x

7 8x

4 13x

5 5x

5 2x

7 8 4 13 5 5 5 2 180( 2)

21 18 180(4 2)

21 18 360

21 18 378

3 210

18

x x x x n

x

x

x

x

x

4 13

4 18 13

72 13

59

m JKL x

Properties of Parallelograms Objectives: SWBAT use properties of parallelograms in real-life situations.

A. Parallelogram and its Properties

Parallelogram~

A quadrilateral where both opposite sides are parallel to each other.

Opposite Side Parallelogram Theorem

If a quadrilateral is a parallelogram,

then opposite sides are congruent

Opposite Angles Parallelogram Theorem

If a quadrilateral is a parallelogram,

then the opposite angles are congruent

Consecutive Angles Parallelogram Theorem

If a quadrilateral is a parallelogram,

then the Consecutive angles are supplementary

Diagonals of a Parallelogram Theorem

If a quadrilateral is a parallelogram, then diagonals bisect each other

Examples:

1. FGHJ is a parallelogram. Find the unknown lengths.

a. JH

b. JK

Since it is a parallelogram, then the opposite sides are congruent, and the diagonals

bisect each other.

If FG=5, then the opposite side will be the same so JH=5.

Since the diagonals bisect each other, if KG = 3 then JK = 3.

F

J

G

H

K

5

3

180

180

180

180

m P m S

m P m Q

m Q m R

m R m S

3

5JH

JK

2. Find the angle measures.

a. mR

b. mQ

Since it is a parallelogram, the opposite angles are congruent.

Since it is a parallelogram, the consecutive (same side) angles are supplementary.

3. Find the value of x and y.

Since it is a parallelogram, the opposite sides are congruent.

Since it is a parallelogram, then the consecutive angles are supplementary.

4. Gracie, Kenny, Teresa, and Billy Bob live at the four corners of a block shaped like a

parallelogram. Gracie lives 3 miles away from Kenny. How far apart do Teresa and Billy Bob

live from each other?

3 miles because opposite sides of parallelograms are

congruent

Q

SP

R

70

Q

SP

R

3x 120

26y 3 10y

70

m R m P

m P

180

3 120 180

3 60

20

m p m s

x

x

x

3 10 26

2 16

8

QP RS

y y

y

y

180

70 180

110

m R m Q

m Q

m Q

Proving Quadrilaterals are Parallelograms Objectives: SWBAT prove that a quadrilateral is a parallelogram.

Definition of a Parallelogram

If both pairs of opposite sides are parallel to the corresponding opposite side in a

quadrilateral, then it is a parallelogram.

Converse Opposite Side Parallelogram Theorem

If both pairs of opposite sides are congruent to the corresponding opposite side in a

quadrilateral, then it is a parallelogram.

Converse Opposite Angle Parallelogram Theorem

If both pairs of opposite angles are congruent to the corresponding opposite angle in a

quadrilateral, then it is a parallelogram.

Converse Consecutive Angles Parallelogram Theorem

If one angle of a quadrilateral is supplementary (add up to 180) to both of the angles

on its sides, then it is a parallelogram.

Converse Diagonals of a Parallelogram Theorem

If the diagonals of a quadrilateral bisect each other, then it is a parallelogram.

Parallel and Congruent Parallelogram Theorem

If one pair of opposite sides are congruent and parallel

then it is a parallelogram

B

D

C

A D

Examples: Determine if the following quadrilaterals are parallelograms. If so why or

why not?

1. 2. 3.

Yes since it has two pairs of opposite No, you only have Yes, since 1 pair of

Congruent sides, then it satisfies the one pair of parallel sides of sides are

Converse Opposite Side Parallelogram sides. parallel and congruent

Theorem then it satisfies the Parallel

and Congruent Parallelogram

Theorem

4. 5. 6.

No, you can not mix

And match theorems to

Prove that a quadrilateral is a

Parallelogram.

Yes since it has an angle that is Yes, since both of the

supplementary , to both of the opposite angles are congruent

other angles it neighbors, then it to their corresponding opposite

satisfies the Converse Consecutive angles, then it satisfies the

Angle of a Parallelogram Theorem. Converse Opposite Angles of a

Parallelogram theorem.

