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New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative
Slide 1 / 189
www.njctl.org
2014-06-03
Quadrilaterals
Geometry
Slide 2 / 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 3 / 189
Angles of Polygons
Return to the Table of Contents
Slide 4 / 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 5 / 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 6 / 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 7 / 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 8 / 189
1 The figure below is a polygon.
True
False
Ans
wer
Slide 9 / 189
1 The figure below is a polygon.
True
False
[This object is a pull tab]
Ans
wer
False
Slide 9 (Answer) / 189
2 The figure below is a polygon.
True
False
Ans
wer
Slide 10 / 189
2 The figure below is a polygon.
True
False
[This object is a pull tab]
Ans
wer
True
Slide 10 (Answer) / 189
3 Indentify the polygon.
A Pentagon
B Octagon
C Quadrilateral
D HexagonE Decagon
F Triangle Ans
wer
Slide 11 / 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 11 (Answer) / 189
4 Is the polygon convex or concave?
A Convex
B Concave
Ans
wer
Slide 12 / 189
4 Is the polygon convex or concave?
A Convex
B Concave
[This object is a pull tab]
Ans
wer
B
Slide 12 (Answer) / 189
5 Is the polygon convex or concave?
A ConvexB Concave
Ans
wer
Slide 13 / 189
5 Is the polygon convex or concave?
A ConvexB Concave
[This object is a pull tab]
Ans
wer
A
Slide 13 (Answer) / 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 14 / 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 / 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 15 (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
Ans
wer
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
[This object is a pull tab]
Ans
wer
F, H
Slide 16 (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
Ans
wer
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
[This object is a pull tab]
Ans
wer
C, F, I
Slide 17 (Answer) / 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 18 / 189
Polygon Number of Sides
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 19 / 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 / 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 20 (Answer) / 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 Sides
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 21 / 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 22 / 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 23 / 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 angles
measure 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 24 / 189
9 What is the sum of the measures of the interior angles of the stop sign?
Ans
wer
Slide 25 / 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 25 (Answer) / 189
10 If the stop sign is a regular polygon. What is the measure of each interior angle?
Ans
wer
Slide 26 / 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 26 (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
Ans
wer
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
[This object is a pull tab]
Ans
wer
C
Slide 27 (Answer) / 189
12 What is the measure of each interior angle of a regular 20-gon?
A 162
B 3240
C 180
D 60 Ans
wer
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
[This object is a pull tab]
Ans
wer
A
Slide 28 (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
Ans
wer
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
[This object is a pull tab]
Ans
wer
D
Slide 29 (Answer) / 189
14 What is the value of x?
(5x+
15)o
(9x-6) o
(8x) o
(11x+16)o
(10x+8)oA
nsw
er
Slide 30 / 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 30 (Answer) / 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 31 / 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 32 / 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 / 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 33 (Answer) / 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 / 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 34 (Answer) / 189
17 What is the sum of the measures of the exterior angles of a pentagon?
Ans
wer
Slide 35 / 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 35 (Answer) / 189
18 If a pentagon is regular, what is the measure of each exterior angle?
Ans
wer
Slide 36 / 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 36 (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.
Ans
wer
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.
[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 37 (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 Ans
wer
o
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
[This object is a pull tab]
Ans
wer
D
Slide 38 (Answer) / 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 / 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 39 (Answer) / 189
Properties of Parallelograms
Return to the Table of Contents
Slide 40 / 189
Lab - Investigating Parallelograms
Lab - Properties of Parallelograms
Click on the links below and complete the two labs before the Parallelogram lesson.
Slide 41 / 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 42 / 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 43 / 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 44 / 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 45 / 189
Example:
ABCD is parallelogram.
Find w, x, y, and z.
A B
CD
12
2y
x-5
9
65o
5zo
wo
Slide 46 / 189
A B
CD
12
2y
x-5
9
65o
5zo
wo
The opposite sides are congruent.
Ans
wer
Slide 47 / 189
A B
CD
12
2y
x-5
9
65o
5zo
wo
The opposite sides are congruent.
[This object is a pull tab]
Ans
wer AB = DC
2y = 92 2y = 4.5
BC = ADx-5 = 12 +5 +5 x = 17
Slide 47 (Answer) / 189
A B
CD
12
2y
x-5
9
65o
5zo
wo
The opposite angles are congruent.
Ans
wer
Slide 48 / 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 48 (Answer) / 189
A B
CD
12
2y
x-5
9
65o
5zo
wo
The consecutive angles are supplementary.
