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Slide 1 / 189 This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others. Click to go to website: www.njctl.org New Jersey Center for Teaching and Learning Progressive Mathematics Initiative Slide 2 / 189 www.njctl.org 2014-06-03 Quadrilaterals Geometry Slide 3 / 189 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 Slide 4 / 189 Angles of Polygons Return to the Table of Contents Slide 5 / 189 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 B C 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 Slide 6 / 189 Types of Polygons Polygons are named by their number of sides. Number of Sides Type of Polygon 3 triangle 4 quadrilateral 5 pentagon 6 hexagon 7 heptagon 8 octagon 9 nonagon 10 decagon 11 11-gon 12 dodecagon n n-gon
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  • Slide 1 / 189

    This material is made freely available at www.njctl.org and is intended for the non-commercial use of students and teachers. These materials may not be used for any commercial purpose without the written permission of the owners. NJCTL maintains its website for the convenience of teachers who wish to make their work available to other teachers, participate in a virtual professional learning community, and/or provide access to course materials to parents, students and others.

    Click to go to website:www.njctl.org

    New Jersey Center for Teaching and Learning

    Progressive Mathematics Initiative

    Slide 2 / 189

    www.njctl.org

    2014-06-03

    Quadrilaterals

    Geometry

    Slide 3 / 189

    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

    Slide 4 / 189

    Angles of Polygons

    Return to the Table of Contents

    Slide 5 / 189

    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

    Slide 6 / 189

    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

    http://www.njctl.orghttp://www.njctl.orghttp://www.njctl.orghttp://www.njctl.orghttp://www.njctl.orgnextPage();page4svgpage56svgpage75svgpage6svgpage115svgpage143svgpage163svgpage172svgpage157svgpage1svg

  • Slide 7 / 189

    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

    Slide 8 / 189

    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

    Slide 9 / 189

    1 The figure below is a polygon.

    True

    False

    Ans

    wer

    Slide 9 (Answer) / 189

    1 The figure below is a polygon.

    True

    False

    [This object is a pull tab]A

    nsw

    er

    False

    Slide 10 / 189

    2 The figure below is a polygon.

    True

    False

    Ans

    wer

    Slide 10 (Answer) / 189

    2 The figure below is a polygon.

    True

    False

    [This object is a pull tab]

    Ans

    wer

    True

  • Slide 11 / 189

    3 Indentify the polygon.

    A Pentagon

    B Octagon

    C Quadrilateral

    D HexagonE Decagon

    F Triangle Ans

    wer

    Slide 11 (Answer) / 189

    3 Indentify the polygon.

    A Pentagon

    B Octagon

    C Quadrilateral

    D HexagonE Decagon

    F Triangle

    [This object is a pull tab]

    Ans

    wer

    D

    Slide 12 / 189

    4 Is the polygon convex or concave?

    A Convex

    B Concave

    Ans

    wer

    Slide 12 (Answer) / 189

    4 Is the polygon convex or concave?

    A Convex

    B Concave

    [This object is a pull tab]

    Ans

    wer

    B

    Slide 13 / 189

    5 Is the polygon convex or concave?

    A ConvexB Concave

    Ans

    wer

    Slide 13 (Answer) / 189

    5 Is the polygon convex or concave?

    A ConvexB Concave

    [This object is a pull tab]

    Ans

    wer

    A

  • Slide 14 / 189

    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

    Slide 15 / 189

    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

    Ans

    wer

    Slide 15 (Answer) / 189

    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

    [This object is a pull tab]

    Ans

    wer

    E, F, H, I, J

    Slide 16 / 189

    7 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

    Ans

    wer

    Slide 16 (Answer) / 189

    7 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

    [This object is a pull tab]

    Ans

    wer

    F, H

    Slide 17 / 189

    8 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

    Ans

    wer

  • Slide 17 (Answer) / 189

    8 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

    [This object is a pull tab]

    Ans

    wer

    C, F, I

    Slide 18 / 189

    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

    Slide 19 / 189

    Polygon Number of SidesNumber of Triangular

    RegionsSum 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

    Slide 20 / 189

    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

    _____________

    _____________

    _____________

    Ans

    wer

    Slide 20 (Answer) / 189

    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

    _____________

    _____________

    _____________

    [This object is a pull tab]

    Ans

    wer

    Heptagon

    The sum of the interior angles is 5(180o) = 900o

    F

    A B

    C

    DE

    F

    G

    Slide 21 / 189

    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 SidesSum 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

  • Slide 22 / 189

    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.

