Quadrilaterals

1
Quadrilaterals

• Eleanor Roosevelt High School
• Chin-Sung Lin

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Definitions of the Quadrilaterals
ERHS Math Geometry
Mr. Chin-Sung Lin
3
Quadrilaterals
ERHS Math Geometry
A quadrilateral is a polygon with four sides
Mr. Chin-Sung Lin
4
Parts Properties of the Quadrilaterals
ERHS Math Geometry
Mr. Chin-Sung Lin
5
Consecutive (Adjacent) Vertices
ERHS Math Geometry
Consecutive vertices or adjacent vertices are
vertices that are endpoints of the same side P
and Q, Q and R, R and S, S and P
Q
P
R
S
Mr. Chin-Sung Lin
6
Consecutive (Adjacent) Sides
ERHS Math Geometry
Consecutive sides or adjacent sides are sides
that have a common endpoint PQ and QR, QR and RS,
RS and SP, SP and PQ
Q
P
R
S
Mr. Chin-Sung Lin
7
Opposite Sides
ERHS Math Geometry
Opposite sides of a quadrilateral are sides that
do not have a common endpoint PQ and RS, SP and QR
Q
P
R
S
Mr. Chin-Sung Lin
8
Consecutive angles
ERHS Math Geometry
Consecutive angles of a quadrilateral are angles
whose vertices are consecutive ?P and ?Q, ?Q and
?R, ?R and ?S, ?S and ?P
Q
P
R
S
Mr. Chin-Sung Lin
9
Opposite Angles
ERHS Math Geometry
Opposite angles of a quadrilateral are angles
whose vertices are not consecutive ?P and ?R, ?Q
and ?S
Q
P
R
S
Mr. Chin-Sung Lin
10
Diagonals
ERHS Math Geometry
A diagonal of a quadrilateral is a line segment
whose endpoints are two nonadjacent vertices of
the quadrilateral PR and QS
Q
P
R
S
Mr. Chin-Sung Lin
11
Sum of the Measures of Angles
ERHS Math Geometry
The sum of the measures of the angles of a
quadrilateral is 360 degrees m?P m?Q m?R
m?S 360
Q
P
R
S
Mr. Chin-Sung Lin
12
Parallelograms
ERHS Math Geometry
Mr. Chin-Sung Lin
13
Parallelogram
ERHS Math Geometry
A parallelogram is a quadrilateral in which two
pairs of opposite sides are parallel AB CD,
AD BC A parallelogram can be denoted by the
symbol ABCD The use of arrowheads, pointing
in the same direction, to show sides that are
parallel in the figure
Mr. Chin-Sung Lin
14
Theorems of Parallelogram
ERHS Math Geometry
Mr. Chin-Sung Lin
15
Theorems of Parallelogram
ERHS Math Geometry
Theorem of Dividing Diagonals Theorem of
Opposite Sides Theorem of Opposite Angles Theorem
of Bisecting Diagonals Theorem of Consecutive
Angles
Mr. Chin-Sung Lin
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Theorem of Dividing Diagonals
ERHS Math Geometry
A diagonal divides a parallelogram into two
congruent triangles If ABCD is a parallelogram,
then ? ABD ? ? CDB
Mr. Chin-Sung Lin
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Theorem of Dividing Diagonals
ERHS Math Geometry
Statements Reasons 1. ABCD is a
parallelogram 1. Given 2. AB DC and AD
BC 2. Definition of parallelogram 3. ?1 ? ?2
and ?3 ? ?4 3. Alternate interior angles 4. BD
? BD 4. Reflexive property 5. ? ABD ? ?
CDB 5. ASA postulate
Mr. Chin-Sung Lin
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Theorem of Opposite Sides
ERHS Math Geometry
Opposite sides of a parallelogram are
congruent If ABCD is a parallelogram, then AB ?
CD, and BC ? DA
Mr. Chin-Sung Lin
19
Theorem of Opposite Sides
ERHS Math Geometry
Statements Reasons 1. ABCD is a
parallelogram 1. Given 2. Connect BD 2. Form
two triangles 3. AB DC and AD BC 3.
Definition of parallelogram 4. ?1 ? ?2 and ?3 ?
?4 4. Alternate interior angles 5.
BD ? BD 5. Reflexive property 6. ? ABD ? ?
CDB 6. ASA postulate 7. AB ? CD and BC ? DA
7. CPCTC
Mr. Chin-Sung Lin
20
Application Example 1
ERHS Math Geometry
ABCD is a parallelogram, whats the perimeter of
ABCD ?
A
B
15
10
D
C
Mr. Chin-Sung Lin
21
Application Example 1
ERHS Math Geometry
ABCD is a parallelogram, whats the perimeter of
ABCD ? perimeter 50
A
B
15
10
D
C
Mr. Chin-Sung Lin
22
Application Example 2
ERHS Math Geometry
ABCD is a parallelogram, if the perimeter of ABCD
is 80, solve for x
A
B
x-20
10
D
C
Mr. Chin-Sung Lin
23
Application Example 2
ERHS Math Geometry
ABCD is a parallelogram, if the perimeter of ABCD
is 80, solve for x x 50
A
B
x-20
10
D
C
Mr. Chin-Sung Lin
24
Theorem of Opposite Angles
ERHS Math Geometry
Opposite angles of a parallelogram are
congruent If ABCD is a parallelogram, then ?A ?
?C, and ?B ? ?D
Mr. Chin-Sung Lin
25
Theorem of Opposite Angles
ERHS Math Geometry
Statements Reasons 1. ABCD is a
parallelogram 1. Given 2. AB DC and AD
BC 2. Definition of parallelogram 3. ?A and ?B
are supplementary 3. Same side interior angles
?A and ?D are supplementary ?C and ?B are
supplementary 4. ?A ? ?C 4. Supplementary
angle theorem ?B ? ?D
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Application Example 3
ERHS Math Geometry
ABCD is a parallelogram, what are the values of
x and y?
A
B
120o
60o
x
y
D
C
Mr. Chin-Sung Lin
27
Application Example 3
ERHS Math Geometry
ABCD is a parallelogram, what are the values of
x and y? x 120o y 60o
A
B
120o
60o
x
y
D
C
Mr. Chin-Sung Lin
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Application Example 4
ERHS Math Geometry
ABCD is a parallelogram, what are the values of
x and y?
A
B
X20
y - 20
2x - 60
180 - y
D
C
Mr. Chin-Sung Lin
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Application Example 4
ERHS Math Geometry
ABCD is a parallelogram, what are the values of
x and y? x 80o y 100o
A
B
X20
y - 20
2x - 60
180 - y
D
C
Mr. Chin-Sung Lin
30
Theorem of Bisecting Diagonals
ERHS Math Geometry
The diagonals of a parallelogram bisect each
other If ABCD is a parallelogram, then AC and BD
bisect each other at O
Mr. Chin-Sung Lin
31
Theorem of Bisecting Diagonals
ERHS Math Geometry