42°138°

42°

78°

78°102°

102°

Solve for the following variables so that the quadrilateral is a parallelogram.

7. 8. 9.

.

7 12 5 6

2 18

9

Oppo Sides Are

x x

x

x

.

3 4

3 36 4

108 4

27

Oppo Angles Are

y x

x

x

x

sec

72 3 180

3 108

36

Con utive Angles

Are Supplementary

y

y

y

sec

15 30

1

2

Diagonals Bi t Each other

x

x

sec

20 4

6

Diagonals Bi t Each other

y

y

Trapezoids and Kites Objectives: SWBAT use properties of trapezoids and kites.

A. Trapezoids

1 pair of parallel sides, 2 bases, 2 legs, and

two pair of base angles

Isosceles Trapezoid

Legs are congruent

B. Trapezoid Theorems

Isosceles Trapezoid Angle Theorem

If a trapezoid is isosceles, then the base angles are congruent

Converse of the Isosceles Trapezoid Angle Theorem

If a pair of base angles are congruent, then it is Isosceles trapezoid

Isosceles Trapezoid Diagonals Theorem

A trapezoid is isosceles if and only if the diagonals are congruent

Examples:

PQRS is an isosceles trapezoid.

Find mP: 110

Find mQ: 110

Find mR: 70

P

S

Q

R

leg leg

base

base

70

360 70 70

360 2 140

220 2

110

m P m Q

x

x

x

BD AC

AE BE

DE CE

A B

D C

Midsegment of a Trapezoid

All three segments are parallel

3. M and N are the midpoints on PS and QR, respectively.

Find the length of the midsegment.

4. A riser is designed to elevate a speaker. The riser

consists of 4 trapezoidal sections that can be stacked

one on top of the other to produce trapezoids of

varying heights.

If only the bottom two stages are used, what is

the width of the top resulting riser?

P Q

M

S R

N

12

8

D

C

|| ||BC MN AD

1( )

2MN AD BC

1

2

112 8

2

120

2

10

MN AD BC

1

2

110 20

2

130

2

15

Middle Bottom Top

1

2

115 20

2

135

2

17.5

Top of Bottom Two Risers Bottom New Top

KITE ~ A quadrilateral with exactly two consecutive congruent sides.

Kite’s Perpendicular Diagonals Theorem

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

Kite Opposite Angle Theorem

If a quadrilateral is a kite, then exactly one pair of

Opposite angles are congruent.

Examples:

1. MQNP is a kite (so the diagonals are perpendicular).

Find MN, NQ, NP, PQ, and the Perimeter

If it is a kite, then the diagonals are congruent (so we can use Pythagorean theor3em)

2. Find a. 3. Find x

Kite is a quad, so internal angles add up to 360 Kites have one pair of Congruent Angles

2 2 2

2 2 2

2

8 6

100

10

10

MN RN MR

MN

MN

MN

MN

M

MQ

Q

2 2 2

2 2 2

2

6

2 10

2

2 10

40

NP NR RP

NP

NP

NP PQ

NP

PQ

10 2 10 2 10 1

2 10

0

0 4

Perimeter MN NP PQ QM

Perimet

Perimet

r

er

e

116 48 116 360

280 360

80

a

a

a

60 90 360

2 150 360

2 210

105

x x

x

x

x

4. Given Kite ABCD, find x, y, and z.

The diagonal 𝑅𝐵̅̅ ̅̅ of kite 𝑅𝐻𝐵𝑊 forms an equilateral triangle with two of the sides, and 𝑚 <

𝐵𝑊𝑅 = 40. Draw and label a diagram showing the diagonals, and the measures of all the angles.

90

AC BD

z

35

BDC is isosceles

BC DC

CBD CDB

y

18

BAD is isosceles

BA DA

BAC DAC

x

Rhombuses and Rectangles

Objectives: SWBAT use properties of sides and angles of rhombuses and rectangles

SWBAT use properties of diagonals of rhombuses and rectangles

A. Properties

Rhombus~ A parallelogram with four congruent sides. Diagonals are perpendicular and bisect each angle

Rectangle~ A parallelogram with four right angles. Diagonals are all congruent.