    Slide 23 / 189

    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 oclick to reveal

    Slide 24 / 189

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

    180(n-2)n

    Complete the table.

    regular polygon number of sides sum 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

    Slide 25 / 189

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

    Ans

    wer

    Slide 25 (Answer) / 189

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

    [This object is a pull tab]

    Ans

    wer

    1080o

    Slide 26 / 189

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

    Ans

    wer

  • Slide 26 (Answer) / 189

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

    [This object is a pull tab]

    Ans

    wer

    135o

    Slide 27 / 189

    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

    Ans

    wer

    Slide 27 (Answer) / 189

    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

    [This object is a pull tab]

    Ans

    wer

    C

    Slide 28 / 189

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

    A 162

    B 3240

    C 180

    D 60 Answ

    er

    Slide 28 (Answer) / 189

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

    A 162

    B 3240

    C 180

    D 60

    [This object is a pull tab]

    Ans

    wer

    A

    Slide 29 / 189

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

    A 2520 B 2880 C 3240 D 157.5

    Ans

    wer

  • Slide 29 (Answer) / 189

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

    A 2520 B 2880 C 3240 D 157.5

    [This object is a pull tab]

    Ans

    wer

    D

    Slide 30 / 189

    14 What is the value of x?

    (5x+

    15)o

    (9x-6) o

    (8x) o

    (11x+16)

    o

    (10x+8)o

    Ans

    wer

    Slide 30 (Answer) / 189

    14 What is the value of x?

    (5x+

    15)o

    (9x-6) o

    (8x) o

    (11x+16)

    o

    (10x+8)o

    [This object is a pull tab]

    Ans

    wer

    14

    Slide 31 / 189

    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

    Slide 32 / 189

    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

    Slide 33 / 189

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

    Ans

    wer

  • Slide 33 (Answer) / 189

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

    [This object is a pull tab]

    Ans

    wer

    B

    Slide 34 / 189

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

    A 72

    B 60C 51.43

    D 45 Ans

    wer

    Slide 34 (Answer) / 189

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

    A 72

    B 60C 51.43

    D 45

    [This object is a pull tab]

    Ans

    wer

    C

    Slide 35 / 189

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

    Ans

    wer

    Slide 35 (Answer) / 189

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

    [This object is a pull tab]

    Ans

    wer

    360o

    Slide 36 / 189

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

    Ans

    wer

  • Slide 36 (Answer) / 189

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

    [This object is a pull tab]

    Ans

    wer

    72o

    Slide 37 / 189

    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.

    Ans

    wer

    Slide 37 (Answer) / 189

    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.

    [This object is a pull tab]

    Ans

    wer

    180(n-2)n

    = 172(n) (n)

    180(n-2) = 172n

    180n-360 = 172n-180n -180n

    -360 = -8n-8 -845 = n

    Slide 38 / 189

    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 Ans

    wer

    o

    Slide 38 (Answer) / 189

    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

    [This object is a pull tab]

    Ans

    wer

    D

    Slide 39 / 189

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

    Ans

    wer

    o

  • Slide 39 (Answer) / 189

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

    o

    [This object is a pull tab]

    Ans

    wer

    20

    Slide 40 / 189

    Properties of Parallelograms

    Return to the Table of Contents

    Slide 41 / 189

    Lab - Investigating Parallelograms

    Lab - Properties of Parallelograms

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

    Slide 42 / 189

    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

    Slide 43 / 189

    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.

    Slide 44 / 189

    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

    page1svghttps://njctl.org/courses/math/geometry/quadrilaterals/investigating-parallelograms/https://njctl.org/courses/math/geometry/quadrilaterals/properties-of-parallelograms/

  • Slide 45 / 189

    If a quadrilateral is a parallelogram, then the consecutive angles are

    supplementary.

    yo

    xo

    xo

    yoA B

    CD

    If ABCD is a parallelogram, then xo + yo = 180o

    Theorem Q5

    Slide 46 / 189

    Example:

    ABCD is parallelogram.

    Find w, x, y, and z.

    A B

    CD

    12

    2y

    x-5

    9

    65o

    5zo

    wo

    Slide 47 / 189

    A B

    CD

    12

    2y

    x-5

    9

    65o

    5zo

    wo

    The opposite sides are congruent.

    Ans

    wer

    Slide 47 (Answer) / 189

    A B

    CD

    12

    2y

    x-5

    9

    65o

    5zo

    wo

    The opposite sides are congruent.

    [This object is a pull tab]A

    nsw

    er AB = DC2y = 92 2y = 4.5

    BC = ADx-5 = 12 +5 +5 x = 17

    Slide 48 / 189

    A B

    CD

    12

    2y

    x-5

    9

    65o

    5zo

    wo

    The opposite angles are congruent.

    Ans

    wer

    Slide 48 (Answer) / 189

    A B

    CD

    12

    2y

    x-5

    9

    65o

    5zo

    wo

    The opposite angles are congruent.

    [This object is a pull tab]

    Ans

    wer m C = m A

    5z = 65 5 5 z = 13

  • Slide 49 / 189

    A B

    CD

    12

    2y

    x-5

    9

    65o

    5zo

    woThe consecutive angles are supplementary.

    Ans

    wer

    Slide 49 (Answer) / 189

    A B

    CD

    12

    2y

    x-5

    9

    65o

    5zo

    woThe consecutive angles are supplementary.