Statements Reasons 1. ABCD is a
parallelogram 1. Given 2. AB DC 2.
Definition of parallelogram 3. ?1 ? ?2 and ?3 ?
?4 3. Alternate interior angles 4. AB ?
DC 4. Opposite sides congruent 5. ? AOB ? ?
COD 5. ASA postulate 6. AO OC and BO OD 6.
CPCTC 7. AC and BD bisect each other 7.
Definition of segment bisector
Mr. Chin-Sung Lin
32
Application Example 5
ERHS Math Geometry
ABCD is a parallelogram, if AO 3, BO 4 AB
6, AC BD ?
A
B
6
3
4
O
D
C
Mr. Chin-Sung Lin
33
Application Example 5
ERHS Math Geometry
ABCD is a parallelogram, if AO 3, BO 4 AB
6, AC BD ? AC BD 24
A
B
6
3
4
O
D
C
Mr. Chin-Sung Lin
34
Application Example 6
ERHS Math Geometry
ABCD is a parallelogram, if AO x4, BO 2y-6,
CO 3x-4, an DO y2, solve for x and y
A
B
x4
2y-6
O
y2
3x-4
D
C
Mr. Chin-Sung Lin
35
Application Example 6
ERHS Math Geometry
ABCD is a parallelogram, if AO x4, BO 2y-6,
CO 3x-4, an DO y2, solve for x and y x 4
y 8
A
B
x4
2y-6
O
y2
3x-4
D
C
Mr. Chin-Sung Lin
36
Theorem of Consecutive Angles
ERHS Math Geometry
The consecutive angles of a parallelogram are
supplementary If ABCD is a parallelogram,
then ?A and ?B are supplementary ?C and ?D are
supplementary ?A and ?D are supplementary ?B and
?C are supplementary
Mr. Chin-Sung Lin
37
Theorem of Consecutive Angles
ERHS Math Geometry
A
B
Statements Reasons 1. ABCD is a
parallelogram 1. Given 2. AB DC and AD
BC 2. Definition of parallelogram 3. ?A and ?B,
?C and ?D 3. Same-side interior angles ?A
and ?D, ?B and ?C are supplementary are
supplementary
D
C
Mr. Chin-Sung Lin
38
Application Example 7
ERHS Math Geometry
ABCD is a parallelogram, what are the values of
x, y and z?
A
B
120o
x
y
z
D
C
Mr. Chin-Sung Lin
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Application Example 7
ERHS Math Geometry
ABCD is a parallelogram, what are the values of
x, y and z? x 60o y 120o z 60o
A
B
120o
x
y
z
D
C
Mr. Chin-Sung Lin
40
Application Example 8
ERHS Math Geometry
ABCD is a parallelogram, what are the values of
x and y?
A
B
X30
X-30
Y20
D
C
Mr. Chin-Sung Lin
41
Application Example 8
ERHS Math Geometry
ABCD is a parallelogram, what are the values of
x and y? x 90o y 100o
A
B
X30
X-30
Y20
D
C
Mr. Chin-Sung Lin
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Group Work
ERHS Math Geometry
Mr. Chin-Sung Lin
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Question 1
ERHS Math Geometry
ABCD is a parallelogram, calculate the perimeter
of ABCD
A
B
x30
2y-10
y10
D
C
2x-10
Mr. Chin-Sung Lin
44
Question 1
ERHS Math Geometry
ABCD is a parallelogram, calculate the perimeter
of ABCD perimeter 200
A
B
x30
2y-10
y10
D
C
2x-10
Mr. Chin-Sung Lin
45
Question 2
ERHS Math Geometry
ABCD is a parallelogram, solve for x
A
B
X30
X-10
O
X10
2X
D
C
Mr. Chin-Sung Lin
46
Question 2
ERHS Math Geometry
ABCD is a parallelogram, solve for x x 30
A
B
X30
X-10
O
X10
2X
D
C
Mr. Chin-Sung Lin
47
Question 3
ERHS Math Geometry
Given ABCD is a parallelogram Prove XO ? YO
Mr. Chin-Sung Lin
48
Question 4
ERHS Math Geometry
Given ABCD is a parallelogram, BO ? OD Prove EO
? OF
A
B
E
O
D
C
F
Mr. Chin-Sung Lin
49
Question 5
ERHS Math Geometry
Given ABCD is a parallelogram, AF CE Prove
?FAB ? ?ECD
A
B
E
F
D
C
Mr. Chin-Sung Lin
50
Review Theorems of Parallelogram
ERHS Math Geometry
Theorem of Dividing Diagonals Theorem of
Opposite Sides Theorem of Opposite Angles Theorem
of Bisecting Diagonals Theorem of Consecutive
Angles
Mr. Chin-Sung Lin
51
Prove Quadrilaterals are Parallelograms
ERHS Math Geometry
Mr. Chin-Sung Lin
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Criteria for Proving Parallelograms
ERHS Math Geometry
Parallel opposite sides Congruent opposite
sides Congruent parallel opposite
sides Congruent opposite angles Supplementary
consecutive angles Bisecting diagonals
Mr. Chin-Sung Lin
53
Parallel Opposite Sides
ERHS Math Geometry
If both pairs of opposite sides of a
quadrilateral are parallel, then the
quadrilateral is a parallelogram If AB CD,
and BC DA then, ABCD is a parallelogram
Mr. Chin-Sung Lin
54
Parallel Opposite Sides
ERHS Math Geometry
Statements Reasons 1. AB CD and BC DA
1. Given 2. ABCD is a parallelogram 2.
Definition of parallelogram
Mr. Chin-Sung Lin
55
Application Example 1
ERHS Math Geometry
If m?1 m?2 m?3, then ABCD is a parallelogram
A
B
1
2
3
D
C
Mr. Chin-Sung Lin
56
Application Example 2
ERHS Math Geometry
ABCD is a quadrilateral as shown below, solve for
x
A
B
3x-20
50o
60o
60o
50o
D
C
2x10
Mr. Chin-Sung Lin
57
Application Example 2
ERHS Math Geometry
ABCD is a quadrilateral as shown below, solve for
x x 30
A
B
3x-20
50o
60o
60o
50o
D
C
2x10
Mr. Chin-Sung Lin
58
Congruent Opposite Sides
ERHS Math Geometry
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram If AB ? CD,
and BC ? DA then, ABCD is a parallelogram
Mr. Chin-Sung Lin
59
Congruent Opposite Sides
ERHS Math Geometry
If both pairs of opposite sides of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram If AB ? CD,
and BC ? DA then, ABCD is a parallelogram
Mr. Chin-Sung Lin
60
Congruent Opposite Sides
ERHS Math Geometry
Statements Reasons 1. Connect BD 1. Form
two triangles 2. AB ? CD and BC ? DA 2. Given
3. BD ? BD 3. Reflexive property 4. ? ABD ?
? CDB 4. SSS postulate 5. ?1 ? ?2 and ?3 ? ?4
5. CPCTC 6. AB DC and AD BC 6.
Converse of alternate interior angles
theorem 7. ABCD is a parallelogram 7.
Definition of parallelogram
Mr. Chin-Sung Lin
61
Application Example 3
ERHS Math Geometry
ABCD is a quadrilateral, solve for x
A
B
15
X50
10
10
2x-30
D
C
15
Mr. Chin-Sung Lin
62
Application Example 3
ERHS Math Geometry
