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Quadrilaterals Eleanor Roosevelt High School Chin-Sung Lin.
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Transcript of Quadrilaterals Eleanor Roosevelt High School Chin-Sung Lin.
Quadrilaterals
Eleanor Roosevelt High School
Chin-Sung Lin
Definitions of the Quadrilaterals
Mr. Chin-Sung Lin
ERHS Math Geometry
Quadrilaterals
Mr. Chin-Sung Lin
A quadrilateral is a polygon with four sides
ERHS Math Geometry
Parts & Properties of the Quadrilaterals
Mr. Chin-Sung Lin
ERHS Math Geometry
Consecutive (Adjacent) Vertices
Mr. Chin-Sung Lin
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
ERHS Math Geometry
S R
QP
Consecutive (Adjacent) Sides
Mr. Chin-Sung Lin
Consecutive sides or adjacent sides are sides that have a common endpoint
PQ and QR, QR and RS, RS and SP, SP and PQ
ERHS Math Geometry
S R
QP
Opposite Sides
Mr. Chin-Sung Lin
Opposite sides of a quadrilateral are sides that do not have a common endpoint
PQ and RS, SP and QR
ERHS Math Geometry
S R
QP
Consecutive angles
Mr. Chin-Sung Lin
Consecutive angles of a quadrilateral are angles whose vertices are consecutive
P and Q, Q and R, R and S, S and P
ERHS Math Geometry
S R
QP
Opposite Angles
Mr. Chin-Sung Lin
Opposite angles of a quadrilateral are angles whose vertices are not consecutive
P and R, Q and S
ERHS Math Geometry
S R
QP
Diagonals
Mr. Chin-Sung Lin
A diagonal of a quadrilateral is a line segment whose endpoints are two nonadjacent vertices of the quadrilateral
PR and QS
ERHS Math Geometry
S R
QP
Sum of the Measures of Angles
Mr. Chin-Sung Lin
The sum of the measures of the angles of a quadrilateral is 360 degrees
mP + mQ + mR + mS = 360
ERHS Math Geometry
S R
QP
Parallelograms
Mr. Chin-Sung Lin
ERHS Math Geometry
Parallelogram
Mr. Chin-Sung Lin
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
ERHS Math GeometryA B
D C
Theorems of Parallelogram
Mr. Chin-Sung Lin
ERHS Math Geometry
Theorems of Parallelogram
Mr. Chin-Sung Lin
Theorem of Dividing Diagonals
Theorem of Opposite Sides
Theorem of Opposite Angles
Theorem of Bisecting Diagonals
Theorem of Consecutive Angles
ERHS Math Geometry
Theorem of Dividing Diagonals
Mr. Chin-Sung Lin
A diagonal divides a parallelogram into two congruent triangles
If ABCD is a parallelogram, then∆ ABD ∆ CDB
A B
D C
ERHS Math Geometry
Theorem of Dividing Diagonals
Mr. Chin-Sung Lin
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
1
2
3
4
A B
D C
ERHS Math Geometry
Theorem of Opposite Sides
Mr. Chin-Sung Lin
Opposite sides of a parallelogram are congruent
If ABCD is a parallelogram, thenAB CD, and BC DA
A B
D C
ERHS Math Geometry
Theorem of Opposite Sides
Mr. Chin-Sung Lin
Statements Reasons
1. ABCD is a parallelogram 1. Given 2. Connect BD 2. Form two triangles3. AB || DC and AD || BC 3. Definition of parallelogram4. 1 2 and 3 4 4. Alternate interior angles5. BD BD 5. Reflexive property6. ∆ ABD ∆ CDB 6. ASA postulate7. AB CD and BC DA 7. CPCTC
1
2
3
4
A B
D C
ERHS Math Geometry
Application Example 1
Mr. Chin-Sung Lin
ABCD is a parallelogram, what’s the perimeter of ABCD ?