    [This object is a pull tab]

    Ans

    wer m B + m A = 180o

    w + 65 = 180 - 65 -65 w = 115o

    Slide 50 / 189

    21 DEFG is a parallelogram. Find w.

    D E

    FG

    70o

    15

    3x-32w z+12

    21

    y2 An

    swer

    Slide 50 (Answer) / 189

    21 DEFG is a parallelogram. Find w.

    D E

    FG

    70o

    15

    3x-32w z+12

    21

    y2

    [This object is a pull tab]

    Ans

    wer

    55o

    Slide 51 / 189

    22 DEFG is a parallelogram. Find x.

    D E

    FG

    70o

    15

    3x-32w z+12

    21

    y2

    Ans

    wer

    Slide 51 (Answer) / 189

    22 DEFG is a parallelogram. Find x.

    D E

    FG

    70o

    15

    3x-32w z+12

    21

    y2

    [This object is a pull tab]

    Ans

    wer

    8

  • Slide 52 / 189

    23 DEFG is a parallelogram. Find y.

    D E

    FG

    70o

    15

    3x-32w z+12

    21

    y2

    Ans

    wer

    Slide 52 (Answer) / 189

    23 DEFG is a parallelogram. Find y.

    D E

    FG

    70o

    15

    3x-32w z+12

    21

    y2

    [This object is a pull tab]

    Ans

    wer

    30

    Slide 53 / 189

    24 DEFG is a parallelogram. Find z.

    D E

    FG

    70o

    15

    3x-32w z+12

    21

    y2

    Ans

    wer

    Slide 53 (Answer) / 189

    24 DEFG is a parallelogram. Find z.

    D E

    FG

    70o

    15

    3x-32w z+12

    21

    y2

    [This object is a pull tab]

    Ans

    wer

    58

    Slide 54 / 189

    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

    Slide 55 / 189

    Example:

    LMNP is a parallelogram. Find QN and MP.

    L M

    NP

    Q

    4

    6(The diagonals bisect each other)

    Ans

    wer

  • Slide 55 (Answer) / 189

    Example:

    LMNP is a parallelogram. Find QN and MP.

    L M

    NP

    Q

    4

    6(The diagonals bisect each other)

    [This object is a pull tab]

    Ans

    wer

    QN = LQLQ = 4

    MQ + QP = MP (Segment Addition)MQ = QPMQ = 4 4 + 4 = MP 8 = MP

    Slide 56 / 189

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

    A

    B E

    R

    S

    x 4y

    8 10

    Ans

    wer

    Slide 56 (Answer) / 189

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

    A

    B E

    R

    S

    x 4y

    8 10

    [This object is a pull tab]

    Ans

    wer x = SA = 104y = RS = 8

    y = 2ER = RS + SE = 16

    Slide 57 / 189

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

    A sometimesB always

    C never Answ

    er

    Slide 57 (Answer) / 189

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

    A sometimesB always

    C never

    [This object is a pull tab]

    Ans

    wer

    B

    Slide 58 / 189

    26 MATH is a parallelogram. Find RT.

    A 6

    B 7

    C 8

    D 9 12

    M A

    TH

    R

    7

    Ans

    wer

  • Slide 58 (Answer) / 189

    26 MATH is a parallelogram. Find RT.

    A 6

    B 7

    C 8

    D 9 12

    M A

    TH

    R

    7

    [This object is a pull tab]

    Ans

    wer

    B

    Slide 59 / 189

    27 MATH is a parallelogram. Find AR.

    A 6

    B 7

    C 8

    D 912

    M A

    TH

    R

    7

    Ans

    wer

    Slide 59 (Answer) / 189

    27 MATH is a parallelogram. Find AR.

    A 6

    B 7

    C 8

    D 912

    M A

    TH

    R

    7

    [This object is a pull tab]

    Ans

    wer

    A

    Slide 60 / 189

    28 MATH is a parallelogram. Find m H.

    M A

    TH98o

    2x-4

    14

    (3y+8)o

    Ans

    wer

    Slide 60 (Answer) / 189

    28 MATH is a parallelogram. Find m H.

    M A

    TH98o

    2x-4

    14

    (3y+8)o

    [This object is a pull tab]

    Ans

    wer

    82

    Slide 61 / 189

    29 MATH is a parallelogram. Find x.

    M A

    TH98o

    2x-4

    14

    (3y+8)o

    Ans

    wer

  • Slide 61 (Answer) / 189

    29 MATH is a parallelogram. Find x.

    M A

    TH98o

    2x-4

    14

    (3y+8)o

    [This object is a pull tab]

    Ans

    wer

    9

    Slide 62 / 189

    30 MATH is a parallelogram. Find y.

    M A

    TH98o

    2x-4

    14

    (3y+8)o

    Ans

    wer

    Slide 62 (Answer) / 189

    30 MATH is a parallelogram. Find y.

    M A

    TH98o

    2x-4

    14

    (3y+8)o

    [This object is a pull tab]

    Ans

    wer

    30

    Slide 63 / 189

    Proving Quadrilaterals are

    Parallelograms

    Return to the Table of Contents

    Slide 64 / 189

    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.

