Download Congruent Angle Pairs - Eleanor Roosevelt High School

Quadrilaterals
Eleanor Roosevelt High School
Chin-Sung Lin
ERHS Math Geometry
Definitions of the
Quadrilaterals
Mr. Chin-Sung Lin
ERHS Math Geometry
Quadrilaterals
A quadrilateral is a polygon with four sides
Mr. Chin-Sung Lin
ERHS Math Geometry
Parts & Properties of the
Quadrilaterals
Mr. Chin-Sung Lin
ERHS Math Geometry
Consecutive (Adjacent) Vertices
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
S
R
Mr. Chin-Sung Lin
ERHS Math Geometry
Consecutive (Adjacent) Sides
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
S
R
Mr. Chin-Sung Lin
ERHS Math Geometry
Opposite Sides
Opposite sides of a quadrilateral are sides that do
not have a common endpoint
PQ and RS, SP and QR
Q
P
S
R
Mr. Chin-Sung Lin
ERHS Math Geometry
Consecutive angles
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
S
R
Mr. Chin-Sung Lin
ERHS Math Geometry
Opposite Angles
Opposite angles of a quadrilateral are angles
whose vertices are not consecutive
P and
R,
Q and
S
Q
P
S
R
Mr. Chin-Sung Lin
ERHS Math Geometry
Diagonals
A diagonal of a quadrilateral is a line segment
whose endpoints are two nonadjacent vertices
of the quadrilateral
PR and QS
Q
P
S
R
Mr. Chin-Sung Lin
ERHS Math Geometry
Sum of the Measures of Angles
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
S
R
Mr. Chin-Sung Lin
ERHS Math Geometry
Parallelograms
Mr. Chin-Sung Lin
ERHS Math Geometry
A
B
Parallelogram
D
C
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
ERHS Math Geometry
Theorems of
Parallelogram
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorems of Parallelogram
Theorem of Dividing Diagonals
Theorem of Opposite Sides
Theorem of Opposite Angles
Theorem of Bisecting Diagonals
Theorem of Consecutive Angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorem of Dividing Diagonals
A diagonal divides a parallelogram into two
congruent triangles
A
If ABCD is a parallelogram, then
∆ ABD  ∆ CDB
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorem of Dividing Diagonals
A
B
1
3
4
D
2
Statements
C
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
ERHS Math Geometry
Theorem of Opposite Sides
Opposite sides of a parallelogram are congruent
If ABCD is a parallelogram, then
AB  CD, and
BC  DA
A
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorem of Opposite Sides
A
B
1
3
4
D
2
C
Statements
1.
2.
3.
4.
5.
6.
7.
ABCD is a parallelogram
Connect BD
AB || DC and AD || BC
1  2 and 3  4
BD  BD
∆ ABD  ∆ CDB
AB  CD and BC  DA
Reasons
1.
2.
3.
4.
5.
6.
7.
Given
Form two triangles
Definition of parallelogram
Alternate interior angles
Reflexive property
ASA postulate
CPCTC
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 1
ABCD is a parallelogram, what’s the perimeter of
ABCD ?
A
B
15
10
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 1
ABCD is a parallelogram, what’s the perimeter of
ABCD ?
A
B
15
perimeter = 50
10
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 2
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
ERHS Math Geometry
Application Example 2
ABCD is a parallelogram, if the perimeter of ABCD
is 80, solve for x
A
B
x-20
x = 50
10
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorem of Opposite Angles
Opposite angles of a parallelogram are congruent
If ABCD is a parallelogram, then
A  C, and
B  D
A
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorem of Opposite Angles
A
Statements
D
1. ABCD is a parallelogram
2. AB || DC and AD || BC
3. A and B are supplementary
A and D are supplementary
C and B are supplementary
4. A  C
B  D
B
C
Reasons
1. Given
2. Definition of parallelogram
3. Same side interior angles
4. Supplementary angle theorem
ERHS Math Geometry
Application Example 3
ABCD is a parallelogram, what are the values of x
and y?
A
B
120o
y
D
60o
x
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 3
ABCD is a parallelogram, what are the values of x
and y?
A
x = 120o
y = 60o
B
120o
y
D
60o
x
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 4
ABCD is a parallelogram, what are the values of x
and y?
A
X+20
180 - y
D
B
y - 20
2x - 60
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 4
ABCD is a parallelogram, what are the values of x
and y?
A
x = 80o
y = 100o
X+20
180 - y
D
B
y - 20
2x - 60