Ans
wer
Slide 49 / 189
A B
CD
12
2y
x-5
9
65o
5zo
wo
The 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 49 (Answer) / 189
21 DEFG is a parallelogram. Find w.
D E
FG
70o
15
3x-32w z+12
21
y2 A
nsw
er
Slide 50 / 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 50 (Answer) / 189
22 DEFG is a parallelogram. Find x.
D E
FG
70o
15
3x-32w z+12
21
y2
Ans
wer
Slide 51 / 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 51 (Answer) / 189
23 DEFG is a parallelogram. Find y.
D E
FG
70o
15
3x-32w z+12
21
y2
Ans
wer
Slide 52 / 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 52 (Answer) / 189
24 DEFG is a parallelogram. Find z.
D E
FG
70o
15
3x-32w z+12
21
y2
Ans
wer
Slide 53 / 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 53 (Answer) / 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 54 / 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 / 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 55 (Answer) / 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 / 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 = 10
4y = RS = 8y = 2
ER = RS + SE = 16
Slide 56 (Answer) / 189
25 In a parallelogram, the opposite sides are ________ parallel.
A sometimesB always
C never Ans
wer
Slide 57 / 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 57 (Answer) / 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 / 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 58 (Answer) / 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 / 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 59 (Answer) / 189
28 MATH is a parallelogram. Find m H.
M A
TH98o
2x-4
14
(3y+8)o
Ans
wer
Slide 60 / 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 60 (Answer) / 189
29 MATH is a parallelogram. Find x.
M A
TH98o
2x-4
14
(3y+8)o
Ans
wer
Slide 61 / 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 61 (Answer) / 189
30 MATH is a parallelogram. Find y.
M A
TH98o
2x-4
14
(3y+8)o
Ans
wer
Slide 62 / 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 62 (Answer) / 189
Proving Quadrilaterals are
Parallelograms
Return to the Table of Contents
Slide 63 / 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 64 / 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 65 / 189
Example
Tell whether PQRS is a parallelogram. Explain.
P
Q
R
S6
6
4
4 Ans
wer
Slide 66 / 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 66 (Answer) / 189
Example
Tell whether PQRS is a parallelogram. Explain.P Q
RS
Ans
wer
Slide 67 / 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 67 (Answer) / 189
31 Tell whether the quadrilateral is a parallelogram.
Yes
No
78o
136o
2
Ans
wer
Slide 68 / 189
31 Tell whether the quadrilateral is a parallelogram.
Yes
No
78o
136o
2
[This object is a pull tab]
Ans
wer
No
Slide 68 (Answer) / 189
32 Tell whether the quadrilateral is a parallelogram.
Yes
No3 3
5
4.99
Ans
wer
Slide 69 / 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 69 (Answer) / 189
33 Tell whether the quadrilateral is a parallelogram.
Yes
No
Ans
wer
Slide 70 / 189
33 Tell whether the quadrilateral is a parallelogram.
Yes
No
[This object is a pull tab]
Ans
wer
No
Slide 70 (Answer) / 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 71 / 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 72 / 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 73 / 189
34 Tell whether the quadrilateral is a parallelogram.
Yes
No
Ans
wer
Slide 74 / 189
34 Tell whether the quadrilateral is a parallelogram.
Yes
No
[This object is a pull tab]
Ans
wer
No
Slide 74 (Answer) / 189
35 Tell whether the quadrilateral is a parallelogram.
Yes
No141o
39o
49o
Ans
wer
Slide 75 / 189
35 Tell whether the quadrilateral is a parallelogram.
Yes
No141o
39o
49o
[This object is a pull tab]
Ans
wer
No
Slide 75 (Answer) / 189
36 Tell whether the quadrilateral is a parallelogram.
Yes
No
89.5
819
Ans
wer
Slide 76 / 189
36 Tell whether the quadrilateral is a parallelogram.
Yes
No
89.5
819
[This object is a pull tab]
Ans
wer
Yes
Slide 76 (Answer) / 189
37 Tell whether the quadrilateral is a parallelogram.
Yes
No
Ans
wer
Slide 77 / 189
37 Tell whether the quadrilateral is a parallelogram.
Yes
No
[This object is a pull tab]
Ans
wer
Yes
Slide 77 (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
Ans
wer
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
[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 78 (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
Ans
wer
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
[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 79 (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 blankA
nsw
er
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
[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 80 (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)
Ans
wer
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)
[This object is a pull tab]
Ans
wer
E
Slide 81 (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.
Ans
wer
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. [This object is a pull tab]
Ans
wer
F
Slide 82 (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)
Ans
wer
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)
[This object is a pull tab]
Ans
wer
F
Slide 83 (Answer) / 189
Constructing Parallelograms
Return to the Table of Contents
Slide 84 / 189
To construct a parallelogram, there are 3 steps.