ABCD is a quadrilateral, solve for x x 80
A
B
15
X50
10
10
2x-30
D
C
15
Mr. Chin-Sung Lin
63
Application Example 4
ERHS Math Geometry
ABCD is a parallelogram, if DF BE, then AECF is
also a parallelogram
Mr. Chin-Sung Lin
64
Congruent Parallel Opposite Sides
ERHS Math Geometry
If one pair of opposite sides of a quadrilateral
are both congruent and parallel, then the
quadrilateral is a parallelogram If AB ? CD,
and AB CD then, ABCD is a parallelogram
Mr. Chin-Sung Lin
65
Congruent Parallel Opposite Sides
ERHS Math Geometry
Statements Reasons 1. Connect BD 1. Form
two triangles 2. AB ? CD and AB CD 2. Given
3. BD ? BD 3. Reflexive property 4. ?1 ? ?2
4. Alternate interior angles 5. ? ABD ? ?
CDB 5. SAS postulate 6. ?3 ? ?4 6.
CPCTC 7. AD BC 7. Converse of alternate
interior angles theorem 8. ABCD is a
parallelogram 8. Definition of parallelogram
Mr. Chin-Sung Lin
66
Application Example 5
ERHS Math Geometry
ABCD is a quadrilateral, solve for x and y
A
B
X5
y50
30o
10
10
30o
2y-20
D
C
Mr. Chin-Sung Lin
67
Application Example 5
ERHS Math Geometry
ABCD is a quadrilateral, solve for x and y x
5 y 70o
A
B
X5
y50o
30o
10
10
30o
2y-20o
D
C
Mr. Chin-Sung Lin
68
Application Example 6
ERHS Math Geometry
ABCD is a parallelogram, if m?1 m?2, then AECF
is also a parallelogram
1
2
Mr. Chin-Sung Lin
69
Congruent Opposite Angles
ERHS Math Geometry
If both pairs of opposite angles of a
quadrilateral are congruent, then the
quadrilateral is a parallelogram If ?A ? ?C,
and ?B ? ?D Then, ABCD is a parallelogram
Mr. Chin-Sung Lin
70
Congruent Opposite Angles
ERHS Math Geometry
Statements Reasons 1. Connect BD 1. Form
two triangles 2. m?1 m?4 m?A ? 180
2. Triangle angle-sum theorem m?2 m?3 m?B
? 180 3. m?1 m?4 m?A 3. Addition
property m?2 m?3 m?C ? 360 4. m?1 m?3
m?B 4. Partition property m?4 m?2
m?D 5. m?A m?B m?C m?D 5. Substitution
property 360
Mr. Chin-Sung Lin
71
Congruent Opposite Angles
ERHS Math Geometry
Statements Reasons 6. ?A ? ?C and ?B ? ?D
6. Given 7. 2m?A 2m?B 360 7.
Substitution property 2m?A 2m?D 360 8.
m?A m?B 180 8. Division property m?A
m?D 180 9. AD BC, AB DC 9. Converse of
same-side interior angles 10. ABCD is
a parallelogram 10. Definition of parallelogram
Mr. Chin-Sung Lin
72
Application Example 7
ERHS Math Geometry
ABCD is a quadrilateral, solve for x
A
B
X30
130o
50o
50o
130o
D
C
2x-40
Mr. Chin-Sung Lin
73
Application Example 7
ERHS Math Geometry
ABCD is a quadrilateral, solve for x x 70
A
B
X30
130o
50o
50o
130o
D
C
2x-40
Mr. Chin-Sung Lin
74
Application Example 8
ERHS Math Geometry
if m?1 m?2, m?3 m?4, then ABCD is a
parallelogram
A
B
1
4
3
D
C
2
Mr. Chin-Sung Lin
75
Bisecting Diagonals
ERHS Math Geometry
If the diagonals of a quadrilateral bisect each
other, then the quadrilateral is a
parallelogram If AC and BD bisect each other at
O, then, ABCD is a parallelogram
Mr. Chin-Sung Lin
76
Bisecting Diagonals
ERHS Math Geometry
Statements Reasons 1. AC and BD bisect at
O 1. Given 2. AO ? CO and BO ? DO 2. Def. of
segment bisector 3. ?AOB ? ?COD, ?AOD ? ?COB
3. Vertical angles 4. ?AOB ? ?COD, ?AOD ? ?COB
4. SAS postulate 5. ?1 ? ?2 and ?3 ? ?4
5. CPCTC 6. AB DC and AD BC 6.
Converse of alternate interior angles
theorem 7. ABCD is a parallelogram 7.
Definition of parallelogram
Mr. Chin-Sung Lin
77
Application Example 9
ERHS Math Geometry
? AOB ? ? COD, then ABCD is a parallelogram
Mr. Chin-Sung Lin
78
Supplementary Consecutive Angles
ERHS Math Geometry
If an angle of a quadrilateral is supplementary
to both of its consecutive angles, then the
quadrilateral is a parallelogram If ?A and ?B
are supplementary ?A and ?D are
supplementary then, ABCD is a parallelogram
Mr. Chin-Sung Lin
79
Supplementary Consecutive Angles
ERHS Math Geometry
Statements Reasons 1. ?A and ?B, ?A and ?D
1. Given are supplementary 2. AB
DC and AD BC 2. Converse of
same-side interior angles theorem 3.
ABCD is a parallelogram 3. Definition
of parallelogram
Mr. Chin-Sung Lin
80
Application Example 10
ERHS Math Geometry
ABCD is a quadrilateral, solve for x
Mr. Chin-Sung Lin
81
Application Example 10
ERHS Math Geometry
ABCD is a quadrilateral, solve for x x 20
Mr. Chin-Sung Lin
82
Review Proving Parallelograms
ERHS Math Geometry
Parallel opposite sides Congruent opposite
sides Congruent parallel opposite
sides Congruent opposite angles Supplementary
consecutive angles Bisecting diagonals
Mr. Chin-Sung Lin
83
Rectangles
ERHS Math Geometry
Mr. Chin-Sung Lin
84
Rectangles
ERHS Math Geometry
A rectangle is a parallelogram containing one
right angle
Mr. Chin-Sung Lin
85
All Angles Are Right Angles
ERHS Math Geometry
All angles of a rectangle are right angles Given
ABCD is a rectangle with ?A 90o Prove ?B
90o, ?C 90o, ?D 90o
Mr. Chin-Sung Lin
86
All Angles Are Right Angles
ERHS Math Geometry
Statements Reasons 1. ABCD is a rectangle
?A 90o 1. Given 2. ?C 90o 2. Opposite
angles 3. m?A m?D 180 3. Consecutive angles
m?A m?B 180 4. 90 m?D 180 4.
Substitution 90 m?B 180 5. m?B 90, m?D
90 5. Subtraction 6. ?B 90o, ?D
90o 6. Def. of measurement of angles
Mr. Chin-Sung Lin
87
All Angles Are Right Angles
ERHS Math Geometry
The diagonals of a rectangle are congruent Given
ABCD is a rectangle Prove AC ? BD
Mr. Chin-Sung Lin
88
All Angles Are Right Angles
ERHS Math Geometry
Statements Reasons 1. ABCD is a rectangle
1. Given 2. ?C 90o, ?D 90o 2. All angles
are right angles 3. ?C ? ?D 3. Substitution 4.
DC ? DC 4. Reflexive 5. AD ? BC 5.
Opposite sides 6. ?ADC ? ?BCD 6. SAS
postulate 7. AC ? BD 7. CPCTC
Mr. Chin-Sung Lin
89
Properties of Rectangle
ERHS Math Geometry