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorem of Bisecting Diagonals
The diagonals of a parallelogram bisect each other
If ABCD is a parallelogram, then
AC and BD bisect each other at O
A
B
O
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorem of Bisecting Diagonals
A
B
1
3
O
4
Statements
2
D
C
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
ERHS Math Geometry
Application Example 5
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
ERHS Math Geometry
Application Example 5
ABCD is a parallelogram, if AO = 3, BO = 4 AB = 6,
AC + BD = ?
A
B
6
AC + BD = 14
3
4
O
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 6
ABCD is a parallelogram, if AO = x+4, BO = 2y-6,
CO = 3x-4, an DO = y+2, solve for x and y
A
B
x+4
O
y+2
D
2y-6
3x-4
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 6
ABCD is a parallelogram, if AO = x+4, BO = 2y-6,
CO = 3x-4, an DO = y+2, solve for x and y
A
x=4
y=8
B
x+4
O
y+2
D
2y-6
3x-4
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorem of Consecutive Angles
The consecutive angles of a parallelogram are
supplementary
If ABCD is
A and
C and
A and
B and
a parallelogram, then
B are supplementary
D are supplementary
D are supplementary
C are supplementary
A
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorem of Consecutive Angles
A
Statements
D
B
C
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
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 7
ABCD is a parallelogram, what are the values of x,
y and z?
A
B
120o
z
D
x
y
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 7
ABCD is a parallelogram, what are the values of x,
y and z?
A
x = 60o
y = 120o
z = 60o
B
120o
z
D
x
y
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 8
ABCD is a parallelogram, what are the values of x
and y?
A
B
X+30
X-30
Y+20
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 8
ABCD is a parallelogram, what are the values of x
and y?
A
x = 90o
y = 100o
B
X+30
X-30
Y+20
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Group Work
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 1
ABCD is a parallelogram, calculate the perimeter of
ABCD
A
B
x+30
2y-10
D
y+10
2x-10
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 1
ABCD is a parallelogram, calculate the perimeter of
ABCD
perimeter = 200
A
B
x+30
2y-10
D
y+10
2x-10
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 2
ABCD is a parallelogram, solve for x
A
B
X+30
X-10
O
X+10
D
2X
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 2
ABCD is a parallelogram, solve for x
A
x = 30
B
X+30
X-10
O
X+10
D
2X
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 3
Given: ABCD is a parallelogram
Prove: XO  YO
A
X
B
O
D
Y
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 4
Given: ABCD is a parallelogram, BO  OD
Prove: EO  OF
A
E
B
O
D
F
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 5
Given: ABCD is a parallelogram, AF || CE
Prove: FAB  ECD
A
B
E
F
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Review: Theorems of Parallelogram
Theorem of Dividing Diagonals
Theorem of Opposite Sides
Theorem of Opposite Angles
Theorem of Bisecting Diagonals
Theorem of Consecutive Angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Prove Quadrilaterals
are Parallelograms
Mr. Chin-Sung Lin
ERHS Math Geometry
Criteria for Proving Parallelograms
Parallel opposite sides
Congruent opposite sides
Congruent & parallel opposite sides
Congruent opposite angles
Supplementary consecutive angles
Bisecting diagonals
Mr. Chin-Sung Lin
ERHS Math Geometry
Parallel Opposite Sides
If both pairs of opposite sides of a quadrilateral are
parallel, then the quadrilateral is a
parallelogram
A
AB || CD, and
BC || DA
then, ABCD is a parallelogram D
B
If
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Parallel Opposite Sides
A
D
Statements
B
C
Reasons
1. AB || CD and BC || DA
1. Given
2. ABCD is a parallelogram
2. Definition of parallelogram
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 1
If m1 = m2 = m3, then ABCD is a
parallelogram
A
B
1
2
3
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 2
ABCD is a quadrilateral as shown below, solve for x
A
60o
D
B
3x-20
50o
50o
2x+10
60o
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 2
ABCD is a quadrilateral as shown below, solve for x
A
x = 30
60o