Construct a Parallelogram
Slide 85 / 189
Step 1 - Use a ruler to draw a segment and its midpoint.
Construct a Parallelogram - Step 1
Slide 86 / 189
Step 2 - Draw another segment such that the midpoints coincide.
Construct a Parallelogram - Step 2
Slide 87 / 189
Why is this a parallelogram?
Step 3 - Connect the endpoints of the segments.
Construct a Parallelogram - Step 3 A
nsw
er
Slide 88 / 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 88 (Answer) / 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 89 / 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 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 3 - Connect the endpoints
Why is this a parallelogram?
3 steps to draw a parallelogram in a coordinate plane
Ans
wer
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
[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 91 (Answer) / 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 92 / 189
41 The opposite angles of a parallelogram are ...
A bisect
B congruent
C parallel
D supplementary
Ans
wer
Slide 93 / 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 93 (Answer) / 189
42 The consecutive angles of a parallelogram are ...
A bisect
B congruent
C parallel
D supplementary
Ans
wer
Slide 94 / 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 94 (Answer) / 189
43 The diagonals of a parallelogram ______ each other.
A bisect
B congruent
C parallel
D supplementary Ans
wer
Slide 95 / 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 95 (Answer) / 189
44 The opposite sides of a parallelogram are ...
A bisect
B congruent
C parallel
D supplementary
Ans
wer
Slide 96 / 189
44 The opposite sides of a parallelogram are ...
A bisect
B congruent
C parallel
D supplementary
[This object is a pull tab]
Ans
wer
B & C
Slide 96 (Answer) / 189
Rhombi, Rectanglesand Squares
Return to the Table of Contents
Slide 97 / 189
three special parallelograms
Rhombus
Rectangle
Square
All the same properties of a parallelogram apply to the rhombus, rectangle,
and square.
Slide 98 / 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 99 / 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 / 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 100 (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
Ans
wer
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[This object is a pull tab]
Ans
wer
A
Slide 101 (Answer) / 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 102 / 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 103 / 189
Example
EFGH is a rhombus.
Find x, y, and z.E F
G H
72o
z
2x-6
5y
10
Slide 104 / 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 105 / 189
Try this ...
The quadrilateral is a rhombus. Find x, y, and z.
8
86o
3x+2
z
12 y2
Ans
wer
Slide 106 / 189
Try this ...
The quadrilateral is a rhombus. Find x, y, and z.
8
86o
3x+2
z
12 y2 [This object is a pull tab]
Ans
wer x = 2
y = 4z = 470
Slide 106 (Answer) / 189
47 This is a rhombus. Find x.
xo
Ans
wer
Slide 107 / 189
47 This is a rhombus. Find x.
xo
[This object is a pull tab]
Ans
wer
90o
Slide 107 (Answer) / 189
48 This is a rhombus. Find x.
13
x-3
9 Ans
wer
Slide 108 / 189
48 This is a rhombus. Find x.
13
x-3
9
[This object is a pull tab]
Ans
wer
36
Slide 108 (Answer) / 189
49 This is a rhombus. Find x.
126ox A
nsw
er
Slide 109 / 189
49 This is a rhombus. Find x.
126ox
[This object is a pull tab]
Ans
wer
27o
Slide 109 (Answer) / 189
50 HJKL is a rhombus. Find the length of HJ.
H J
KL
6 M16
Ans
wer
Slide 110 / 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 110 (Answer) / 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 111 / 189
51 What value of y will make the quadrilateral a rectangle?
6y
12
Ans
wer
Slide 112 / 189
51 What value of y will make the quadrilateral a rectangle?
6y
12
[This object is a pull tab]
Ans
wer
15
Slide 112 (Answer) / 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 113 / 189
Example
RECT is a rectangle. Find x and y.
2x-5 13
63o9yo
R E
CT
Ans
wer
Slide 114 / 189
Example
RECT is a rectangle. Find x and y.