• The properties of a rectangle
• All the properties of a parallelogram
• Four right angles (equiangular)
• Congruent diagonals

Mr. Chin-Sung Lin
90
Proving Rectangles
ERHS Math Geometry
Mr. Chin-Sung Lin
91
Proving Rectangles
ERHS Math Geometry

• To show that a quadrilateral is a rectangle, by
  showing that the quadrilateral is equiangular or
  a parallelogram
• that contains a right angle, or
• with congruent diagonals
• If a parallelogram does not contain a right
  angle, or doesnt have congruent diagonals, then
  it is not a rectangle

Mr. Chin-Sung Lin
92
Proving Rectangles
ERHS Math Geometry
If one angle of a parallelogram is a right angle,
then the parallelogram is a rectangle Given
ABCD is a parallelogram and m?A 90 Prove ABCD
is a rectangle
Mr. Chin-Sung Lin
93
Proving Rectangles
ERHS Math Geometry
If a quadrilateral is equiangular, it is a
rectangle Given ABCD is a quadrangular
m?A m?B m?C m?D Prove ABCD is a rectangle
Mr. Chin-Sung Lin
94
Proving Rectangles
ERHS Math Geometry
The diagonals of a parallelogram are
congruent Given AC ? BD Prove ABCD is a
rectangle
Mr. Chin-Sung Lin
95
Application Example
ERHS Math Geometry

• ABCD is a parallelogram, m?A 6x - 30 and m?C
  4x 10. Show that ABCD is a rectangle

Mr. Chin-Sung Lin
96
Application Example
ERHS Math Geometry

• ABCD is a parallelogram, m?A 6x - 30 and m?C
  4x 10. Show that ABCD is a rectangle
• x 20
• m?A 90
• ABCD is a rectangle

Mr. Chin-Sung Lin
97
Rhombuses
ERHS Math Geometry
Mr. Chin-Sung Lin
98
Rhombus
ERHS Math Geometry
A rhombus is a parallelogram that has two
congruent consecutive sides
Mr. Chin-Sung Lin
99
All Sides Are Congruent
ERHS Math Geometry
All sides of a rhombus are congruent Given ABCD
is a rhombus with AB ? DA Prove AB ? BC ? CD ? DA
Mr. Chin-Sung Lin
100
All Sides Are Congruent
ERHS Math Geometry
Statements Reasons 1. ABCD is a rhombus w.
AB ? DA 1. Given 2. AB ? DC, AD ? BC 2.
Opposite sides are congruent 3. AB ? BC ? CD ?
DA 3. Transitive
Mr. Chin-Sung Lin
101
Perpendicular Diagonals
ERHS Math Geometry
The diagonals of a rhombus are perpendicular to
each other Given ABCD is a rhombus Prove AC ? BD
O
Mr. Chin-Sung Lin
102
Perpendicular Diagonals
ERHS Math Geometry
Statements Reasons 1. ABCD is a rhombus 1.
Given 2. AO ? AO 2. Reflexive 3. AD ? AB 3.
Congruent sides 4. BO ? DO 4. Bisecting
diagonals 5. ?AOD ? ?AOB 5. SSS postulate 6.
?AOD ? ?AOB 6. CPCTC 7. m?AOD m?AOB 180 7.
Supplementary angles 8. 2m?AOD 180 8.
Substitution 9. ?AOD 90o 9.
Division pustulate 10. AC ? BD 10. Definition
of perpendicular
103
Diagonals Bisecting Angles
ERHS Math Geometry
The diagonals of a rhombus bisect its
angles Given ABCD is a rhombus Prove AC bisects
?DAB and ?DCB DB bisects ?CDA and ?CBA
Mr. Chin-Sung Lin
104
Diagonals Bisecting Angles
ERHS Math Geometry
Statements Reasons 1. ABCD is a rhombus 1.
Given 2. AD ? AB, DC ? BC 2. Congruent sides
AD ? DC, AB ? BC 3. AC ? AC, DB ? DB 3.
Reflexive postulate 4. ?ACD ? ?ACB, ?BAD ?
?BCD 4. SSS postulate 5. ?DAC ? ?BAC, ?DCA ?
?BCA 5. CPCTC ?ADB ? ?CDB, ?ABD ? ?CBD 6.
AC bisects ?DAB and ?DCB 6. Definition of angle
bisector DB bisects ?CDA and ?CBA
Mr. Chin-Sung Lin
105
Properties of Rhombus
ERHS Math Geometry