True or false, if false explain why it is false.

1. A rectangle is a parallelogram. True

2. A parallelogram is always a rhombus. False, but a rhombus is always a parallelogram.

3. A rhombus is always a Rectangle. False, it they have different properties.

4. A rhombus is always a Kite. False, Kites Congruent Sides are consecutive.

5. A rhombus is always a parallelogram. True

Matching: Which of the following quadrilaterals has the given property?

1. All sides are congruent. C A. Parallelogram

2. All angles are congruent. B B. Rectangle

3. The diagonals are congruent. B C. Rhombus

4. Opposite angles are congruent. A, B, C D. Quadrilateral

5. Sum of Interior angles equals 360 degrees. A,B,C,D

What is the value of 𝑥 𝑎𝑛𝑑 𝑦 in the rectangle to the right?

Answer is A

.

4 20

5

Oppo Sides Are

PS QR

x

x

90

90

90 3 5 5

90 3 5 5 4

90 3 5 20

75 3

25

Corner Angles Are

m PSR

m PSR m PSQ m QSR

y x

y

y

y

y

Use the rhombus PRYZ, to find the measurements of the following given that

PK = 4y + 1 RK = 3x - 1

KY = 7y – 14 KZ = 2x + 6

A) PY

4 1 7 14

1 3 14

15 3

5

y y

y

y

y

4 1

4 5 1

20 1

21

y PK

PK

PK

PK

2

2 21

42

PY PK

PY

PY

B) RZ

3 1 2 6

1 6

7

x x

x

x

3 1

3 7 1

21 1

20

x RK

RK

RK

RK

2

2 20

40

RZ RK

RZ

RZ

C) m YKZ Diagonals are perpendicular, therefore 90m YKZ

D) YZ

2 2 2

2 2 2

2

2

2

21 20

441 400

841

841

29

KY KZ YZ

YZ

YZ

YZ

YZ

YZ

14. Based on the figure below, which statements are true?

I. The figure is a rectangle (False, not all diagonals are congruent)

II. The figure is a parallelogram (True opposite sides are congruent

III. 6𝑥 − 4 = 9𝑥 + 3 (False, it is not a rectangle so not all diagonals are congruent)

IV. 9𝑥 + 3 = 10𝑥 − 2 (True, it is a

parallelogram so diagonals bisect each other).

V. 𝑥 = 8 (False, x = 5)

VI.

The longest side has a length of 60. True,

12(5)=60

Since it is a Parellelogram

6 4 5 1

5 5

2 4 1

4 4

5

x x

x x

x

x

Since it is a Parellelogram

9 3 10 2

9 9

3 2

2 2

5

x x

x x

x

x

If is is a Rectangle, all Diagonals are congruent

9 3 9 5 3 48

10 2 10 5 2 48

6 4 6 5 4 26

5 1 5 5 1 26

Not A Rectangle

x

x

x

x

Squares

Objectives: SWBAT use properties of sides and angles of Squares

SWBAT use properties of diagonals of Squares

A. Properties

Square~ a parallelogram with four congruent

sides and four right angles.

Examples:

Identify each figure as a quadrilateral, parallelogram, rhombus, rectangle,

trapezoid, kite, square or none of the above.

Square Rectangle Quadrilateral

Parallelogram Kite Trapezoid

1. Given that the figure to the right is a square, find the length of a

side.