    Slide 65 / 189

    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

    page1svg

  • Slide 66 / 189

    Example

    Tell whether PQRS is a parallelogram. Explain.

    P

    Q

    R

    S6

    6

    4

    4 Ans

    wer

    Slide 66 (Answer) / 189

    Example

    Tell whether PQRS is a parallelogram. Explain.

    P

    Q

    R

    S6

    6

    4

    4

    [This object is a pull tab]

    Ans

    wer

    Yes, PQRS is a quadrilateral. The opposite sides are congruent.

    Slide 67 / 189

    Example

    Tell whether PQRS is a parallelogram. Explain.P Q

    RS

    Ans

    wer

    Slide 67 (Answer) / 189

    Example

    Tell whether PQRS is a parallelogram. Explain.P Q

    RS[This object is a pull tab]

    Ans

    wer

    Because PQRS is a quadrilateral, m Q + m R = 180o. But, we can't assume that Q and R are right angles. We can't prove PQRS is a

    parallelogram.

    Slide 68 / 189

    31 Tell whether the quadrilateral is a parallelogram.

    Yes

    No

    78o

    136o 2

    Ans

    wer

    Slide 68 (Answer) / 189

    31 Tell whether the quadrilateral is a parallelogram.

    Yes

    No

    78o

    136o 2

    [This object is a pull tab]

    Ans

    wer

    No

  • Slide 69 / 189

    32 Tell whether the quadrilateral is a parallelogram.

    Yes

    No3 3

    5

    4.99

    Ans

    wer

    Slide 69 (Answer) / 189

    32 Tell whether the quadrilateral is a parallelogram.

    Yes

    No3 3

    5

    4.99

    [This object is a pull tab]

    Ans

    wer

    No

    Slide 70 / 189

    33 Tell whether the quadrilateral is a parallelogram.

    Yes

    No

    Ans

    wer

    Slide 70 (Answer) / 189

    33 Tell whether the quadrilateral is a parallelogram.

    Yes

    No

    [This object is a pull tab]

    Ans

    wer

    No

    Slide 71 / 189

    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

    Slide 72 / 189

    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

  • Slide 73 / 189

    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

    Slide 74 / 189

    34 Tell whether the quadrilateral is a parallelogram.

    Yes

    No

    Ans

    wer

    Slide 74 (Answer) / 189

    34 Tell whether the quadrilateral is a parallelogram.

    Yes

    No

    [This object is a pull tab]

    Ans

    wer

    No

    Slide 75 / 189

    35 Tell whether the quadrilateral is a parallelogram.

    Yes

    No141o

    39o

    49o

    Ans

    wer

    Slide 75 (Answer) / 189

    35 Tell whether the quadrilateral is a parallelogram.

    Yes

    No141o

    39o

    49o

    [This object is a pull tab]

    Ans

    wer

    No

    Slide 76 / 189

    36 Tell whether the quadrilateral is a parallelogram.

    Yes

    No

    89.5

    819

    Ans

    wer

  • Slide 76 (Answer) / 189

    36 Tell whether the quadrilateral is a parallelogram.

    Yes

    No

    89.5

    819

    [This object is a pull tab]

    Ans

    wer

    Yes

    Slide 77 / 189

    37 Tell whether the quadrilateral is a parallelogram.

    Yes

    No

    Ans

    wer

    Slide 77 (Answer) / 189

    37 Tell whether the quadrilateral is a parallelogram.

    Yes

    No

    [This object is a pull tab]

    Ans

    wer

    Yes

    Slide 78 / 189

    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

    Ans

    wer

    Slide 78 (Answer) / 189

    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

    [This object is a pull tab]

    Ans

    wer

    NO, the question did not state the position of the measurements in the quadrilateral. We cannot assume their position.

    67 67

    113

    This is not a parallelogram

    o o o

    Slide 79 / 189

    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

    Ans

    wer

  • Slide 79 (Answer) / 189

    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

    [This object is a pull tab]

    Ans

    wer

    In a parallelogram...the opposite sides are parallel and congruent,the opposite angles are congruent, the consecutive angles are supplementary and the diagonals bisect each other.

    Slide 80 / 189

    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

    Ans

    wer

    Slide 80 (Answer) / 189

    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

    [This object is a pull tab]

    Ans

    wer

    To prove a quadrilateral is a parallelogram...both pairs of opposite sides of a quadrilateral must be congruent,both pairs of opposite angles of a quadrilateral must be congruent,an angle of the quadrilateral must be supplementary to its consecutive angles, the diagonals of the quadrilateral bisect each other, or one pair of opposite sides of a quadrilateral are parallel and congruent.

    Slide 81 / 189

    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)

    Ans

    wer

    Slide 81 (Answer) / 189

    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)

    [This object is a pull tab]

    Ans

    wer

    E

    Slide 82 / 189

    39 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.

    Ans

    wer

  • Slide 82 (Answer) / 189

    39 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. [This object is a pull tab]

    Ans

    wer

    F

    Slide 83 / 189

    40 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.