D
B
3x-20
50o
50o
2x+10
60o
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Opposite Sides
If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a
parallelogram
A
AB  CD, and
BC  DA
then, ABCD is a parallelogram
If
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Opposite Sides
If both pairs of opposite sides of a quadrilateral are
congruent, then the quadrilateral is a
parallelogram
A
AB  CD, and
BC  DA
then, ABCD is a parallelogram
If
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Opposite Sides
A
B
1
3
4
Statements
1.
2.
3.
4.
5.
6.
D
2
Connect BD
AB  CD and BC  DA
BD  BD
∆ ABD  ∆ CDB
1  2 and 3  4
AB || DC and AD || BC
7. ABCD is a parallelogram
C
Reasons
1. Form two triangles
2. Given
3. Reflexive property
4. SSS postulate
5. CPCTC
6. Converse of alternate interior
angles theorem
7. Definition of parallelogram
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 3
ABCD is a quadrilateral, solve for x
A
B
15
X+50
10
10
2x-30
D
15
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 3
ABCD is a quadrilateral, solve for x
A
x = 80
B
15
X+50
10
10
2x-30
D
15
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 4
ABCD is a parallelogram, if DF = BE, then AECF is
also a parallelogram
A
D
E
F
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent & Parallel Opposite Sides
If one pair of opposite sides of a quadrilateral are
both congruent and parallel, then the
quadrilateral is a parallelogram
AB  CD, and
AB || CD
then, ABCD is a parallelogram
A
If
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent & Parallel Opposite Sides
A
B
1
3
4
Statements
1.
2.
3.
4.
5.
6.
7.
D
2
Connect BD
AB  CD and AB || CD
BD  BD
1  2
∆ ABD  ∆ CDB
3  4
AD || BC
C
Reasons
1. Form two triangles
2. Given
3. Reflexive property
4. Alternate interior angles
5. SAS postulate
6. CPCTC
7. Converse of alternate interior
angles theorem
8. ABCD is a parallelogram 8. Definition of parallelogram
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 5
ABCD is a quadrilateral, solve for x and y
A
B
X+5
y+50o
30o
10
30o
D
10
2y-20o
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 5
ABCD is a quadrilateral, solve for x and y
A
x=5
y = 70o
B
X+5
y+50o
30o
10
30o
D
10
2y-20o
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 6
ABCD is a parallelogram, if m1 = m2, then AECF
is also a parallelogram
A
E
B
1
2
D
F
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Opposite Angles
If both pairs of opposite angles of a quadrilateral
are congruent, then the quadrilateral is a
parallelogram
A
A  C, and
B  D
Then, ABCD is a parallelogram D
B
If
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Opposite Angles
A
B
1
3
4
Statements
D
2
1. Connect BD
2. m1 +m4 + mA  180
m2 +m3 + mC  180
3. m1 +m4 + mA +
m2 +m3 + mC  360
4. m1 +m3 = mB
m4 +m2 = mD
5. mA +mB + mC + mD
= 360
C
Reasons
1. Form two triangles
2. Triangle angle-sum theorem
3. Addition property
4. Partition property
5. Substitution property
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Opposite Angles
A
B
1
3
4
Statements
D
2
6. A  C and B  D
7. 2mA + 2mB = 360
2mA + 2mD = 360
8. mA + mB = 180
mA + mD = 180
9. AD || BC, AB || DC
10. ABCD is a parallelogram
C
Reasons
6. Given
7. Substitution property
8. Division property
9. Converse of same-side
interior angles
10. Definition of parallelogram
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 7
ABCD is a quadrilateral, solve for x
A
X+30
130o
50o
D
B
50o
130o
2x-40
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 7
ABCD is a quadrilateral, solve for x
A
x = 70
X+30
130o
50o
D
B
50o
130o
2x-40
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 8
if m1 = m2, m3 = m4, then ABCD is a
parallelogram
1
A
4
B
3
D
C
2
Mr. Chin-Sung Lin
ERHS Math Geometry
Bisecting Diagonals
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
A
B
O
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Bisecting Diagonals
A
B
1
O
4
Statements
3
2
D
C
Reasons
1. AC and BD bisect at O
1. Given
2. AO  CO and BO  DO
2. Def. of segment bisector
3.
4.
5.
6.
3.
4.
5.
6.
AOB  COD, AOD  COB
∆AOB  ∆COD, ∆AOD  ∆COB
1  2 and 3  4
AB || DC and AD || BC
7. ABCD is a parallelogram
Vertical angles
SAS postulate
CPCTC
Converse of alternate interior
angles theorem
7. Definition of parallelogram
