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Topic 4: Quadrilaterals and Coordinate Proof

This Packet Belongs to ________________________(Student Name)

Unit 4 – Quadrilaterals and Coordinate ProofModule 9: Properties of Quadrilaterals

9.1 Properties of Parallelograms9.2 Conditions of Parallelograms9.3 Properties of Rectangles, Rhombi, and Squares9.4 Conditions of Rectangles, Rhombi, and Squares

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Please follow along with notes.

Topic 1,2,3,4,etc…

Please follow along with notes. At the end of quarter 2 there will be a BINDER CHECK to check

Topic 1,2,3,4,etc…

Module 10: Coordinate Proof Using Slope and Distance10.1 Slope and Parallel Lines

10.2 Slope and Perpendicular Lines10.3 Coordinate Proof Using Distance with Segments and Triangles10.4 Coordinate Proof Using Distance with Quadrilaterals10.5 Perimeter and Area on the Coordinate Plane

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Module 9Properties of Quadrilaterals

Part 1:

Parallelograms2

2

Definition

• A parallelogram is a quadrilateral whose opposite sides are parallel.

• Its symbol is a small figure:

CB

A D

AB CD and BC AD

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Naming a Parallelogram

• A parallelogram is named using all four vertices.

• You can start from any one vertex, but you must continue in a clockwise or counterclockwise direction.

• For example, this can be either ABCD or ADCB. CB

A D4

Basic Properties

• There are four basic properties of all parallelograms.

– Opposite Sides

– Opposite Angles

– Consecutive Angles

– Diagonals

• These properties have to do with the angles, the sides and the diagonals.

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Opposite Sides

Theorem Opposite sides of a parallelogram are congruent.

• That means that .

• So, if AB = 7, then _____ = 7?

CB

A D

AB CD and BC AD

CD

66 7

2

Opposite Angles

One pair of opposite angles is A and C. The other pair is B and D.

Theorem Opposite angles of a parallelogram are congruent.

• Complete: If m A = 75 and m B = 105, then m C = ______ and m D = ______ .

CB

A D8

75° 105°

8 9

Consecutive Angles

• Each angle is consecutive to two other angles. A is consecutive with B and D.

CB

A D

1010

Consecutive Angles in Parallelograms

Theorem Consecutive angles in a parallelogram are supplementary.

• Therefore, m A + m B = 180 and m A + m D = 180.

• If m<C = 46, then m B = _____?

CB

A D

Consecutive INTERIOR Angles are

Supplementary!

134°

11

Diagonals• Diagonals are segments that join non‐consecutive vertices.

• For example, in this diagram, the only two diagonals are .

AC and BD

CB

A D

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Diagonal PropertyWhen the diagonals of a parallelogram intersect, they meet at the midpoint of each diagonal.

• So, P is the midpoint of .

• Therefore, they bisect each other; so and .

• But, the diagonals are not congruent!

AC and BD

AP PC BP PD

P

CB

A D

AC BD

1313

3

Diagonal PropertyTheorem The diagonals of a parallelogram bisect each other.

P

CB

A D

1414

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Parallelogram Summary

• By its definition, opposite sides are parallel.

Other properties (theorems):

• Opposite sides are congruent.

• Opposite angles are congruent.

• Consecutive angles are supplementary.

• The diagonals bisect each other.

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Examples

• 1. Draw HKLP.

• 2. Complete: HK = _______ and HP = ________ .

• 3. m<K = m<______ .

• 4. m<L + m<______ = 180.• 5. If m<P = 65, then m<H = ____,

m<K = ______ and m<L =______ .

PL

KL

P

P or <K

11565 115

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Examples (cont’d)

• 6. Draw in the diagonals. They intersect at M.

• 7. Complete: If HM = 5, then ML = ____ .

• 8. If KM = 7, then KP = ____ .

• 9. If HL = 15, then ML = ____ .

• 10. If m<HPK = 36, then m<PKL = _____ .

5

14

7.5

36

1818

1919

4

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Part 2

Tests for Parallelograms

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Review: Properties of Parallelograms

• Opposite sides are parallel.

• Opposite sides are congruent.

• Opposite angles are congruent.

• Consecutive angles are supplementary.

• The diagonals bisect each other.

2222

How can you tell if a quadrilateral is a parallelogram?

• Defn: A quadrilateral is a parallelogram iffopposite sides are parallel.

• Property If a quadrilateral is a parallelogram, then opposite sides are parallel.

• Test If opposite sides of a quadrilateral are parallel, then it is a parallelogram.

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Proving Quadrilaterals as Parallelograms

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

Theorem 1:

H G

E F

If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram .

Theorem 2:

If EF GH; FG EH, then Quad. EFGH is a parallelogram.