• The properties of a rhombus
• All the properties of a parallelogram
• Four congruent sides (equilateral)
• Perpendicular diagonals
• Diagonals that bisect opposite pairs of angles

Mr. Chin-Sung Lin
106
Proving Rhombus
ERHS Math Geometry
Mr. Chin-Sung Lin
107
Proving Rhombus
ERHS Math Geometry

• To show that a quadrilateral is a rhombus, by
  showing that the quadrilateral is equilateral or
  a parallelogram
• that contains two congruent consecutive sides
• with perpendicular diagonals, or
• with diagonals bisecting opposite angles
• If a parallelogram does not contain two congruent
  consecutive sides, or doesnt have perpendicular
  diagonals, then it is not a rectangle

Mr. Chin-Sung Lin
108
Proving Rhombus
ERHS Math Geometry
If a parallelogram has two congruent consecutive
sides, then the parallelogram is a
rhombus Given ABCD is a parallelogram and AB ?
DA Prove ABCD is a rhombus
Mr. Chin-Sung Lin
109
Proving Rhombus
ERHS Math Geometry
If a quadrilateral is equilateral, it is a
rhombus Given ABCD is a parallelogram and
AB ? BC ? CD ? DA Prove ABCD is a rhombus
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Proving Rhombus
ERHS Math Geometry
The diagonals of a parallelogram are
perpendicular Given AC ? BD Prove ABCD is a
rhombus
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Proving Rhombus
ERHS Math Geometry
Each diagonal of a rhombus bisects two angles of
the rhombus Given AC bisects ?DAB and
?DCB Prove ABCD is a rhombus
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Application Example
ERHS Math Geometry
ABCD is a parallelogram. AB 2x 1, DC 3x -
11, AD x 13 Prove ABCD is a rhombus
A
B
2x1
x13
D
C
3x-11
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Application Example
ERHS Math Geometry
ABCD is a parallelogram. AB 2x 1, DC 3x -
11, AD x 13 Prove ABCD is a rhombus x
12 AB AD 25 ABCD is a rhombus
A
B
2x1
x13
D
C
3x-11
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Application Example
ERHS Math Geometry

• ABCD is a parallelogram, AB 3x - 2, BC 2x
  2, and CD x 6. Show that ABCD is a rhombus

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Application Example
ERHS Math Geometry

• ABCD is a parallelogram, AB 3x - 2, BC 2x
  2, and CD x 6. Show that ABCD is a rhombus
• x 4
• AB BC 10
• ABCD is a rhombus

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Squares
ERHS Math Geometry
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Squares
ERHS Math Geometry
A square is a rectangle that has two congruent
consecutive sides
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Squares
ERHS Math Geometry
A square is a rectangle with four congruent sides
(an equilateral rectangle)
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Squares
ERHS Math Geometry
A square is a rhombus with four right angles (an
equiangular rhombus)
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Squares
ERHS Math Geometry
A square is an equilateral quadrilateral A square
is an equiangular quadrilateral
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Squares
ERHS Math Geometry
A square is a rhombus A square is a rectangle
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Properties of Square
ERHS Math Geometry

• The properties of a square
• All the properties of a parallelogram
• All the properties of a rectangle
• All the properties of a rhombus

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Proving Squares
ERHS Math Geometry
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Proving Squares
ERHS Math Geometry
If a rectangle has two congruent consecutive
sides, then the rectangle is a
square Given ABCD is a rectangle and AB ?
DA Prove ABCD is a square
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Proving Squares
ERHS Math Geometry
If one of the angles of a rhombus is a right
angle, then the rhombus is a square Given ABCD
is a rhombus and ?A 90o Prove ABCD
is a square
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Proving Squares
ERHS Math Geometry

• To show that a quadrilateral is a square, by
  showing that the quadrilateral is a
• rectangle with a pair of congruent consecutive
  sides, or
• a rhombus that contains a right angle

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Application Example
ERHS Math Geometry

• ABCD is a square, m?A 4x - 30, AB 3x 10
  and BC 4y. Solve x and y

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Application Example
ERHS Math Geometry

• ABCD is a square, m?A 4x - 30, AB 3x 10
  and BC 4y. Solve x and y
• 4x 30 90
• x 30
• y 25

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Review Questions
ERHS Math Geometry
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Question 1
ERHS Math Geometry

• A parallelogram where all angles are right angles
  (90o) is a _________?

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Question 1 Answer
ERHS Math Geometry

• A parallelogram where all angles are right angles
  (90o) is a _________?

Rectangle
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Question 2
ERHS Math Geometry

• A parallelogram where all sides are congruent is
  a _________?

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Question 2 Answer
ERHS Math Geometry

• A parallelogram where all sides are congruent is
  a _________?

Rhombus
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Question 3
ERHS Math Geometry

• A rectangle with four congruent sides is a
  _________?

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Question 3 Answer
ERHS Math Geometry

• A rectangle with four congruent sides is a
  _________?

Square
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Question 4
ERHS Math Geometry

• A rhombus with four right angles is a _________?

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Question 4 Answer
ERHS Math Geometry

• A rhombus with four right angles is a _________?

Square
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Question 5
ERHS Math Geometry

• A parallelogram with congruent diagonals is a
  _________?

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Question 5 Answer
ERHS Math Geometry

• A parallelogram with congruent diagonals is a
  _________?

Rectangle
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Question 6
ERHS Math Geometry

• A parallelogram where all angles are right angles
  and all sides are congruent is a _________?

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Question 6 Answer
ERHS Math Geometry

• A parallelogram where all angles are right angles
  and all sides are congruent is a _________?

Square
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Question 7
ERHS Math Geometry

• A parallelogram with perpendicular diagonals is a
  _________?

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Question 7 Answer
ERHS Math Geometry

• A parallelogram with perpendicular diagonals is a
  _________?

Rhombus
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Question 8
ERHS Math Geometry

• A parallelogram whose diagonals bisect opposite
  pairs of angles is a ______?

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Question 8 Answer
ERHS Math Geometry

• A parallelogram whose diagonals bisect opposite
  pairs of angles is a ______?

Rhombus
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Question 9
ERHS Math Geometry

• A quadrilateral which is both rectangle and
  rhombus is a _________?

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Question 9 Answer
ERHS Math Geometry

• A quadrilateral which is both rectangle and
  rhombus is a _________?