    6

    63(6-4)

    Ans

    wer

    Slide 83 (Answer) / 189

    40 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.

    6

    63(6-4)

    [This object is a pull tab]

    Ans

    wer

    F

    Slide 84 / 189

    Constructing Parallelograms

    Return to the Table of Contents

    Slide 85 / 189

    To construct a parallelogram, there are 3 steps.

    Construct a Parallelogram

    Slide 86 / 189

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

    Construct a Parallelogram - Step 1

    page1svg

  • Slide 87 / 189

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

    Construct a Parallelogram - Step 2

    Slide 88 / 189

    Why is this a parallelogram?

    Step 3 - Connect the endpoints of the segments.

    Construct a Parallelogram - Step 3

    Ans

    wer

    Slide 88 (Answer) / 189

    Why is this a parallelogram?

    Step 3 - Connect the endpoints of the segments.

    Construct a Parallelogram - Step 3

    [This object is a pull tab]

    Ans

    wer The diagonals

    bisect each other

    Slide 89 / 189

    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.

    Slide 90 / 189

    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.

    3 steps to draw a parallelogram in a coordinate plane

    Slide 91 / 189

    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?

    3 steps to draw a parallelogram in a coordinate plane

    Ans

    wer

  • Slide 91 (Answer) / 189

    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?

    3 steps to draw a parallelogram in a coordinate plane

    [This object is a pull tab]

    Ans

    wer

    Remember all horizontal lines have

    a slope of zero.One pair of opposite

    sides are parallel and congruent.

    Slide 92 / 189

    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

    Slide 93 / 189

    41 The opposite angles of a parallelogram are ...

    A bisect

    B congruent

    C parallel

    D supplementary

    Ans

    wer

    Slide 93 (Answer) / 189

    41 The opposite angles of a parallelogram are ...

    A bisect

    B congruent

    C parallel

    D supplementary

    [This object is a pull tab]

    Ans

    wer

    B

    Slide 94 / 189

    42 The consecutive angles of a parallelogram are ...

    A bisect

    B congruent

    C parallel

    D supplementary

    Ans

    wer

    Slide 94 (Answer) / 189

    42 The consecutive angles of a parallelogram are ...

    A bisect

    B congruent

    C parallel

    D supplementary

    [This object is a pull tab]

    Ans

    wer

    D

  • Slide 95 / 189

    43 The diagonals of a parallelogram ______ each other.

    A bisect

    B congruent

    C parallel

    D supplementary Answ

    er

    Slide 95 (Answer) / 189

    43 The diagonals of a parallelogram ______ each other.

    A bisect

    B congruent

    C parallel

    D supplementary

    [This object is a pull tab]

    Ans

    wer

    A

    Slide 96 / 189

    44 The opposite sides of a parallelogram are ...

    A bisect

    B congruent

    C parallel

    D supplementary

    Ans

    wer

    Slide 96 (Answer) / 189

    44 The opposite sides of a parallelogram are ...

    A bisect

    B congruent

    C parallel

    D supplementary

    [This object is a pull tab]A

    nsw

    er

    B & C

    Slide 97 / 189

    Rhombi, Rectanglesand Squares

    Return to the Table of Contents

    Slide 98 / 189

    three special parallelograms

    Rhombus

    Rectangle

    Square

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

    and square.

    page1svg

  • Slide 99 / 189

    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

    Slide 100 / 189

    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

    Ans

    wer

    Slide 100 (Answer) / 189

    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[This object is a pull tab]

    Ans

    wer

    C

    Slide 101 / 189

    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

    Ans

    wer

    Slide 101 (Answer) / 189

    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[This object is a pull tab]

    Ans

    wer

    A

    Slide 102 / 189

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

    A B

    CD

    If ABCD is a rhombus,

    then AC BD.

    Theorem Q11

  • Slide 103 / 189

    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

    Slide 104 / 189

    Example

    EFGH is a rhombus.

    Find x, y, and z.E F

    G H

    72oz

    2x-6

    5y

    10

    Slide 105 / 189

    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.

    Slide 106 / 189

    Try this ...

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

    8

    86o

    3x+2

    z

    12 y

    2

    Ans

    wer

    Slide 106 (Answer) / 189

    Try this ...

