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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
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
Slide 10 / 189
2 The figure below is a polygon.
True
False
Slide 11 / 189
3 Indentify the polygon.
A Pentagon
B Octagon
C Quadrilateral
D Hexagon
E Decagon
F Triangle
Slide 12 / 189
4 Is the polygon convex or concave?
A Convex
B Concave
Slide 13 / 189
5 Is the polygon convex or concave?
A ConvexB Concave
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
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
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
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 S ides
Number of Triangular Regions
Sum of the Interior Angles
triangle 3 1 1(180o) = 180o
quadrilateral 4 2 2(180o) = 360o
pentagon 5 3 3(180o) = 540o
hexagon 6 4 4(180o) = 720o
Complete the table
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
_____________
_____________
_____________
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 S ides
Sum of the measures of the interior angles .
hexagon 6 180(6-2) = 720o
heptagon 7 180(7-2) = 900o
octagon 8 180(8-2) = 1080o
nonagon 9 180(9-2)=1260o
decagon 10 180(10-2)=1440o
11-gon 11 180(11-2) = 1620o
dodecagon 12 180(12-2) = 1800o
Polygon Interior Angles Theorem Q1
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 o
click 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 s idessum of interior
anglesmeasure of each
angle
triangle 3 180o 60o
quadrilateral 4 360o 90o
pentagon 5 540o 108o
hexagon 6 720o 120o
octagon 8 1080o 135o
decagon 10 1440o 144o
15-gon 15 2340o 156o
Polygon Interior Angles Theorem Corollary
Slide 25 / 189
9 What is the sum of the measures of the interior angles of the stop sign?
Slide 26 / 189
10 If the stop sign is a regular polygon. What is the measure of each interior angle?
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
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
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
Slide 30 / 189
14 What is the value of x?
(5x+
15)o
(9x-6) o
(8x) o
(11x+16)o
(10x+8)o
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
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
Slide 35 / 189
17 What is the sum of the measures of the exterior angles of a pentagon?
Slide 36 / 189
18 If a pentagon is regular, what is the measure of each exterior angle?
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.
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
o
Slide 39 / 189
20 The measure of each angle of a regular convex polygon is 162 . Find the number of sides of the polygon.
o
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
Slide 45 / 189
If a quadrilateral is a parallelogram, then the consecutive angles are
supplementary.
yo
xo
xo
yo
A B
CD
If ABCD is a parallelogram, then xo + yo = 180o
Theorem Q5
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.
Slide 48 / 189
A B
CD
12
2y
x-5
9
65o
5zo
wo
The opposite angles are congruent.
Slide 49 / 189
A B
CD
12
2y
x-5
9
65o
5zo
wo
The consecutive angles are supplementary.
Slide 50 / 189
21 DEFG is a parallelogram. Find w.
D E
FG
70o
15
3x-32w z+12
21
y2
Slide 51 / 189
22 DEFG is a parallelogram. Find x.
D E
FG
70o
15
3x-32w z+12
21
y2
Slide 52 / 189
23 DEFG is a parallelogram. Find y.
D E
FG
70o
15
3x-32w z+12
21
y2
Slide 53 / 189
24 DEFG is a parallelogram. Find z.
D E
FG
70o
15
3x-32w z+12
21
y2
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)
Slide 56 / 189
Try this...BEAR is a parallelogram. Find x, y, and ER.
A
B E
R
S
x 4y
8 10
Slide 57 / 189
25 In a parallelogram, the opposite sides are ________ parallel.
A sometimes
B always
C never
Slide 58 / 189
26 MATH is a parallelogram. Find RT.
A 6
B 7
C 8
D 9 12
M A
TH
R
7
Slide 59 / 189
27 MATH is a parallelogram. Find AR.
A 6
B 7
C 8
D 912
M A
TH
R
7
Slide 60 / 189
28 MATH is a parallelogram. Find m H.
M A
TH98o
2x-4
14
(3y+8)o
Slide 61 / 189
29 MATH is a parallelogram. Find x.
M A
TH98o
2x-4
14
(3y+8)o
Slide 62 / 189
30 MATH is a parallelogram. Find y.
M A
TH98o
2x-4
14
(3y+8)o
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
Slide 66 / 189
Example
Tell whether PQRS is a parallelogram. Explain.
P
Q
R
S6
6
4
4
Slide 67 / 189
Example
Tell whether PQRS is a parallelogram. Explain.P Q
RS
Slide 68 / 189
31 Tell whether the quadrilateral is a parallelogram.
Yes
No
78o
136o
2
Slide 69 / 189
32 Tell whether the quadrilateral is a parallelogram.
Yes
No3 3
5
4.99
Slide 70 / 189
33 Tell whether the quadrilateral is a parallelogram.
Yes
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
Slide 75 / 189
35 Tell whether the quadrilateral is a parallelogram.
Yes
No141o
39o
49o
Slide 76 / 189
36 Tell whether the quadrilateral is a parallelogram.
Yes
No
89.5
819
Slide 77 / 189
37 Tell whether the quadrilateral is a parallelogram.
Yes
No
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
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
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
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)
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.