Square
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Question 10
ERHS Math Geometry

• Choose the right answer(s)
• A parallelogram is a rhombus
• A rectangle is a square
• A rhombus is a parallelogram

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Question 10 Answer
ERHS Math Geometry

• Choose the right answer(s)
• A parallelogram is a rhombus
• A rectangle is a square
• A rhombus is a parallelogram

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Question 11
ERHS Math Geometry

• Choose the right answer(s)
• A quadrilateral is a parallelogram
• A square is a rhombus
• A rectangle is a rhombus

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Question 11 Answer
ERHS Math Geometry

• Choose the right answer(s)
• A quadrilateral is a parallelogram
• A square is a rhombus
• A rectangle is a rhombus

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Question 12
ERHS Math Geometry

• Choose the right answer(s)
• A rectangle is a parallelogram
• A square is a rectangle
• A rhombus is a square

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Question 12 Answer
ERHS Math Geometry

• Choose the right answer(s)
• A rectangle is a parallelogram
• A square is a rectangle
• A rhombus is a square

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Trapezoids
ERHS Math Geometry
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Definitions of Trapezoids
ERHS Math Geometry
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Trapezoids
ERHS Math Geometry
A trapezoid is a quadrilateral that has exactly
one pair of parallel sides The parallel sides of
a trapezoid are called bases. The nonparallel
sides of a trapezoid are the legs
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Isosceles Trapezoids
ERHS Math Geometry
A trapezoid whose nonparallel sides are congruent
is called an isosceles trapezoid
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Median of a Trapezoid
ERHS Math Geometry
The median of a trapezoid is the line segment
connecting the midpoints of the nonparallel sides
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Examples of Trapezoids
ERHS Math Geometry
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Exercise - Trapezoids
ERHS Math Geometry
Which one is a trapezoid? Why?
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Exercise - Trapezoids
ERHS Math Geometry
Which one is a trapezoid? Why?
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Exercise - Trapezoids
ERHS Math Geometry
Which one is a trapezoid?
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Exercise - Trapezoids
ERHS Math Geometry
Which one is a trapezoid?
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Properties of Isosceles Trapezoids
ERHS Math Geometry
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Properties of Isosceles Trapezoids
ERHS Math Geometry

• The properties of a isosceles trapezoid
• Base angles are congruent
• Diagonals are congruent
• The property of a trapezoid
• Median is parallel to and average of the bases

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Congruent Base Angles
ERHS Math Geometry
In an isosceles trapezoid the two angles whose
vertices are the endpoints of either base are
congruent The upper and lower base angles are
congruent Given Isosceles trapezoid ABCD AB
CD and AD ? BC Prove ?A ? ?B ?C ? ?D
Mr. Chin-Sung Lin
167
Congruent Base Angles
ERHS Math Geometry
Given Isosceles trapezoid ABCD AB CD and
AD ? BC Prove ?A ? ?B ?C ? ?D
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Congruent Diagonals
ERHS Math Geometry
The diagonals of an isosceles trapezoid are
congruent Given Isosceles trapezoid ABCD AB
CD and AD ? BC Prove AC ? BD
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169
Congruent Diagonals
ERHS Math Geometry
Given Isosceles trapezoid ABCD AB CD and
AD ? BC Prove AC ? BD
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170
Parallel and Average Median
ERHS Math Geometry
The median of a trapezoid is parallel to the
bases, and its length is half the sum of the
lengths of the bases Given Isosceles trapezoid
ABCD AB CD and median EF Prove AB EF
, CD EF and EF (1/2)(AB CD)
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171
Parallel and Average Median
ERHS Math Geometry
Given Isosceles trapezoid ABCD AB CD and
median EF Prove AB EF , CD EF and
EF (1/2)(AB CD)
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172
Proving Trapezoids
ERHS Math Geometry
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Proving Trapezoids
ERHS Math Geometry
To prove that a quadrilateral is a trapezoid,
show that two sides are parallel and the other
two sides are not parallel To prove that a
quadrilateral is not a trapezoid, show that both
pairs of opposite sides are parallel or that both
pairs of opposite sides are not parallel
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174
Proving Isosceles Trapezoids
ERHS Math Geometry

• To prove that a trapezoid is an isosceles
  trapezoid, show that one of the following
  statements is true
• The legs are congruent
• The lower/upper base angles are congruent
• The diagonals are congruent