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 9
∆ AOB  ∆ COD, then ABCD is a parallelogram
A
B
O
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Supplementary Consecutive Angles
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
A
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Supplementary Consecutive Angles
A
D
Statements
1. A and B, A and D
B
C
Reasons
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
ERHS Math Geometry
Application Example 10
ABCD is a quadrilateral, solve for x
A
2x+80
3x
D
B
100-2x
2(x+45)-10
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example 10
ABCD is a quadrilateral, solve for x
A
2x+80
x = 20
3x
D
B
100-2x
2(x+45)-10
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Review: Proving Parallelograms
Parallel opposite sides
Congruent opposite sides
Congruent & parallel opposite sides
Congruent opposite angles
Supplementary consecutive angles
Bisecting diagonals
Mr. Chin-Sung Lin
ERHS Math Geometry
Rectangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Rectangles
A rectangle is a parallelogram containing one right
angle
A
B
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
All Angles Are Right Angles
All angles of a rectangle are right angles
Given: ABCD is a rectangle with A = 90o
Prove: B = 90o, C = 90o, D = 90o
A
B
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
All Angles Are Right Angles
Statements
A
B
D
C
Reasons
1. ABCD is a rectangle & A = 90o 1. Given
2. C = 90o
2. Opposite angles
3. mA + mD = 180
mA + mB = 180
4. 90 + mD = 180
90 + mB = 180
5. mB = 90, mD = 90
6. B = 90o, D = 90o
3. Consecutive angles
4. Substitution
5. Subtraction
6. Def. of measurement of angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Diagonals Are Congruent
The diagonals of a rectangle are congruent
Given: ABCD is a rectangle
Prove: AC  BD
A
B
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
All Angles Are Right Angles
Statements
A
B
D
C
Reasons
1. ABCD is a rectangle
1. Given
2. C = 90o, D = 90o
2. All angles are right angles
3.
4.
5.
6.
7.
3.
4.
5.
6.
7.
C  D
DC  DC
AD  BC
∆ADC  ∆BCD
AC  BD
Substitution
Reflexive
Opposite sides
SAS postulate
CPCTC
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Rectangle
The properties of a rectangle
 All the properties of a parallelogram
 Four right angles (equiangular)
A
B
D
C
 Congruent diagonals
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Rectangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Rectangles
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 doesn’t
have congruent diagonals, then it is not a rectangle
Mr. Chin-Sung Lin
ERHS Math Geometry
A
B
D
C
Proving Rectangles
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
ERHS Math Geometry
A
B
D
C
Proving Rectangles
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
ERHS Math Geometry
B
A
Proving Rectangles
O
D
C
The diagonals of a parallelogram are congruent
Given: AC BD
Prove: ABCD is a rectangle
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example
ABCD is a parallelogram, mA = 6x - 30 and
mC = 4x + 10. Show that ABCD is a
rectangle
A
B
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example
ABCD is a parallelogram, mA = 6x - 30 and
mC = 4x + 10. Show that ABCD is a
rectangle
A
B
D
C
x =20
mA = 90
ABCD is a rectangle
Mr. Chin-Sung Lin
ERHS Math Geometry
Rhombuses
Mr. Chin-Sung Lin
ERHS Math Geometry
Rhombus
A rhombus is a parallelogram that has two
congruent consecutive sides
A
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
All Sides Are Congruent
All sides of a rhombus are congruent
Given: ABCD is a rhombus with AB
Prove: AB
BC
CD
DA
A
D
DA
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
All Sides Are Congruent
A
D
B
C
Statements
Reasons
1. ABCD is a rhombus w. AB
2. AB
DC, AD
congruent
3. AB
BC
CD
BC
DA
DA 1. Given
2. Opposite sides are
3. Transitive
Mr. Chin-Sung Lin
ERHS Math Geometry
Perpendicular Diagonals
The diagonals of a rhombus are perpendicular to
each other
A
Given: ABCD is a rhombus
Prove: AC  BD
D
O
B
C
Mr. Chin-Sung Lin
A
ERHS Math Geometry
Perpendicular Diagonals
Statements
O
D
Reasons
1. ABCD is a rhombus
1. Given
2. AO  AO
2. Reflexive
3. AD  AB
4. BO  DO
5. ∆AOD  ∆AOB
6. AOD  AOB
7. mAOD + mAOB = 180
8. 2mAOD = 180
9. AOD = 90o
10. AC  BD
3.
4.
5.
6.
7.
8.
B
C