If EF GH and EF || HG, then Quad. EFGH is a parallelogram.24 25

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

Theorem 3:

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

Theorem 4:

H G

EF

M

,If H F and E G

then Quad. EFGH is a parallelogram.

intIf M is themidpo of EG and FH

then Quad. EFGH is a parallelogram.EM = GM and HM = FM

Proving Quadrilaterals as Parallelograms (part 2)

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5 ways to prove that a quadrilateral is a parallelogram.

1. Show that both pairs of opposite sides are || . [definition]

2. Show that both pairs of opposite sides are .

3. Show that one pair of opposite sides are both || and .

4. Show that both pairs of opposite angles are .

5. Show that the diagonals bisect each other .

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Examples ……

Find the values of x and y that ensures the quadrilateral is a parallelogram.

Example 1:

6x4x+8

y+2

2y

6x = 4x + 8

2x = 8

x = 4

2y = y + 2

y = 2

Example 2: Find the value of x and y that ensure the quadrilateral is a parallelogram.

120°

5y°(2x + 8)°2x + 8 = 120

2x = 112

x = 56

5y + 120 = 180

5y = 60

y = 12

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3030

3131

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9.1‐9.2 ClassworkPAGE 426

• GO ONLINE and complete 9.1‐9.2 hw.

• Alternative:Honors: 9.1: 3, 5‐6, 14, 17‐18, 23, 24

9.2: 1, 5, 8, 11‐12, 18‐19

• Regular: 9.1: 5‐6, 8, 17‐18

9.2: 1, 5, 8, 12, 18

Reminders:

…32

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Part 3

Rectangles

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Rectangles

• Opposite sides are parallel.• Opposite sides are congruent.• Opposite angles are congruent.• Consecutive angles are supplementary.• Diagonals bisect each other.

Definition: A rectangle is a quadrilateral with four right angles.

Is a rectangle a parallelogram?

Thus a rectangle has all the properties of a parallelogram.

Yes, since opposite angles are congruent.

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Properties of Rectangles

Therefore, ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles.

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

E

D C

BA

Theorem:

Converse: If the diagonals of a parallelogram are congruent , then the parallelogram is a rectangle.

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Properties of Rectangles

E

D C

BA

Parallelogram Properties: Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other.Plus: All angles are right angles. Diagonals are congruent.

Also: ∆AEB, ∆BEC, ∆CED, and ∆AED are isosceles triangles

36 37

7

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Examples

1. If AE = 3x +2 and BE = 29, find the value of x.

2. If AC = 21, then BE = _______.

3. If m<1 = 4x and m<4 = 2x, find the value of x.

4. If m<2 = 40, find m<1, m<3, m<4, m<5 and m<6.

m<1=50, m<3=40, m<4=80, m<5=100, m<6=40

10.5 units

x = 9 units

x = 18 units

6

54

321

E

D C

BA

3839

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Part 4

Rhombi and

Squares40

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Rhombus

Definition: A rhombus is a quadrilateral with four congruent sides.

Since a rhombus is a parallelogram the following are true:• Opposite sides are parallel.• Opposite sides are congruent.• Opposite angles are congruent.• Consecutive angles are supplementary.• Diagonals bisect each other.

Is a rhombus a parallelogram?

Yes, since opposite sides are congruent.

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Rhombus

Note: The four small triangles are congruent, by SSS.

This means the diagonals form four angles that are congruent, and must measure 90 degrees each.

So the diagonals are perpendicular.

This also means the diagonals bisect each of the four angles of the rhombus

So the diagonals bisect opposite angles.

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Properties of a RhombusTheorem: The diagonals of a rhombus are perpendicular.

Theorem: Each diagonal of a rhombus bisects a pair of opposite angles.

Note: The small triangles are RIGHT and CONGRUENT!

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8

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Your Turn: Rhombus Examples

Given: ABCD is a rhombus. Complete the following.

1. If AB = 9, then AD = ______.

2. If m<1 = 65, the m<2 = _____.

3. m<3 = ______.

4. If m<ADC = 80, the m<DAB = ______.

5. If m<1 = 3x -7 and m<2 = 2x +3, then x = _____.

54

3

21E

D C

BA9 units

65°

90°

100°

10

4445

Properties of a Rhombus

.Since a rhombus is a parallelogram the following are true:• Opposite sides are parallel.• Opposite sides are congruent.• Opposite angles are congruent.• Consecutive angles are supplementary.• Diagonals bisect each other.Plus:• All four sides are congruent.• Diagonals are perpendicular.• Diagonals bisect opposite angles.• Also remember: the small triangles are RIGHT and

CONGRUENT!45

4646 47

4848

4949

9

5050 51

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Square

• Opposite sides are parallel.• Opposite sides are congruent.• Opposite angles are congruent.• Consecutive angles are supplementary.• Diagonals bisect each other.Plus:• Four right angles.• Four congruent sides.• Diagonals are congruent.• Diagonals are perpendicular.• Diagonals bisect opposite angles.