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Application Examples
ERHS Math Geometry
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Numeric Example of Trapezoids
ERHS Math Geometry
Isosceles Trapezoid ABCD, AB CD and AD ? BC
Solve for x and y
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Numeric Example of Trapezoids
ERHS Math Geometry
Isosceles Trapezoid ABCD, AB CD and AD ? BC
Solve for x and y x 60 y 20
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Numeric Example of Trapezoids
ERHS Math Geometry
Trapezoid ABCD, AB CD and median EF Solve for x
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Numeric Example of Trapezoids
ERHS Math Geometry
Trapezoid ABCD, AB CD and median EF Solve for
x x 6
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Proving Isosceles Trapezoids
ERHS Math Geometry
Given Trapezoid ABCD and ?A ? ?B Prove ABCD is
an isosceles trapezoid
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181
Proving Isosceles Trapezoids
ERHS Math Geometry
Given Trapezoid ABCD and AC ? BD Prove ABCD is
an isosceles trapezoid
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182
Proving Isosceles Trapezoids
ERHS Math Geometry
Given Trapezoid ABCD, AB CD and AE ? BE
Prove ABCD is an isosceles trapezoid
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183
Summary of Quadrilaterals
ERHS Math Geometry
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184
Properties of Quadrilaterals - 1
ERHS Math Geometry
Properties
Cong. Oppo. Sides (1 P)
Cong. Oppo. Sides (2 P)
Cong. Four Sides
Parallel Oppo. Sides (1P)
Parallel Oppo. Sides (2P)
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185
Properties of Quadrilaterals - 1
ERHS Math Geometry
Properties
Cong. Oppo. Sides (1 P) ? ? ? ? ?
Cong. Oppo. Sides (2 P)
Cong. Four Sides
Parallel Oppo. Sides (1P)
Parallel Oppo. Sides (2P)
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Properties of Quadrilaterals - 1
ERHS Math Geometry
Properties
Cong. Oppo. Sides (1 P) ? ? ? ? ?
Cong. Oppo. Sides (2 P) ? ? ? ?
Cong. Four Sides
Parallel Oppo. Sides (1P)
Parallel Oppo. Sides (2P)
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187
Properties of Quadrilaterals - 1
ERHS Math Geometry
Properties
Cong. Oppo. Sides (1 P) ? ? ? ? ?
Cong. Oppo. Sides (2 P) ? ? ? ?
Cong. Four Sides ? ?
Parallel Oppo. Sides (1P)
Parallel Oppo. Sides (2P)
Mr. Chin-Sung Lin
188
Properties of Quadrilaterals - 1
ERHS Math Geometry
Properties
Cong. Oppo. Sides (1 P) ? ? ? ? ?
Cong. Oppo. Sides (2 P) ? ? ? ?
Cong. Four Sides ? ?
Parallel Oppo. Sides (1P) ? ? ? ? ? ?
Parallel Oppo. Sides (2P)
Mr. Chin-Sung Lin
189
Properties of Quadrilaterals - 1
ERHS Math Geometry
Properties
Cong. Oppo. Sides (1 P) ? ? ? ? ?
Cong. Oppo. Sides (2 P) ? ? ? ?
Cong. Four Sides ? ?
Parallel Oppo. Sides (1P) ? ? ? ? ? ?
Parallel Oppo. Sides (2P) ? ? ? ?
Mr. Chin-Sung Lin
190
Properties of Quadrilaterals - 2
ERHS Math Geometry
Properties
Cong. Diagonals
Bisecting Diagonals
Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles
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191
Properties of Quadrilaterals - 2
ERHS Math Geometry
Properties
Cong. Diagonals ? ? ?
Bisecting Diagonals
Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles
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192
Properties of Quadrilaterals - 2
ERHS Math Geometry
Properties
Cong. Diagonals ? ? ?
Bisecting Diagonals ? ? ? ?
Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles
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193
Properties of Quadrilaterals - 2
ERHS Math Geometry
Properties
Cong. Diagonals ? ? ?
Bisecting Diagonals ? ? ? ?
Perpendicular Diagonals ? ?
Cong. Opposite Angles
Supp. Opposite Angles
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194
Properties of Quadrilaterals - 2
ERHS Math Geometry
Properties
Cong. Diagonals ? ? ?
Bisecting Diagonals ? ? ? ?
Perpendicular Diagonals ? ?
Cong. Opposite Angles ? ? ? ?
Supp. Opposite Angles
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195
Properties of Quadrilaterals - 2
ERHS Math Geometry
Properties
Cong. Diagonals ? ? ?
Bisecting Diagonals ? ? ? ?
Perpendicular Diagonals ? ?
Cong. Opposite Angles ? ? ? ?
Supp. Opposite Angles ? ? ?
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196
Properties of Quadrilaterals - 3
ERHS Math Geometry
Properties
Cong. Adj. Angles (1 P)
Cong. Adj. Angles (2 P)
Cong. Four Right Angles
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
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197
Properties of Quadrilaterals - 3
ERHS Math Geometry
Properties
Cong. Adj. Angles (1 P) ? ? ?
Cong. Adj. Angles (2 P)
Cong. Four Right Angles
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
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198
Properties of Quadrilaterals - 3
ERHS Math Geometry
Properties
Cong. Adj. Angles (1 P) ? ? ?
Cong. Adj. Angles (2 P) ? ? ?
Cong. Four Right Angles
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
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Properties of Quadrilaterals - 3
ERHS Math Geometry
Properties
Cong. Adj. Angles (1 P) ? ? ?
Cong. Adj. Angles (2 P) ? ? ?
Cong. Four Right Angles ? ?
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
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200
Properties of Quadrilaterals - 3
ERHS Math Geometry
Properties
Cong. Adj. Angles (1 P) ? ? ?
Cong. Adj. Angles (2 P) ? ? ?
Cong. Four Right Angles ? ?
Diagonals Bisect Angles ? ?
Non-Parallel Oppo. Sides
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201
Properties of Quadrilaterals - 3
ERHS Math Geometry
Properties
Cong. Adj. Angles (1 P) ? ? ?
Cong. Adj. Angles (2 P) ? ? ?
Cong. Four Right Angles ? ?
Diagonals Bisect Angles ? ?
Non-Parallel Oppo. Sides ? ?
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202
Quadrilaterals and Proofs
ERHS Math Geometry
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203
Quadrilaterals and Proofs
ERHS Math Geometry
Given Isosceles trapezoid ABCD AB CD and
AD ? BC Prove ?1 ? ?2
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204
Quadrilaterals and Proofs
ERHS Math Geometry
Given Parallelogram ABCD and ABDE Prove ?
EAD ? ? DBC
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205
Quadrilaterals and Proofs
ERHS Math Geometry
Given ABC is a right ?, O is the midpoint of
AC Prove ?1 ? ?2
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206
Quadrilaterals and Proofs
ERHS Math Geometry
Given ABCD is a rhombus, DBFE is an isosceles
trapezoid Prove CE ? CF
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207
Coordinate Geometry and Quadrilaterals
ERHS Math Geometry
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208
Proving Rectangles
ERHS Math Geometry

• To show that a quadrilateral is a rectangle, by
  showing that the quadrilateral is a parallelogram
• that contains a right angle, or
• with congruent diagonals

Mr. Chin-Sung Lin
209
Proving Rectangles
ERHS Math Geometry
Given The coordinates of the vertices of a
quadrilateral Prove A given quadrilateral is a
rectangle Can be done by . (in terms of
coordinate geometry)
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210
Proving Rectangles
ERHS Math Geometry

• Given The coordinates of the vertices of a
  quadrilateral
• Prove A given quadrilateral is a rectangle
• Can be done by proving a parallelogram and
• the product of the slopes of adjacent sides is
  equal to -1
• the diagonals have the same lengths

Mr. Chin-Sung Lin
211
Proving Rectangle - Parallelogram with a Right
Angle
ERHS Math Geometry
ABCD is a quadrilateral, where A (1, 1), B(7,
5), C(9, 2) and D(3, -2) prove ABCD is a
rectangle by proving that ABCD is a parallelogram
with a right angle
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212
Proving Rectangle - Parallelogram with Congruent
Diagonals
ERHS Math Geometry
ABCD is a quadrilateral, where A (1, 1), B(7,
5), C(9, 2) and D(3, -2) prove ABCD is a
rectangle by proving that ABCD is a parallelogram
with congruent diagonals
Mr. Chin-Sung Lin
213
Proving Rhombuses
ERHS Math Geometry

• To show that a quadrilateral is a rhombus, by
  showing that the quadrilateral
• has four congruent sides, or
• is a parallelogram
• a pair of adjacent sides are congruent
• the diagonals intersect at right angles, or
• the opposite angles are bisected by the diagonals

Mr. Chin-Sung Lin
214
Proving Rhombuses
ERHS Math Geometry
Given The coordinates of the vertices of a
quadrilateral Prove A given quadrilateral is a
rhombus Can be done by . (in terms of
coordinate geometry)
Mr. Chin-Sung Lin
215
Proving Rhombuses
ERHS Math Geometry