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

    8

    86o

    3x+2

    z

    12 y

    2 [This object is a pull tab]

    Ans

    wer x = 2

    y = 4z = 470

    Slide 107 / 189

    47 This is a rhombus. Find x.

    xoA

    nsw

    er

  • Slide 107 (Answer) / 189

    47 This is a rhombus. Find x.

    xo

    [This object is a pull tab]

    Ans

    wer

    90o

    Slide 108 / 189

    48 This is a rhombus. Find x.

    13

    x-3

    9 Ans

    wer

    Slide 108 (Answer) / 189

    48 This is a rhombus. Find x.

    13

    x-3

    9

    [This object is a pull tab]

    Ans

    wer

    36

    Slide 109 / 189

    49 This is a rhombus. Find x.

    126ox An

    swer

    Slide 109 (Answer) / 189

    49 This is a rhombus. Find x.

    126ox

    [This object is a pull tab]

    Ans

    wer

    27o

    Slide 110 / 189

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

    H J

    KL

    6 M16

    Ans

    wer

  • Slide 110 (Answer) / 189

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

    H J

    KL

    6 M16

    [This object is a pull tab]

    Ans

    wer

    10

    Slide 111 / 189

    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

    Slide 112 / 189

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

    6y

    12

    Ans

    wer

    Slide 112 (Answer) / 189

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

    6y

    12

    [This object is a pull tab]A

    nsw

    er

    15

    Slide 113 / 189

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

    If ABCD is a rectangle,

    then AC BD.

    A B

    CD

    Theorem Q13

    Slide 114 / 189

    Example

    RECT is a rectangle. Find x and y.

    2x-513

    63o9yo

    R E

    CT

    Ans

    wer

  • Slide 114 (Answer) / 189

    Example

    RECT is a rectangle. Find x and y.

    2x-513

    63o9yo

    R E

    CT[This object is a pull tab]

    Ans

    wer

    Option A2x - 5 = 13

    2x = 18x = 9

    Option B2x - 5 + 13 = 26

    2x + 8 = 262x = 18

    x = 9The measure of each angle in a

    rectangle is 90o.

    9y + 63 = 909y = 27y = 3

    Slide 115 / 189

    52 RSTU is a rectangle. Find z.R S

    TU8z

    Ans

    wer

    Slide 115 (Answer) / 189

    52 RSTU is a rectangle. Find z.R S

    TU8z

    [This object is a pull tab]

    Ans

    wer

    11.25

    Slide 116 / 189

    53 RSTU is a rectangle. Find z.R S

    TU

    4z-9

    7

    Ans

    wer

    Slide 116 (Answer) / 189

    53 RSTU is a rectangle. Find z.R S

    TU

    4z-9

    7

    [This object is a pull tab]

    Ans

    wer

    4

    Slide 117 / 189

    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

  • Slide 118 / 189

    Example

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

    z - 4

    (5x)o

    6

    3y

    Ans

    wer

    Slide 118 (Answer) / 189

    Example

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

    z - 4

    (5x)o

    6

    3y

    [This object is a pull tab]

    Ans

    wer

    In a rhombus the diagonals are perpendicular.

    In a rhombus the diagonal

    bisect the opposite angles.

    5x = (90)5x = 45

    x = 9

    12

    3y = 90y = 30

    z - 4 = 6z = 10

    Slide 119 / 189

    Try this ...

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

    3y

    12z

    8y - 1

    0

    (x2 + 9)o

    Ans

    wer

    Slide 119 (Answer) / 189

    Try this ...

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

    3y

    12z

    8y - 1

    0

    (x2 + 9)o

    [This object is a pull tab]

    Ans

    wer x = 6

    y = 2 z = 7.5

    Slide 120 / 189

    54 The quadrilateral is a square. Find y.

    A 2

    B 3

    C 4

    D 5

    18y

    Ans

    wer

    Slide 120 (Answer) / 189

    54 The quadrilateral is a square. Find y.

    A 2

    B 3

    C 4

    D 5

    18y

    [This object is a pull tab]

    Ans

    wer

    5

  • Slide 121 / 189

    55 The quadrilateral is a rhombus. Find x.

    A 2

    B 3

    C 4

    D 5

    2x + 6

    4x

    Ans

    wer

    Slide 121 (Answer) / 189

    55 The quadrilateral is a rhombus. Find x.

    A 2

    B 3

    C 4

    D 5

    2x + 6

    4x

    [This object is a pull tab]

    Ans

    wer

    3

    Slide 122 / 189

    112o

    (4x)o

    56 The quadrilateral is parallelogram. Find x.A

    nsw

    er

    Slide 122 (Answer) / 189

    112o

    (4x)o

    56 The quadrilateral is parallelogram. Find x.

    [This object is a pull tab]

    Ans

    wer

    17

    Slide 123 / 189

    57 The quadrilateral is a rectangle. Find x.

    10x

    3x + 7

    Ans

    wer

    Slide 123 (Answer) / 189

    57 The quadrilateral is a rectangle. Find x.

    10x

    3x + 7

    [This object is a pull tab]

    Ans

    wer

    1

  • Slide 124 / 189

    Opposite sidesare

    Diagonals bisectopposite

  • Slide 130 / 189

    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

    Slide 131 / 189

    Slide 131 (Answer) / 189 Slide 132 / 189

    59 The quadrilateral is an isosceles trapezoid. Find x.

    A 3

    B 5

    C 7

    D 9 64o (9x + 1)o

    Ans

    wer

    Slide 132 (Answer) / 189

    59 The quadrilateral is an isosceles trapezoid. Find x.

    A 3

    B 5

    C 7

    D 9 64o (9x + 1)o

    [This object is a pull tab]