2. The quadrilateral at the below is a square. Solve for x and y.

x=_________

y=_________

3. Given Square SQUR, find the following.

𝑬𝑸 = 𝟏𝟔 𝒎𝒎. 𝒎∠𝑺𝑬𝑸 = 𝟗𝟎 𝒅𝒆𝒈𝒓𝒆𝒆𝒔

𝑬𝑼 = 𝟏𝟔 𝒎𝒎. 𝒎∠𝑺𝑸𝑼 = 𝟗𝟎 𝒅𝒆𝒈𝒓𝒆𝒆𝒔

𝑺𝑼 = 𝟑𝟐 𝒎𝒎. 𝒎∠𝑼𝑬𝑸 = 𝟗𝟎 𝒅𝒆𝒈𝒓𝒆𝒆𝒔

𝑹𝑼 = 𝟏𝟔√𝟐 𝒎𝒎. 𝒎∠𝑺𝑸𝑬 = 𝟒𝟓 𝒅𝒆𝒈𝒓𝒆𝒆𝒔

2 14x 15x

2 14 15

29

44

All Sides Are

x x

x

Sides

2 2 16

3 18

6

10

All Sides Are

y y

y

y

Sides

4y 16 y

2 2y

5 10x

10

5 10 10

5 20

4

Since the Sides

x

x

x

16 .RE mm

Quadrilateral Family Tree

Objectives: SWBAT identify special quadrilaterals.

The Family Tree

Quadrilateral (Interior Angles Add up to 360)

Kite Parallelogram Trapezoid

Rectangle Rhombus

Isosceles Trapezoid

Square

Shape

Description of Sides

Description of Angles

Interesting Information

Quad

4 non

congruent nor parallel sides

No equal

angles

All angles add

up to 360 degrees

Trapezoid

1 set of parallel sides (no congruent

sides)

No equal angles but angles by legs

are supp.

Nevada is close to a right trapezoid

Isosceles

Trapezoid

1 pair of parallel sides and 1 pair of

congruent sides (not

same)

Base angles are congruent and angles by

legs are supp.

Diagonals are congruent

Kite

No parallel sides

2 pairs of neighboring

congruent sides

1 set of congruent

angles.

Diagonals are perpendicular.

Parallelogram

2 pairs of congruent and parallel sides

Opposite angles are congruent and

neighboring angles are supp.

Diagonals Bisect Each other

Rectangle

2 pairs of congruent,

parallel and perpendicular

sides

4 Right angles Diagonals are congruent.

It is a

parallelogram.

Rhombus

2 pairs of parallel sides

and all 4 sides are congruent

Opposite angles are

congruent and neighboring

angles are supp.

Diagonals are perpendicular.

It is a

parallelogram.

Square

2 pairs of parallel and perpendicular

sides and all 4 sides are congruent

4 Right angles Diagonals are congruent, an perpendicular.

It is a Rectangle and Rhombus

Fill in the table. Put an X in the box if the shape always has the property.

Property

Parallelogram Rectangle Rhombus Square Trapezoid

Both pairs of

opp. sides

X X X X

Exactly 1 pair of

opp. sides

X

Diagonals are

X X

Diagonals are

X X

Diagonals bisect

each other

X X X X

3. Quadrilateral ABCD has at least one pair of opposite sides congruent. Draw the kinds of

quadrilaterals meet this condition (5).

Parallelogram Square

Rhombus Rectangle

Isosceles Trapezoid

COORDINATE QUADS Objectives: SWBAT identify types of quads using a coordinate plane.

Characteristic Definition Formula

Congruent Equal Sides / Angles

2 2

2 1 2 1d x x y y

Perpendicular Opposite Reciprocal Slope

Negative Reciprocal Slope

2 1

2 1

y ym

x x

Parallel Same Slope

2 1

2 1

y ym

x x

Midsegment / Midpoint

Bisecting a segment or is the

middle 1 2 1 2,

2 2

x x y y

Coordinate Proof ~ Using algebra and formulas on a coordinate plane to prove a

statement.

1. Use the coordinate plane, and the Distance Formula to show that KLMN is a Rhombus.

K(2, 5), L(-2, 3), M(2, 1), N(6, 3)

Since all sides are congruent, the quadrilateral is a rhombus.

2 2

2 2

2,5 2,3

2 2 3 5

4 2

16 4

20

2 5

KL

2 2

2 2

2,3 2,1

2 2 1 3

4 2

16 4

20

2 5

LM

2 2

2 2

2,1 6,3

6 2 3 1

4 2

16 4

20

2 5

MN

2 2

2 2

6,3 2,5

2 6 5 3

4 2

16 4

20

2 5

NK

Use slope or the distance formula to determine the most precise name for the figure:

𝐴(−1, −4), 𝐵(1, −1), 𝐶(4, 1), 𝐷(2, −2).