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)
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
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
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
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
Slide 94 / 189
42 The consecutive angles of a parallelogram are ...
A bisect
B congruent
C parallel
D supplementary
Slide 95 / 189
43 The diagonals of a parallelogram ______ each other.
A bisect
B congruent
C parallel
D supplementary
Slide 96 / 189
44 The opposite sides of a parallelogram are ...
A bisect
B congruent
C parallel
D supplementary
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.
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
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
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
72o
z
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 y2
Slide 107 / 189
47 This is a rhombus. Find x.
xo
Slide 108 / 189
48 This is a rhombus. Find x.
13
x-3
9
Slide 109 / 189
49 This is a rhombus. Find x.
126ox
Slide 110 / 189
50 HJKL is a rhombus. Find the length of HJ.
H J
KL
6 M16
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
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-5 13
63o9yo
R E
CT
Slide 115 / 189
52 RSTU is a rectangle. Find z.R S
TU8z
Slide 116 / 189
53 RSTU is a rectangle. Find z.R S
TU
4z-9
7
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
Slide 119 / 189
Try this ...
The quadrilateral is a square. Find x, y, and z.
3y
12z
8y - 1
0
(x2 + 9)o
Slide 120 / 189
54 The quadrilateral is a square. Find y.
A 2
B 3
C 4
D 5
18y
Slide 121 / 189
55 The quadrilateral is a rhombus. Find x.
A 2
B 3
C 4
D 5
2x + 6
4x
Slide 122 / 189
112o
(4x)o
56 The quadrilateral is parallelogram. Find x.
Slide 123 / 189
57 The quadrilateral is a rectangle. Find x.
10x
3x + 7
Slide 124 / 189
Opposite sidesare
Diagonals bisectopposite <'s
Has 4 sides
Has 4 right <'s
Diagonals are
Slide the description under the correct special parallelogram.
rhombus rectangle square
Diagonals are
Has 4 right <'s
Has 4 sides
Diagonals are
Opposite sidesare
Diagonals are
Slide 125 / 189
Lab - Quadrilaterals in the Coordinate Plane
Click on the link below and complete the lab.
Slide 126 / 189
Trapezoids
Return to theTable of Contents
Slide 127 / 189
A trapezoid is a quadrilateral with one pair of parallel sides. base
legbase angles
base
leg
The parallel sides are called bases.
The nonparallel sides are called legs.
A trapezoid also has two pairs of base angles.
trapezoid
Slide 128 / 189
An isosceles trapezoid is a trapezoid with congruent legs.
isosceles trapezoid
Slide 129 / 189
If a trapezoid is isosceles, then each pair of base angles are congruent.
ABCD is an isosceles trapezoid. <A <B
and <C <D.
A B
CD
Theorem Q14
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 132 / 189
59 The quadrilateral is an isosceles trapezoid. Find x.
A 3
B 5
C 7
D 9 64o (9x + 1)o
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 = 11
OR
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
Slide 138 / 189
60 The trapezoid is isosceles. Find x.
9
4
6x + 3
2x + 2
Slide 139 / 189
61 The trapeziod is isosceles. Find x.
137o
xo
Slide 140 / 189
62 In trapezoid HIJK, can HI and KJ have slopes that are opposite reciprocals?
H I
JK
YesNo
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.
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.
Slide 144 / 189
P
Q R
S
L M
20
14.5
Example
PQRS is a trapezoid. Find PS.
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.
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63 EF is the midsegment of trapezoid HIJK. Find x.
H I
JK
E F
6
x
15
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64 EF is the midsegment of trapezoid HIJK. Find x.
HI
J K
EF
x
19
10
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65 Which of the following is true of every trapezoid? Choose all that apply.
A Exactly 2 sides are congruent.
B Exactly one pair of sides are parallel.
C The diagonals are perpendicular.
D There are 2 pairs of base angles.
Slide 149 / 189
Kites
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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.
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In kite ABCD, <B <D
and <A <D
If a quadrilateral is a kite, then it has one pair of congruent opposite angles.
A
B
C
D
Theorem Q18
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Theorem Q18
If a quadrilateral is a kite, then it has one pair of congruent opposite angles.
In kite ABCD, B D and A D
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Example
LMNP is a kite. Find x.