• Given The coordinates of the vertices of a
  quadrilateral
• Prove A given quadrilateral is a rhombus
• Can be done by proving
• All four sides have the same lengths
• A parallelogram and the adjacent sides have the
  same lengths
• A parallelogram with the product of the slopes of
  the diagonals is equal to -1

Mr. Chin-Sung Lin
216
Proving Rhombus - Quadrilateral with Four
Congruent Sides
ERHS Math Geometry
ABCD is a quadrilateral, where A (3, 7), B(5,
3), C(3, -1) and D(1, 3) prove ABCD is a rhombus
by proving that ABCD is a quadrilateral with four
congruent sides
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217
Proving Rhombus - Parallelogram with Congruent
Adjacent Sides
ERHS Math Geometry
ABCD is a quadrilateral, where A (3, 7), B(5,
3), C(3, -1) and D(1, 3) prove ABCD is a rhombus
by proving that ABCD is a parallelogram with a
pair of congruent adjacent sides
Mr. Chin-Sung Lin
218
Proving Rhombus - Parallelogram with
Perpendicular Diagonals
ERHS Math Geometry
ABCD is a quadrilateral, where A (3, 7), B(5,
3), C(3, -1) and D(1, 3) prove ABCD is a rhombus
by proving that ABCD is a parallelogram with
perpendicular diagonals
Mr. Chin-Sung Lin
219
Proving Squares
ERHS Math Geometry

• To show that a quadrilateral is a square, by
  showing that the quadrilateral is a
• a rhombus that contains a right angle, or
• a rectangle with a pair of congruent adjacent
  sides

Mr. Chin-Sung Lin
220
Proving Squares
ERHS Math Geometry
Given The coordinates of the vertices of a
quadrilateral Prove A given quadrilateral is a
square Can be done by . (in terms of
coordinate geometry)
Mr. Chin-Sung Lin
221
Proving Squares
ERHS Math Geometry

• Given The coordinates of the vertices of a
  quadrilateral
• Prove A given quadrilateral is a square
• Can be done by proving
• A rhombus and the product of the slopes of
  adjacent sides is equal to -1
• A rectangle and two adjacent sides have the same
  lengths

Mr. Chin-Sung Lin
222
Proving Squares - Rhombus with a Right Angle
ERHS Math Geometry
ABCD is a quadrilateral, where A (0, 4), B(3,
5), C(4, 2) and D(1, 1) prove ABCD is a square
by proving that ABCD is a rhombus with a right
angle
Mr. Chin-Sung Lin
223
Proving Squares - Rectangle with Congruent
Adjacent Sides
ERHS Math Geometry
ABCD is a quadrilateral, where A (0, 4), B(3,
5), C(4, 2) and D(1, 1) prove ABCD is a square
by proving that ABCD is a rectangle with
congruent adjacent sides
Mr. Chin-Sung Lin
224
Proving Trapezoids
ERHS Math Geometry
To prove that a quadrilateral is a trapezoid,
show that two sides are parallel and the other
two sides are not parallel
Mr. Chin-Sung Lin
225
Proving Trapezoids
ERHS Math Geometry
Given The coordinates of the vertices of a
quadrilateral Prove A given quadrilateral is a
trapezoid Can be done by . (in terms of
coordinate geometry)
Mr. Chin-Sung Lin
226
Proving Trapezoids
ERHS Math Geometry

• Given The coordinates of the vertices of a
  quadrilateral
• Prove A given quadrilateral is a trapezoid
• Can be done by proving
• the slopes of one pair of opposite sides are
  equal while the slopes of the other pair of
  opposite sides are not equal

Mr. Chin-Sung Lin
227
Proving Trapezoids - Parallel Bases and
Non-Parallel Legs
ERHS Math Geometry
ABCD is a quadrilateral, where A (-3, 5), B(4,
5), C(6, 1) and D(-5, 1) prove ABCD is a
trapezoid by proving that there are two parallel
bases and two non-parallel legs
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228
Proving Isosceles Trapezoids
ERHS Math Geometry

• To prove that a trapezoid is an isosceles
  trapezoid, show that one of the following
  statements is true
• The legs are congruent
• The lower/upper base angles are congruent
• The diagonals are congruent

Mr. Chin-Sung Lin
229
Proving Isosceles Trapezoids
ERHS Math Geometry
Given The coordinates of the vertices of a
quadrilateral Prove A given quadrilateral is
an isosceles trapezoid Can be done by
. (in terms of coordinate geometry)
Mr. Chin-Sung Lin
230
Proving Isosceles Trapezoids
ERHS Math Geometry

• Given The coordinates of the vertices of a
  quadrilateral
• Prove A given quadrilateral is an isosceles
  trapezoid
• Can be done by proving
• A trapezoid whose two legs have the same lengths
• A trapezoid whose two diagonals have the same
  lengths

Mr. Chin-Sung Lin
231
Proving Isosceles Trapezoids - Trapezoid with
Congruent Legs
ERHS Math Geometry
ABCD is a quadrilateral, where A (-3, 5), B(4,
5), C(6, 1) and D(-5, 1) prove ABCD is an
isosceles trapezoid by proving that ABCD is a
trapezoid with congruent legs
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232
Proving Isosceles Trapezoids - Trapezoid w.
Congruent Diagonals
ERHS Math Geometry
ABCD is a quadrilateral, where A (-3, 5), B(4,
5), C(6, 1) and D(-5, 1) prove ABCD is an
isosceles trapezoid by proving that ABCD is a
trapezoid with congruent diagonals
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Application Example
ERHS Math Geometry
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Finding the Type of Quadrilateral
ERHS Math Geometry
Given ABCD is a quadrilateral, where A (3, 6),
B(7, 0), C(1, -4), D(-3, 2) Find the type of
quadrilateral ABCD
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Areas of Polygons
ERHS Math Geometry
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Areas of Polygons
ERHS Math Geometry
The area of a polygon is the unique real number
assigned to any polygon that indicates the number
of non-overlapping square units contained in the
polygons interior
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Areas of Quadrilaterals
ERHS Math Geometry
The area of a quadrilateral is the product of the
length of the base and the length of the altitude
(height)
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Areas of Parallelograms
ERHS Math Geometry
The area of a parallelogram is the product of the
length of the base and the length of the altitude
(height)
altitude
base
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Q A
ERHS Math Geometry
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The End
ERHS Math Geometry
Mr. Chin-Sung Lin