Definition:A square is a quadrilateral with four congruent angles and four congruent sides.

Since every square is a parallelogram as well as a rhombus and rectangle, it has all the properties of these quadrilaterals.

5253

53

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Squares – Examples…...Given: ABCD is a square. Complete the following.

1. If AB = 10, then AD = _______ and DC = _______.

2. If CE = 5, then DE = _____.

3. m<ABC = _____.

4. m<ACD = _____.

5. m<AED = _____.

8 7 65

4321

E

D C

BA10 units 10 units

5 units

90°

45°

90°

5462

62

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9.3‐9.4 ClassworkPAGE 452

• GO ONLINE and complete 9.3‐9.4 hw.

• Alternative:Honors: 9.3: 1‐2, 5‐8, 15

9.4: 5‐13 odds, 22

• Regular: 9.3: 1‐2, 5‐6, 10, 13

9.4: 6, 9, 12, 15, 18, 22

Reminders:

Module 9 Quiz Next Class!

MYA next week!63

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Module 10:Getting Ready

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Definition of a ParallelogramUse Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ),B(2, 6), C (5, 7) and D(3,1) .

I need to show that both pairs of opposite sides are parallel by showing

that their slopes are equal.

A

B C

D

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Definition of a ParallelogramUse Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ),B(2, 6), C (5, 7) and D(3,1) .

AB: m = 6 – 0 = 6 = 32 – 0 2

CD: m = 1 – 7 = - 6 = 33 – 5 - 2

BC: m = 7 – 6 = 15 – 2 3

AD: m = 1 – 0 = 13 – 0 3

ll

ll

ABCD is a Parallelogramby Definition

A

B C

D

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Both Pairs of Opposite Sides Congruent

Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ),B(2, 6), C (5, 7) and D(3,1) .

I need to show that both pairs of opposite sides are congruent by using the distance formula to

find their lengths.

A

B C

D

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Both Pairs of Opposite Sides Congruent

Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ),B(2, 6), C (5, 7) and D(3,1) .AB = (2 – 0)2 + (6 – 0)2

= 4 + 36 = 40

ABCD is a Parallelogram because both pair of opposite sides are congruent.

CD = (3 – 5)2 + (1 – 7)2

= 4 + 36 = 40

AB CDBC = (5 – 2)2 + (7 – 6)2

= 9 + 1 = 10

AD = (3 – 0)2 + (1 – 0)2

= 9 + 1 = 10

A

B C

D

6868

11


I need to show that onepair of opposite sides is

both parallel and congruent.

One Pair of Opposite Sides Both Parallel and Congruent

ll (slope) and (distance)

A

B C

D

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One Pair of Opposite Sides Both Parallel and Congruent

Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3,1) .

BC ADABCD is a Parallelogram because one pair of opposite sides are parallel and congruent.

BC ll ADBC = (5 – 2)2 + (7 – 6)2

= 9 + 1 = 10

AD = (3 – 0)2 + (1 – 0)2

= 9 + 1 = 10

BC: m = 7 – 6 = 15 – 2 3

AD: m = 1 – 0 = 13 – 0 3

A

B C

D

7070


I need to show that each diagonal shares the SAME _________.

Diagonals Bisect Each Other

A

B C

D

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midpoint

Use Coordinate Geometry to show that quadrilateral ABCD is a parallelogram given the vertices A(0, 0 ), B(2, 6), C (5, 7) and D(3,1) .

ABCD is a Parallelogram because the diagonals share the same midpoint, thus bisecting each other.

Diagonals Bisect Each Other

The midpoint of AC is 0 + 5 , 0 + 72 2

5 , 72 2

The midpoint of BD is 2 + 3 , 6 + 12 2

5 , 72 2

A

B C

D

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Module 10:Coordinate Proof Using Slope and

Distance

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7474

12

7575

7676

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Understanding Slope

• Two (non-vertical) lines are parallel if and only if they have the same slope. (All vertical lines are parallel.)

4

2

-2

-4

-6

5

D: (4, -1)

C: (-2, -4)

B: (3, 3)

A: (-1, 1)

7878

Understanding Slope

• The slope of AB is:

• The slope of CD is:

• Since m1=m2, AB || CD

4

2

-2

-4

-6

5

D: (4, -1)

C: (-2, -4)

B: (3, 3)

A: (-1, 1) 1

3 1 2 1

3 1 4 2m

2

1 4 3 1

4 2 6 2m

7979 80

13

8181

8282

8383

10.1 ClassworkPAGE 501

• GO ONLINE and complete 10.1 hw.