72
(x2-1)
48
M
N
P
o
o
oL
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m L + m M +m N +m P = 360 (Remember M ≅ P)
72 + (x2-1) + (x2-1) + 48 = 3602x2 + 118 = 360
2x2 = 242x2 = 121x = ±11
o
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66 READ is a kite. RE is congruent to ____.
A EA
B ADC DR R
E
A
D
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67 READ is a kite. A is congruent to ____.
A EB D
C RR
E
A
D
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68 Find the value of z in the kite.
z 5z-8
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69 Find the value of x in the kite.
68o
(8x+4)o
44o
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70 Find the value of x.
36
(3x 2 + 3)
24
o
o
o
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
Slide 162 / 189
72 Find the value of y in the kite.
12y
Slide 163 / 189
Families of Quadrilaterals
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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.
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Complete the chart by sliding the special quadrilateral next to its description. (There can be more than one answer).
squarerectanglerhombusparallelogram kite
trapezoid isosceles trapezoid
Description Answer(s)
An equilateral quadrilateral
An equiangular quadrilateral
The diagonals are perpendicular
The diagonals are congruent
Has at least 1 pair of parallel sides
rectangle & square
rhombus & square
rhombus, square & isosceles trapezoid
rectangle, square & kite
All except kite
Special Quadrilateral(s)
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QUADRILATERALS
Kite
Trapezoid
IsoscelesTrapezoid
Parallelogram
Rhombus Rectangle
Squa
re
Rhombus
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73 A rhombus is a square.
A alwaysB sometimes
C never
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74 A square is a rhombus.
A alwaysB sometimes
C never
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75 A rectangle is a rhombus.
A alwaysB sometimes
C never
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76 A trapezoid is isosceles.
A alwaysB sometimes
C never
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77 A kite is a quadrilateral.
A alwaysB sometimes
C never
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78 A parallelogram is a kite.
A alwaysB sometimes
C never
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Coordinate Proofs
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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
= 3 2 + 3 2 = (-3) 2 + 3 2 = 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
= 5 2 + (-3) 2 = (-5) 2 + (-3) 2
= 25 + 9 = 25 + 9 = 34 = 34 = 5.83 = 5.83
#
##
##
##
#
So, because SP=PQ and SR=RQ, PQRS is a kite.
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Given: JKLM is a parallelogramProve: JKLM is a square
2
4
6
8
10
-2
-4
-6
-8
-10
2 4 6 8 10-2-4-6-8-10 0
J (1,3)
K (4,-1)
L (0,-4)
(-3,0) M
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
= 3 2 + 4 2 = (-4) 2 + 3 2
= 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
Slide 181 / 189
Proofs
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Slide 182 / 189
Given: TE MA, <1 <2Prove: TEAM is a parallelogram.
T E
AM
1
2
≅ ≅
Slide 183 / 189T E
AM
1
2
Option A
s tatements reasons
1) TE ≅ MA, <1 ≅ <2 1) Given
2) EM ≅ EM 2) Reflexive Property
3) Triangle MTE ≅ Triangle EAM 3) S ide Angle S ide
4) TM ≅ AE 4) CPCTC
5) TEAM is a paralle logram5) The oppos ite s ides of a paralle logram are congruent
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T E
AM
1
2
Option B
We are given that TE MA and 2 3. TE AM, by the alternate interior angles converse.
So, TEAM is a parallelogram because each pair of opposite sides is parallel and congruent.
≅≅
click
click to reveal
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Given: FGHJ is a parallelogram, F is a right angleProve: FGHJ is a rectangle
F G
HJ
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F G
HJ
s tatements reasons
1) FGHJ is a paralle logramand F is a right angle 1) Given
2) J and G are right angles 2) The consecutive angles of a paralle logram are supplementary
3) H is a right angle 3) The oppos ite angles of a paralle logram are congruent
4) TEAM is a rectangle 4) Rectangle Corollary
Slide 187 / 189
Given: COLD is a quadrilateral, m O=140o, m D =40o, m L=60o
Prove: COLD is a trapezoid
C O
LD
140o
40o60o
Slide 188 / 189C O
LD
140o
40o60o
s tatements reasons
1) COLD is a quadrilateral,m O=140,m L=40,m D=60 1) Given
2) m O + m L = 180m L + m D = 100 2) Angle Addition
3) O and D are supplementary 3) Definition of Supplementary Angles
4) L and D are not supplementary 4) Definition of Supplementary Angles
5) CO is paralle l to LD 5) Consecutive Interior Angles Converse
6) CL is not paralle l to OD 6) Consecutive Interior Angles Converse
7) COLD is a trapezoid 7) Definition of a Trapezoid(A trapezoid has one pair of paralle l s ides)
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Try this ...
Given: FCD FEDProve: FD CE
F
C
D
E
≅