    Ans

    wer

    C

    Slide 133 / 189

    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

  • Slide 134 / 189

    Example

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

    112o 147o

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

    P

    R

    Q

    S

    Slide 135 / 189

    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

    Slide 136 / 189

    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 = 11OR

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

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

    Option B

    o

    o o

    o

    Slide 137 / 189

    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

    Ans

    wer

    Slide 137 (Answer) / 189

    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

    [This object is a pull tab]

    Ans

    wer

    123 + 4w + 1 = 180124 + 4w = 180

    4w = 56w = 14

    m S = 4(14) + 1 = 57m Q = 9(14) - 3 = 123m R = 180 - 123 = 57

    o o o

    Slide 138 / 189

    60 The trapezoid is isosceles. Find x.

    9

    4

    6x + 3

    2x + 2 An

    swer

  • Slide 138 (Answer) / 189

    60 The trapezoid is isosceles. Find x.

    9

    4

    6x + 3

    2x + 2

    [This object is a pull tab]

    Ans

    wer

    6x + 3 = 9or 4 = 2x + 2

    or 6x + 3 + 2x + 2 = 4 + 9

    x = 1

    Slide 139 / 189

    61 The trapeziod is isosceles. Find x.

    137o

    xo

    Ans

    wer

    Slide 139 (Answer) / 189

    61 The trapeziod is isosceles. Find x.

    137o

    xo

    [This object is a pull tab]

    Ans

    wer

    143

    Slide 140 / 189

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

    H I

    JK

    Ans

    wer

    YesNo

    Slide 140 (Answer) / 189

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

    H I

    JK

    YesNo

    [This object is a pull tab]

    Ans

    wer

    No, the bases are parallel, and have the same slope

    Slide 141 / 189

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

    midsegment of a trapezoid

    Lab - Midsegments of a Trapezoid

    Click on the link below and complete the lab.

    https://njctl.org/courses/math/geometry/quadrilaterals/midsegments-of-trapezoids/

  • Slide 142 / 189

    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

    Slide 143 / 189

    P

    Q R

    S

    L M

    15

    7

    Example

    PQRS is a trapezoid. Find LM.

    Ans

    wer

    Slide 143 (Answer) / 189

    P

    Q R

    S

    L M

    15

    7

    Example

    PQRS is a trapezoid. Find LM.

    [This object is a pull tab]

    Ans

    wer

    Slide 144 / 189

    P

    Q R

    S

    L M

    20

    14.5

    Example

    PQRS is a trapezoid. Find PS.

    Ans

    wer

    Slide 144 (Answer) / 189

    P

    Q R

    S

    L M

    20

    14.5

    Example

    PQRS is a trapezoid. Find PS.

    [This object is a pull tab]

    Ans

    wer

    Slide 145 / 189

    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.

    Ans

    wer

  • Slide 145 (Answer) / 189

    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.

    [This object is a pull tab]

    Ans

    wer x = 18

    y = 7z = 5

    Slide 146 / 189

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

    H I

    JK

    E F

    6

    x

    15

    Ans

    wer

    Slide 146 (Answer) / 189

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

    H I

    JK

    E F

    6

    x

    15

    [This object is a pull tab]

    Ans

    wer

    10.5

    Slide 147 / 189

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

    HI

    J K

    EF

    x

    19

    10

    Ans

    wer

    Slide 147 (Answer) / 189

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

    HI

    J K

    EF

    x

    19

    10

    [This object is a pull tab]

    Ans

    wer

    1

    Slide 148 / 189

    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.

    Ans

    wer

  • Slide 148 (Answer) / 189

    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.

    [This object is a pull tab]

    Ans

    wer

    B and D

    Slide 149 / 189

    Kites

    Return to the Table of Contents

    Slide 150 / 189

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

    kites

    Lab - Properties of Kites

    Click on the link below and complete the lab.

    Slide 151 / 189

    In kite ABCD,

  • Slide 154 / 189

    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

    Slide 155 / 189

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

    A EA

    B ADC DR R

    E

    A

    D

    Ans

    wer

    Slide 155 (Answer) / 189

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

    A EA

    B ADC DR R

    E

    A

    D

    [This object is a pull tab]

    Ans

    wer

    A

    Slide 156 / 189

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

    A EB D

    C RR

    E

    A

    D

    Ans

    wer

    Slide 156 (Answer) / 189

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

    A EB D

    C RR

    E

    A

    D

    [This object is a pull tab]

    Ans

    wer

    C

    Slide 157 / 189

    68 Find the value of z in the kite.

    z 5z-8

    Ans

    wer

  • Slide 157 (Answer) / 189

    68 Find the value of z in the kite.

    z 5z-8

    [This object is a pull tab]

    Ans

    wer

    2

    Slide 158 / 189

    69 Find the value of x in the kite.

    68o

    (8x+4)o

    44o

    Ans

    wer

    Slide 158 (Answer) / 189

    69 Find the value of x in the kite.

    68o

    (8x+4)o

    44o

    [This object is a pull tab]