Congruent sides
Bisecting diagonals
SSS postulate
CPCTC
Supplementary angles
Substitution
9. Division postulate
10. Definition of perpendicular
ERHS Math Geometry
Diagonals Bisecting Angles
The diagonals of a rhombus bisect its angles
Given: ABCD is a rhombus
A
Prove: AC bisects DAB and DCB
DB bisects CDA and CBA
D
B
C
Mr. Chin-Sung Lin
A
ERHS Math Geometry
Diagonals Bisecting Angles
Statements
D
Reasons
1. ABCD is a rhombus
1. Given
2. AD  AB, DC  BC
2. Congruent sides
B
C
AD  DC, AB  BC
3. AC  AC, DB  DB
4. ∆ACD  ∆ACB, ∆BAD  ∆BCD
5. DAC  BAC, DCA  BCA
ADB  CDB, ABD  CBD
6. AC bisects DAB and DCB
DB bisects CDA and CBA
3. Reflexive postulate
4. SSS postulate
5. CPCTC
6. Definition of angle bisector
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Rhombus
A
The properties of a rhombus
 All the properties of a parallelogram
D
B
 Four congruent sides (equilateral)
 Perpendicular diagonals
C
 Diagonals that bisect opposite pairs of angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Rhombus
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Rhombus
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 doesn’t have perpendicular
diagonals, then it is not a rectangle
Mr. Chin-Sung Lin
A
ERHS Math Geometry
Proving Rhombus
D
B
If a parallelogram has two congruent consecutive
C
sides, then the parallelogram is a rhombus
Given: ABCD is a parallelogram and AB
Prove: ABCD is a rhombus
DA
Mr. Chin-Sung Lin
A
ERHS Math Geometry
Proving Rhombus
D
If a quadrilateral is equilateral, it is a rhombus
B
C
Given: ABCD is a parallelogram and
AB
BC
CD
DA
Prove: ABCD is a rhombus
Mr. Chin-Sung Lin
ERHS Math Geometry
A
Proving Rhombus
D
B
The diagonals of a parallelogram are
perpendicular
C
Given: AC  BD
Prove: ABCD is a rhombus
Mr. Chin-Sung Lin
ERHS Math Geometry
A
1 2
Proving Rhombus
D
B
3 4
If each diagonal of a parallelogram bisects two
opposite angles, then it is a rhombus
C
Given: AC bisects DAB and DCB
BD bisects ADC and ABC
Prove: ABCD is a rhombus
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example
ABCD is a parallelogram. AB = 2x + 1,
DC = 3x - 11, AD = x + 13
Prove: ABCD is a rhombus
A
B
2x+1
x+13
D
3x-11
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example
ABCD is a parallelogram. AB = 2x + 1,
DC = 3x - 11, AD = x + 13
Prove: ABCD is a rhombus
A
x = 12
AB = AD = 25
ABCD is a rhombus
B
2x+1
x+13
D
3x-11
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example
ABCD is a parallelogram, AB = 3x - 2, BC = 2x +
2, and CD = x + 6. Show that ABCD is a
rhombus
A
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example
ABCD is a parallelogram, AB = 3x - 2, BC = 2x +
2, and CD = x + 6. Show that ABCD is a
rhombus
A
x=4
AB = BC = 10
D
B
ABCD is a rhombus
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Squares
Mr. Chin-Sung Lin
ERHS Math Geometry
Squares
A square is a rectangle that has two congruent
consecutive sides
A
B
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Squares
A square is a rectangle with four congruent sides
(an equilateral rectangle)
A
B
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Squares
A square is a rhombus with four right angles (an
equiangular rhombus)
A
B
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Squares
A square is an equilateral quadrilateral
A square is an equiangular quadrilateral
A
B
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Squares
A square is a rhombus
A square is a rectangle
A
B
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Square
The properties of a square
 All the properties of a parallelogram
 All the properties of a rectangle
A
B
D
C
 All the properties of a rhombus
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Squares
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Squares
If a rectangle has two congruent
consecutive sides, then the
rectangle is a square
A
B
D
C
Given: ABCD is a rectangle and AB
DA
Prove: ABCD is a square
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Squares
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
A
B
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Squares
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
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example
ABCD is a square, mA = 4x - 30, AB = 3x +
10 and BC = 4y. Solve x and y