• Alternative:Honors: 3, 4, 7‐10, 13, 17‐19, 22, 23‐26

• Regular: 2, 4, 7, 8, 16, 17, 22, 25

Reminders:

…

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Distance

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Perpendicular Lines

• (┴)Perpendicular Lines‐ 2 lines that intersect forming 4 right angles

Right angle

8686

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Slopes of Lines

• In a coordinate plane, 2 non vertical lines are iff the product of their slopes is ‐1.

• This means, if 2 lines are their slopes are opposite reciprocals of each other; such as ½ and ‐2.

• Vertical and horizontal lines are to each other.

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Example• Line l passes through (0,3) and (3,1).

• Line m passes through (0,3) and (‐4,‐3).

Are they ?

Slope of line l =

Slope of line m =

l m

30

13

3

2-or

3

2

40

33

2

3or

4

6

Opposite Reciprocals!

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Equation of a line in slope intercept form (y = mx+b)

Now that we know how to find slope given any two points, we cangenerate an equation of the line connecting the two points.

Example : points (3,2) and (6,9)

8989

9090

9191



• Alternative:Honors: 1, 2, 4‐6, 9‐18, 20, 22

• Regular: 2, 4, 6, 9‐15, 18, 20

Reminders:

…

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15


Distance

9393

9494

9595

9696

9797

9898

16

9999 100



• Alternative:Honors: 1, 3, 5, 8, 12, 18

• Regular: 3, 5, 8, 12,18

Reminders:

…

101101


Distance

102102

103103

104104

17

105105

106106

107108

108

109109

110110

18

111111

112112



• Alternative:Honors: 1, 3, 8, 11‐13, 17‐19, 26

• Regular: 3, 8, 11, 13, 17‐18, 19, 26

Reminders:

…

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Module 10.5:Perimeter and Area in the Coordinate

Plane

115115

Finding Perimeter and Area in the Coordinate Plane

Concept: Distance in the Coordinate Plane

EQ: How do we find area & perimeter in the coordinate plane?

Vocabulary: distance formula, polygon, area, perimeter

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Area Formulas

Perimeter for any polygon = sum of all sides.

•A parallelogram includes shapes such as squares, rectangles, rhombi. •The length of the base and height are found using the distance formula. •The final answer must include the appropriate label (units², feet², inches², meters², centimeters², etc.) 118

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Guided practice, Example 1Parallelogram ABCD has vertices A (‐5, 4), B (3, 4), C (5, ‐1), and D (‐3, ‐1).

Calculate the perimeter and area of parallelogramABCD.

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Example 1, continuedWe need to find the length of all four sides before we can find the area and the

perimeter. So we will use the distance formula:

, , , ,

The length of is 8 units The length of is 8 units

120120



, , , ,

.

The length of is 5.39 units

.

The length of is 5.39 units

121121

Example 1, continued

• 8units 5.39units 8units 5.39units• Find the perimeter by adding up all the sides:

• . . . •Find the area by using the formula

– or is the base and they are the same length so 8

– The height can be found by drawing a perpendicularline straight up from D to side and down from B to side .

• You can do this by counting the units or using the distance formula

• Finding the distance from D to the point 3, 4 and the distance from B to the point where the perpendicular line touches at 4, 1

122122


Area of a Parallelogram =

Base = 8 units

Height = 5 units

Area of a parallelogram = ∗

123123

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Area of a triangle

• The area of a triangle is found by using the formula:

Area = ∗ ∗

• The height of a triangle is the perpendicular distance from a vertex to the base of the triangle.

• Determining the lengths of the base and the height is necessary if these lengths are not stated in the problem.

• The final answer must include the appropriate label (units², feet², inches², meters², centimeters², etc.)

124124

Guided Practice, Example 2

Triangle ABC has vertices

A (2, 1), B (4, 5), and C (7, 1).

Calculate the perimeterand area of triangle ABC.

125125



, , ,

.

The length of is 4.47 units The length of is 5 units.

126

5

The length of is 5 units.

126


• . • Find the perimeter by adding up all the sides:

• . .

•Find the area by using the formula

– is the base so 5– The height can be found by drawing a

perpendicularline straight down from B to side .

– Then find the distance from B to the point where the perpendicular line touches at 4,1

• You can do this by counting the units or

using the distance formula

127

Area of a Triangle = Base = 5 unitsHeight = the distance from , to

, = 4

Area of a triangle =

127

128128



• Alternative:Honors: 1, 2, 5, 7, 9 , 11, 13, 15‐18

• Regular: 1, 5, 7, 9 , 11, 15, 18

Reminders:

Topic 4 Review Next Class

Topic 4 Test next week!

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This Packet Belongs to (Student Name) Topic 4