2x-5 13
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 114 (Answer) / 189
52 RSTU is a rectangle. Find z.R S
TU8z
Ans
wer
Slide 115 / 189
52 RSTU is a rectangle. Find z.R S
TU8z
[This object is a pull tab]
Ans
wer
11.25
Slide 115 (Answer) / 189
53 RSTU is a rectangle. Find z.R S
TU
4z-9
7
Ans
wer
Slide 116 / 189
53 RSTU is a rectangle. Find z.R S
TU
4z-9
7
[This object is a pull tab]
Ans
wer
4
Slide 116 (Answer) / 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 117 / 189
Example
The quadrilateral is a square. Find x, y, and z.
z - 4
(5x)o
6
3y
Ans
wer
Slide 118 / 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 118 (Answer) / 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 / 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 119 (Answer) / 189
54 The quadrilateral is a square. Find y.
A 2
B 3
C 4
D 5
18y
Ans
wer
Slide 120 / 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 120 (Answer) / 189
55 The quadrilateral is a rhombus. Find x.
A 2
B 3
C 4
D 5
2x + 6
4x
Ans
wer
Slide 121 / 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 121 (Answer) / 189
112o
(4x)o
56 The quadrilateral is parallelogram. Find x.
Ans
wer
Slide 122 / 189
112o
(4x)o
56 The quadrilateral is parallelogram. Find x.
[This object is a pull tab]
Ans
wer
17
Slide 122 (Answer) / 189
57 The quadrilateral is a rectangle. Find x.
10x
3x + 7
Ans
wer
Slide 123 / 189
57 The quadrilateral is a rectangle. Find x.
10x
3x + 7
[This object is a pull tab]
Ans
wer
1
Slide 123 (Answer) / 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 124 / 189
Lab - Quadrilaterals in the Coordinate Plane
Click on the link below and complete the lab.
Slide 125 / 189
Trapezoids
Return to theTable of Contents
Slide 126 / 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 127 / 189
An isosceles trapezoid is a trapezoid with congruent legs.
isosceles trapezoid
Slide 128 / 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 129 / 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 130 / 189
Slide 131 / 189
Slide 131 (Answer) / 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 / 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 132 (Answer) / 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 133 / 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 134 / 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 135 / 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 136 / 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 / 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 137 (Answer) / 189
60 The trapezoid is isosceles. Find x.
9
4
6x + 3
2x + 2
Ans
wer
Slide 138 / 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 138 (Answer) / 189
61 The trapeziod is isosceles. Find x.
137o
xo
Ans
wer
Slide 139 / 189
61 The trapeziod is isosceles. Find x.
137o
xo
[This object is a pull tab]
Ans
wer
143
Slide 139 (Answer) / 189
62 In trapezoid HIJK, can HI and KJ have slopes that are opposite reciprocals?
H I
JK
Ans
wer
YesNo
Slide 140 / 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 140 (Answer) / 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 141 / 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 142 / 189
P
Q R
S
L M
15
7
Example
PQRS is a trapezoid. Find LM.A
nsw
er
Slide 143 / 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 143 (Answer) / 189
P
Q R
S
L M
20
14.5
Example
PQRS is a trapezoid. Find PS.
Ans
wer
Slide 144 / 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 144 (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.
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.
[This object is a pull tab]
Ans
wer x = 18
y = 7z = 5
Slide 145 (Answer) / 189
63 EF is the midsegment of trapezoid HIJK. Find x.
H I
JK
E F
6
x
15
Ans
wer
Slide 146 / 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 146 (Answer) / 189
64 EF is the midsegment of trapezoid HIJK. Find x.
HI
J K
EF
x
19
10
Ans
wer
Slide 147 / 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 147 (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.
Ans
wer
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.
[This object is a pull tab]
Ans
wer
B and D
Slide 148 (Answer) / 189
Kites
Return to the Table of Contents
Slide 149 / 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 150 / 189
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
Slide 151 / 189
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
Slide 152 / 189
Example
LMNP is a kite. Find x.
72
(x2-1)
48
M
N
P
o
o
oL
Slide 153 / 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 154 / 189
66 READ is a kite. RE is congruent to ____.
A EA
B ADC DR R
E
A
D
Ans
wer
Slide 155 / 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 155 (Answer) / 189
67 READ is a kite. A is congruent to ____.
A EB D
C RR
E
A
D
Ans
wer
Slide 156 / 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 156 (Answer) / 189
68 Find the value of z in the kite.
z 5z-8
Ans
wer
Slide 157 / 189
68 Find the value of z in the kite.
z 5z-8
[This object is a pull tab]
Ans
wer
2
Slide 157 (Answer) / 189
69 Find the value of x in the kite.
68o
(8x+4)o
44o
Ans
wer
Slide 158 / 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 158 (Answer) / 189
70 Find the value of x.
36
(3x 2 + 3)
24 Ans
wer
o
o
o
Slide 159 / 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 159 (Answer) / 189
Theorem Q19
If a quadrilateral is a kite then the diagonals are perpendicular.