A
B
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example
ABCD is a square, mA = 4x - 30, AB = 3x +
10 and BC = 4y. Solve x and y
A
B
D
C
4x – 30 = 90
x = 30
y = 25
Mr. Chin-Sung Lin
ERHS Math Geometry
Review Questions
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 1
A parallelogram where all angles are right angles
(90o) is a _________?
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 1 Answer
A parallelogram where all angles are right angles
(90o) is a _________?
Rectangle
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 2
A parallelogram where all sides are congruent is
a _________?
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 2 Answer
A parallelogram where all sides are congruent is
a _________?
Rhombus
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 3
A rectangle with four congruent sides is a
_________?
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 3 Answer
A rectangle with four congruent sides is a
_________?
Square
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 4
A rhombus with four right angles is a
_________?
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 4 Answer
A rhombus with four right angles is a
_________?
Square
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 5
A parallelogram with congruent diagonals is a
_________?
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 5 Answer
A parallelogram with congruent diagonals is a
_________?
Rectangle
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 6
A parallelogram where all angles are right angles
and all sides are congruent is a _________?
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 6 Answer
A parallelogram where all angles are right angles
and all sides are congruent is a _________?
Square
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 7
A parallelogram with perpendicular diagonals is a
_________?
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 7 Answer
A parallelogram with perpendicular diagonals is a
_________?
Rhombus
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 8
A parallelogram whose diagonals bisect opposite
pairs of angles is a ______?
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 8 Answer
A parallelogram whose diagonals bisect opposite
pairs of angles is a ______?
Rhombus
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 9
A quadrilateral which is both rectangle and
rhombus is a _________?
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 9 Answer
A quadrilateral which is both rectangle and
rhombus is a _________?
Square
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 10
Choose the right answer(s):
1.
2.
3.
A parallelogram is a rhombus
A rectangle is a square
A rhombus is a parallelogram
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 10 Answer
Choose the right answer(s):
1. A parallelogram is a rhombus
2. A rectangle is a square
3. A rhombus is a parallelogram
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 11
Choose the right answer(s):
1.
2.
3.
A quadrilateral is a parallelogram
A square is a rhombus
A rectangle is a rhombus
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 11 Answer
Choose the right answer(s):
1. A quadrilateral is a parallelogram
2. A square is a rhombus
3. A rectangle is a rhombus
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 12
Choose the right answer(s):
1.
2.
3.
A rectangle is a parallelogram
A square is a rectangle
A rhombus is a square
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 12 Answer
Choose the right answer(s):
1. A rectangle is a parallelogram
2. A square is a rectangle
3. A rhombus is a square
Mr. Chin-Sung Lin
ERHS Math Geometry
Trapezoids
Mr. Chin-Sung Lin
ERHS Math Geometry
Definitions of
Trapezoids
Mr. Chin-Sung Lin
ERHS Math Geometry
Trapezoids
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
A
Upper base
Leg
D
B
Leg
Lower base
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Isosceles Trapezoids
A trapezoid whose nonparallel sides are congruent
is called an isosceles trapezoid
A
Upper base
Leg
D
B
Leg
Lower base
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Median of a Trapezoid
The median of a trapezoid is the line segment
connecting the midpoints of the nonparallel
sides
A
Upper base
B