Graph it first, that will give you some insight into the shape, and allow you to eliminate obviously wrong

answers.

A. Kite

B. Rhombus

C. Trapezoid

D. Square

Now we know that it is a kite or rhombus – use the slope to see if the opposite sides are parallel or not.

3. Given points 𝐵 (−3,3), 𝐶(3, 4), and 𝐷(4, −2). Which of the following points must be point A in

order for the quadrilateral 𝐴𝐵𝐶𝐷 to be a parallelogram?

A. 𝐴(−2, −1)

B. 𝐴(−1, −2)

C. 𝐴(−2, −3)

D. 𝐴(−3, −2)

Since it is a parallelogram, it has the same slope as CD which is 𝒎 =𝒓𝒊𝒔𝒆

𝒓𝒖𝒏=

−𝟔

𝟏. So start

at point B, move the same slope of 𝒎 =−𝟔

𝟏, which lands you at 𝑨(−𝟐, −𝟑), so the answer is

C.

1, 1 4,1

1 1

4 1

2

3

BC

m

m

1, 4 2, 2

2 4

2 1

2

3

AD

m

m

1, 4 1, 1

1 4

1 1

3

2

AB

m

m

4,1 2, 2

2 1

2 4

3

2

CD

m

m

Since

BC AD

and

AB CD

it must be B

4. Given a Trapezoid (−3,4), 𝐵(−5, −2), 𝐶(5, −2), and 𝐷(3,4). Find the following

a) Is the trapezoid Isosceles?

b) What are coordinates of the midsegment for the trapezoid?

c) What is the length of the midsegment?

2 2

2 2

3, 4 5, 2

5 3 2 4

2 6

4 36

40

2 10

AB

22

2 2

3, 4 5, 2

5 3 4 2

2 6

4 36

40

2 10

DC

Since

AB DC

then

Yes

3,4 5, 2

3 5 4 2,

2 2

8 2,

2 2

4,1

AB

3, 4 5, 2

4 23 5,

2 2

8 2,

2 2

4,1

DC

2 2

2 2

4,1 4,1

4 4 1 1

8 0

64 0

64

8

Midsegment

Classifying

Quadrilaterals

How many pairs of parallel sides?

YOU HAVE

A KITE

YOU HAVE

A

TRAPEZOID

ARE THE 4 CORNERS

90 DEGREES?

ARE THE 4 CORNERS

90 DEGREES?

ARE THE 4 SIDES

EQUAL?

YOU HAVE

A

RHOMBUS

YOU HAVE

A SQUARE

YOU HAVE A

RECTANGLE YOU HAVE A

PARALLELOGRAM

ZERO

ONE

TWO

YES!

YES!YES!

NO

NO

NO

Distance between a Line and a point Objectives: SWBAT find the distance between a line and a point

Shortest distance between two points is always a _____straight__________________ line

that is ___perpendicular________________ to both points.

Steps for Success:

1) Find the equation perpendicular to the given line through the point given.

2) Find the point where the two lines intersection using substitution.

3) Use the distance formula to find the distance between the given point and

Just watch an example:

Find the distance between the line 𝑦 = 2𝑥 + 5, and the point (7, −1)

Step 1 Step 2 Step 3

1

2

int (7, 1)

m

po

1 1

11 7

2

1 71

2 2

1 1

1 5

2 2

y y m x x

y x

y x

y x

1 5

2 2

2 5

Since they both equal y

1 52 5

2 2

2

1 52 5 2

2 2

5 4 10

5 5 10

10 10

5 5

1

int

y x

y x

x x

Multiply everything by to

cancel out the fraction

x x

x x

x x

x

x

x

now plug the x value back

o one of

2 5

2 1 5

2 5

3

int

1,3

the equations

and solve for y

y x

y

y

y

Po