In kite ABCDAC BD
A
B
C
D
Slide 160 / 189
71 Find the value of x in the kite.
x
Ans
wer
Slide 161 / 189
71 Find the value of x in the kite.
x
[This object is a pull tab]
Ans
wer
90o
Slide 161 (Answer) / 189
72 Find the value of y in the kite.
12y
Ans
wer
Slide 162 / 189
72 Find the value of y in the kite.
12y
[This object is a pull tab]
Ans
wer
7.5
Slide 162 (Answer) / 189
Families of Quadrilaterals
Return to the Table of Contents
Slide 163 / 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.
Slide 164 / 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 165 / 189
QUADRILATERALS
Kite
Trapezoid
IsoscelesTrapezoid
Parallelogram
Rhombus Rectangle
Squa
re
Rhombus
Slide 166 / 189
73 A rhombus is a square.
A alwaysB sometimes
C never
Ans
wer
Slide 167 / 189
73 A rhombus is a square.
A alwaysB sometimes
C never
[This object is a pull tab]
Ans
wer
B
Slide 167 (Answer) / 189
74 A square is a rhombus.
A alwaysB sometimes
C never
Ans
wer
Slide 168 / 189
74 A square is a rhombus.
A alwaysB sometimes
C never
[This object is a pull tab]
Ans
wer
A
Slide 168 (Answer) / 189
75 A rectangle is a rhombus.
A alwaysB sometimes
C never
Ans
wer
Slide 169 / 189
75 A rectangle is a rhombus.
A alwaysB sometimes
C never
[This object is a pull tab]
Ans
wer
C
Slide 169 (Answer) / 189
76 A trapezoid is isosceles.
A alwaysB sometimes
C never
Ans
wer
Slide 170 / 189
76 A trapezoid is isosceles.
A alwaysB sometimes
C never
[This object is a pull tab]
Ans
wer
B
Slide 170 (Answer) / 189
77 A kite is a quadrilateral.
A alwaysB sometimes
C never
Ans
wer
Slide 171 / 189
77 A kite is a quadrilateral.
A alwaysB sometimes
C never
[This object is a pull tab]
Ans
wer
A
Slide 171 (Answer) / 189
78 A parallelogram is a kite.
A alwaysB sometimes
C never
Ans
wer
Slide 172 / 189
78 A parallelogram is a kite.
A alwaysB sometimes
C never
[This object is a pull tab]
Ans
wer
C
Slide 172 (Answer) / 189
Coordinate Proofs
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Slide 173 / 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 174 / 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 175 / 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.
Slide 176 / 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 177 / 189
J (1,3)
K (4,-1)
L (0,-4)
(-3,0) M
Since JKLM is a parallelogram, we know the opposite sides are parallel and congruent.
We also know that a square is a rectangle and a rhombus.We need to prove the sides are congruent and perpendicular.
MJ = (3-0)2 + (1-(-3))2 JK = (-1-3)2 + (4-1)2 = 32 + 42 = (-4)2 + 32
= 9 + 16 = 9 + 16 = 25 = 25 = 5 = 5
√ √√√√ √
√√
Slide 178 / 189
J (1,3)
K (4,-1)
L (0,-4)
(-3,0) M
mMJ = = mJK = =
3 - 0 31-(-3) 4
-1-3 -4 4-1 3
MJ JK and MJ JKWhat else do you know?
MJ LK and JK LM (Opposite sides are congruent)MJ LM and JK LK (Perpendicular Transversal Theorem)
JKLM is a square
Slide 179 / 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 / 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 180 (Answer) / 189
Proofs
Return to the Table of Contents
Slide 181 / 189
Given: TE MA, <1 <2Prove: TEAM is a parallelogram.
T E
AM
1
2
≅ ≅
Slide 182 / 189
T E
AM
1
2
Option A
statements reasons
1) TE ≅ MA, <1 ≅ <2 1) Given
2) EM ≅ EM 2) Reflexive Property
3) Triangle MTE ≅ Triangle EAM 3) Side Angle Side
4) TM ≅ AE 4) CPCTC
5) TEAM is a parallelogram 5) The opposite sides of a parallelogram are congruent
Slide 183 / 189
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
Slide 184 / 189
Given: FGHJ is a parallelogram, F is a right angleProve: FGHJ is a rectangle
F G
HJ
Slide 185 / 189
F G
HJ
statements reasons
1) FGHJ is a parallelogramand F is a right angle 1) Given
2) J and G are right angles 2) The consecutive angles of a parallelogram are supplementary
3) H is a right angle 3) The opposite angles of a parallelogram are congruent
4) TEAM is a rectangle 4) Rectangle Corollary
Slide 186 / 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 187 / 189
C 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 188 / 189
Try this ...
Given: FCD FEDProve: FD CE
F
C
D
E
≅H
int
Slide 189 / 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.
Slide 189 (Answer) / 189