Median
D
Lower base
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Examples of Trapezoids
B
A
D
A
80o
100o
100o
80o
70o
135o
110o
D
120o
90o
90o
C
A
B
60o
D
B
A
C
70o
45o
C
60o
B
110o 120o
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise - Trapezoids
Which one is a trapezoid? Why?
A
D
B
75o
130o
C
110o
D
45o
C
105o
A
75o
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise - Trapezoids
Which one is a trapezoid? Why?
A
D
B
75o
130o
C
110o
D
45o
C
105o
A
75o
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise - Trapezoids
Which one is a trapezoid?
D
80o
100o
90o
C
90o
A
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Exercise - Trapezoids
Which one is a trapezoid?
D
80o
100o
90o
C
90o
A
B
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of
Isosceles Trapezoids
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Isosceles Trapezoids
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
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Base Angles
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
A
B
Prove: A  B; C  D
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Base Angles
Given: Isosceles trapezoid ABCD
AB || CD and AD  BC
Prove: A  B; C  D
B
A
D
E
A
C
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Diagonals
The diagonals of an isosceles trapezoid are congruent
Given: Isosceles trapezoid ABCD
AB || CD and AD  BC
A
B
Prove: AC  BD
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Congruent Diagonals
Given: Isosceles trapezoid ABCD
AB || CD and AD  BC
A
B
Prove: AC  BD
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Parallel and Average Median
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)
A
E
D
B
F
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Parallel and Average Median
Given: Isosceles trapezoid ABCD
AB || CD and median EF
Prove: AB || EF , CD || EF and
EF = (1/2)(AB + CD)
A
E
D
B
F
C
G
H
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Trapezoids
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Trapezoids
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
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Isosceles Trapezoids
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
ERHS Math Geometry
Application Examples
Mr. Chin-Sung Lin
ERHS Math Geometry
Numeric Example of Trapezoids
Isosceles Trapezoid ABCD, AB || CD and AD  BC
Solve for x and y
A
D
xo
2xo
B
3yo
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Numeric Example of Trapezoids
Isosceles Trapezoid ABCD, AB || CD and AD  BC
Solve for x and y
A
x = 60
y = 20
D
xo
2xo
B
3yo
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Numeric Example of Trapezoids
Trapezoid ABCD, AB || CD and median EF
Solve for x
A
E
D
2x
2x + 4
3x + 2
B
F
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Numeric Example of Trapezoids
Trapezoid ABCD, AB || CD and median EF
Solve for x
A
E
x=6
D
2x
2x + 4
3x + 2
B
F
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Isosceles Trapezoids
Given: Trapezoid ABCD and A  B
Prove: ABCD is an isosceles trapezoid
A
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Isosceles Trapezoids
Given: Trapezoid ABCD and AC  BD
Prove: ABCD is an isosceles trapezoid
B
A
O
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Isosceles Trapezoids
Given: Trapezoid ABCD, AB || CD and AE  BE
Prove: ABCD is an isosceles trapezoid
E
A
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Summary of
Quadrilaterals
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 1
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
ERHS Math Geometry
Properties of Quadrilaterals - 1
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
ERHS Math Geometry
Properties of Quadrilaterals - 1
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
ERHS Math Geometry
Properties of Quadrilaterals - 1
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
ERHS Math Geometry
Properties of Quadrilaterals - 1
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
ERHS Math Geometry
Properties of Quadrilaterals - 1
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
ERHS Math Geometry
Properties of Quadrilaterals - 2
Properties
Cong. Diagonals
Bisecting Diagonals
Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 2
Properties
Cong. Diagonals