    Ans

    wer

    15

    Slide 159 / 189

    70 Find the value of x.

    36

    (3x 2 + 3)

    24 Ans

    wer

    o

    o

    o

    Slide 159 (Answer) / 189

    70 Find the value of x.

    36

    (3x 2 + 3)

    24

    o

    o

    o

    [This object is a pull tab]

    Ans

    wer

    7

    Slide 160 / 189

    Theorem Q19

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

    In kite ABCDAC BD

    A

    B

    C

    D

  • Slide 161 / 189

    71 Find the value of x in the kite.

    x

    Ans

    wer

    Slide 161 (Answer) / 189

    71 Find the value of x in the kite.

    x

    [This object is a pull tab]

    Ans

    wer

    90o

    Slide 162 / 189

    72 Find the value of y in the kite.

    12y

    Ans

    wer

    Slide 162 (Answer) / 189

    72 Find the value of y in the kite.

    12y

    [This object is a pull tab]A

    nsw

    er

    7.5

    Slide 163 / 189

    Families of Quadrilaterals

    Return to the Table of Contents

    Slide 164 / 189

    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.

    page1svg

  • Slide 165 / 189

    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)

    Slide 166 / 189

    QUADRILATERALS

    Kite

    Trapezoid

    IsoscelesTrapezoid

    Parallelogram

    Rhombus Rectangle

    Squa

    re

    Rhombus

    Slide 167 / 189

    73 A rhombus is a square.

    A alwaysB sometimes

    C never

    Ans

    wer

    Slide 167 (Answer) / 189

    73 A rhombus is a square.

    A alwaysB sometimes

    C never

    [This object is a pull tab]A

    nsw

    er

    B

    Slide 168 / 189

    74 A square is a rhombus.

    A alwaysB sometimes

    C never

    Ans

    wer

    Slide 168 (Answer) / 189

    74 A square is a rhombus.

    A alwaysB sometimes

    C never

    [This object is a pull tab]

    Ans

    wer

    A

  • Slide 169 / 189

    75 A rectangle is a rhombus.

    A alwaysB sometimes

    C never

    Ans

    wer

    Slide 169 (Answer) / 189

    75 A rectangle is a rhombus.

    A alwaysB sometimes

    C never

    [This object is a pull tab]

    Ans

    wer

    C

    Slide 170 / 189

    76 A trapezoid is isosceles.

    A alwaysB sometimes

    C never

    Ans

    wer

    Slide 170 (Answer) / 189

    76 A trapezoid is isosceles.

    A alwaysB sometimes

    C never

    [This object is a pull tab]A

    nsw

    er

    B

    Slide 171 / 189

    77 A kite is a quadrilateral.

    A alwaysB sometimes

    C never

    Ans

    wer

    Slide 171 (Answer) / 189

    77 A kite is a quadrilateral.

    A alwaysB sometimes

    C never

    [This object is a pull tab]

    Ans

    wer

    A

  • Slide 172 / 189

    78 A parallelogram is a kite.

    A alwaysB sometimes

    C never

    Ans

    wer

    Slide 172 (Answer) / 189

    78 A parallelogram is a kite.

    A alwaysB sometimes

    C never

    [This object is a pull tab]

    Ans

    wer

    C

    Slide 173 / 189

    Coordinate Proofs

    Return to the Table of Contents

    Slide 174 / 189

    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 175 / 189

    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 = 32 + 32 = (-3)2 + 32 = 9 + 9 = 9 + 9 = 18 = 18 = 4.24 = 4.24

    √√

    √√

    √√

    Slide 176 / 189

    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 = 52 + (-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.

    page1svg

  • Slide 177 / 189

    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

    Slide 178 / 189J (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 = 32 + 42 = (-4)2 + 32

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

    √ √√√√ √

    √√

    Slide 179 / 189J (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

    Slide 180 / 189

    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

    Hin

    t

    Slide 180 (Answer) / 189

    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

    [This object is a pull tab]

    Hin

    t

    Remember, the midsegment is the segment that joins the

    midpoints of the legs and is parallel to the bases.

    You need to show that:1. SL = LP2. QM = MR

    3. slope of LM = slope of SR

    Slide 181 / 189

    Proofs

    Return to the Table of Contents

    page1svg

  • Slide 182 / 189

    Given: TE MA,

  • Slide 188 / 189C O

    LD

    140o

    40o60o

    statements 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 parallel to LD 5) Consecutive Interior Angles Converse

    6) CL is not parallel to OD 6) Consecutive Interior Angles Converse

    7) COLD is a trapezoid 7) Definition of a Trapezoid(A trapezoid has one pair of parallel sides)

    Slide 189 / 189

    Try this ...

    Given: FCD FEDProve: FD CE

    F

    C

    D

    E

    Hin

    t

    Slide 189 (Answer) / 189

    Try this ...

    Given: FCD FEDProve: FD CE

    F

    C

    D

    E

    [This object is a pull tab]

    Hin

    t

    If you prove CDEF is kite, then the diagonals must be perpendicular.