Bisecting Diagonals
Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 2
Properties

Cong. Diagonals
Bisecting Diagonals






Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 2
Properties

Cong. Diagonals
Bisecting Diagonals
Perpendicular Diagonals








Cong. Opposite Angles
Supp. Opposite Angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 2
Properties

Cong. Diagonals
Bisecting Diagonals


Perpendicular Diagonals
Cong. Opposite Angles










Supp. Opposite Angles
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 2
Properties

Cong. Diagonals
Bisecting Diagonals


Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles













Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 3
Properties
Cong. Adj. Angles (1 P)
Cong. Adj. Angles (2 P)
Cong. Four Right Angles
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 3
Properties
Cong. Adj. Angles (1 P)



Cong. Adj. Angles (2 P)
Cong. Four Right Angles
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 3
Properties
Cong. Adj. Angles (1 P)



Cong. Adj. Angles (2 P)



Cong. Four Right Angles
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 3
Properties
Cong. Adj. Angles (1 P)



Cong. Adj. Angles (2 P)



Cong. Four Right Angles


Diagonals Bisect Angles
Non-Parallel Oppo. Sides
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 3
Properties
Cong. Adj. Angles (1 P)



Cong. Adj. Angles (2 P)



Cong. Four Right Angles


Diagonals Bisect Angles


Non-Parallel Oppo. Sides
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 3
Properties
Cong. Adj. Angles (1 P)



Cong. Adj. Angles (2 P)



Cong. Four Right Angles


Diagonals Bisect Angles
Non-Parallel Oppo. Sides




Mr. Chin-Sung Lin
ERHS Math Geometry
Quadrilaterals
and Proofs
Mr. Chin-Sung Lin
ERHS Math Geometry
Quadrilaterals and Proofs
Given: Isosceles trapezoid ABCD
AB || CD and AD  BC
A
B
1
2
Prove: 1  2
D
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Quadrilaterals and Proofs
Given: Parallelogram ABCD and ABDE
Prove:  EAD   DBC
E
A
D
B
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Quadrilaterals and Proofs
Given: ABC is a right , O is the midpoint of AC
Prove: 1  2
A
O
1
B
2
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Quadrilaterals and Proofs
Given: ABCD is a rhombus, DBFE is an isosceles
trapezoid
A
Prove: CE  CF
D
E
B
C
F
Mr. Chin-Sung Lin
ERHS Math Geometry
Coordinate Geometry and
Quadrilaterals
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Rectangles
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
ERHS Math Geometry
Proving Rectangles
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)
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Rectangles
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
ERHS Math Geometry
Proving Rectangle - Parallelogram
with a Right Angle
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
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Rectangle - Parallelogram
with Congruent Diagonals
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
ERHS Math Geometry
Proving Rhombuses
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
ERHS Math Geometry
Proving Rhombuses
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
ERHS Math Geometry
Proving Rhombuses
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
ERHS Math Geometry
Proving Rhombus - Quadrilateral
with Four Congruent Sides
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
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Rhombus - Parallelogram
with Congruent Adjacent Sides
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
ERHS Math Geometry
Proving Rhombus - Parallelogram
with Perpendicular Diagonals
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
ERHS Math Geometry
Proving Squares
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
ERHS Math Geometry
Proving Squares
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
ERHS Math Geometry
Proving Squares
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
ERHS Math Geometry
Proving Squares - Rhombus with a
Right Angle
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
ERHS Math Geometry
Proving Squares - Rectangle with
Congruent Adjacent Sides
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
ERHS Math Geometry
Proving Trapezoids
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
ERHS Math Geometry
Proving Trapezoids
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
ERHS Math Geometry
Proving Trapezoids
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
ERHS Math Geometry
Proving Trapezoids - Parallel Bases
and Non-Parallel Legs
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
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Isosceles Trapezoids
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
ERHS Math Geometry
Proving Isosceles Trapezoids
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
ERHS Math Geometry
Proving Isosceles Trapezoids
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
ERHS Math Geometry
Proving Isosceles Trapezoids Trapezoid with Congruent Legs
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
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Isosceles Trapezoids Trapezoid w. Congruent Diagonals
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
Mr. Chin-Sung Lin
ERHS Math Geometry
Application Example
Mr. Chin-Sung Lin
ERHS Math Geometry
Finding the Type of Quadrilateral
Given ABCD is a quadrilateral,
where A (3, 6), B(7, 0), C(1, -4), D(-3, 2)
Find the type of quadrilateral ABCD
Mr. Chin-Sung Lin
ERHS Math Geometry
Areas of Polygons
Mr. Chin-Sung Lin
ERHS Math Geometry
Areas of Polygons
The area of a polygon is the unique real number assigned
to any polygon that indicates the number of nonoverlapping square units contained in the polygon’s
interior
Mr. Chin-Sung Lin
ERHS Math Geometry
Areas of Quadrilaterals
The area of a quadrilateral is the product of the length
of the base and the length of the altitude (height)
B
A
altitude
D
base
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Areas of Parallelograms
The area of a parallelogram is the product of the length
of the base and the length of the altitude (height)
B
A
altitude
D
base
C
Mr. Chin-Sung Lin
ERHS Math Geometry
Q&A
Mr. Chin-Sung Lin
ERHS Math Geometry
The End
Mr. Chin-Sung Lin