A B
D C
10
15
ERHS Math Geometry
Application Example 1
Mr. Chin-Sung Lin
ABCD is a parallelogram, what’s the perimeter of ABCD ?
perimeter = 50
A B
D C
10
15
ERHS Math Geometry
Application Example 2
Mr. Chin-Sung Lin
ABCD is a parallelogram, if the perimeter of ABCD is 80, solve for x
A B
D C
10
x-20
ERHS Math Geometry
Application Example 2
Mr. Chin-Sung Lin
ABCD is a parallelogram, if the perimeter of ABCD is 80, solve for x
x = 50
A B
D C
10
x-20
ERHS Math Geometry
Theorem of Opposite Angles
Mr. Chin-Sung Lin
Opposite angles of a parallelogram are congruent
If ABCD is a parallelogram, thenA C, and B D
A B
D C
ERHS Math Geometry
Theorem of Opposite Angles
Statements Reasons
1. ABCD is a parallelogram 1. Given 2. AB || DC and AD || BC 2. Definition of parallelogram3. A and B are supplementary 3. Same side interior angles A and D are supplementary C and B are supplementary4. A C 4. Supplementary angle theorem B D
A B
D C
ERHS Math Geometry
Application Example 3
Mr. Chin-Sung Lin
ABCD is a parallelogram, what are the values of x and y?
A B
D C
x
120o
y
60o
ERHS Math Geometry
Application Example 3
Mr. Chin-Sung Lin
ABCD is a parallelogram, what are the values of x and y?
x = 120o y = 60o
A B
D C
x
120o
y
60o
ERHS Math Geometry
Application Example 4
Mr. Chin-Sung Lin
ABCD is a parallelogram, what are the values of x and y?
A B
D C
2x - 60
X+20
180 - y
y - 20
ERHS Math Geometry
Application Example 4
Mr. Chin-Sung Lin
ABCD is a parallelogram, what are the values of x and y?
x = 80o y = 100o
A B
D C
2x - 60
X+20
180 - y
y - 20
ERHS Math Geometry
Theorem of Bisecting Diagonals
Mr. Chin-Sung Lin
The diagonals of a parallelogram bisect each other
If ABCD is a parallelogram, thenAC and BD bisect each other at O
A B
D C
O
ERHS Math Geometry
Theorem of Bisecting Diagonals
Mr. Chin-Sung Lin
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
1
2
3
4
A B
D C
O
ERHS Math Geometry
Application Example 5
Mr. Chin-Sung Lin
ABCD is a parallelogram, if AO = 3, BO = 4 AB = 6, AC + BD = ?
A B
D C
O3 4
ERHS Math Geometry
6
Application Example 5
Mr. Chin-Sung Lin
ABCD is a parallelogram, if AO = 3, BO = 4 AB = 6, AC + BD = ?
AC + BD = 24
A B
D C
O3 4
ERHS Math Geometry
6
Application Example 6
Mr. Chin-Sung Lin
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
D C
Ox+4
3x-4y+2
2y-6
ERHS Math Geometry
Application Example 6
Mr. Chin-Sung Lin
ABCD is a parallelogram, if AO = x+4, BO = 2y-6, CO = 3x-4, an DO = y+2, solve for x and y
x = 4 y = 8
A B
D C
Ox+4
3x-4y+2
2y-6
ERHS Math Geometry
Theorem of Consecutive Angles
Mr. Chin-Sung Lin
The consecutive angles of a parallelogram are supplementary
If ABCD is a parallelogram, thenA and B are supplementaryC and D are supplementaryA and D are supplementaryB and C are supplementary
A B
D C
ERHS Math Geometry
Theorem of Consecutive Angles
Mr. Chin-Sung Lin
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
A B
D C
ERHS Math Geometry
Application Example 7
Mr. Chin-Sung Lin
ABCD is a parallelogram, what are the values of x, y and z?
A B
D C
y
120o
z
x
ERHS Math Geometry
Application Example 7
Mr. Chin-Sung Lin
ABCD is a parallelogram, what are the values of x, y and z?
x = 60o
y = 120o
z = 60o
A B
D C
y
120o
z
x
ERHS Math Geometry
Application Example 8
Mr. Chin-Sung Lin
ABCD is a parallelogram, what are the values of x and y?
A B
D C
Y+20
X+30 X-30
ERHS Math Geometry
Application Example 8
Mr. Chin-Sung Lin
ABCD is a parallelogram, what are the values of x and y?
x = 90o
y = 100o
A B
D C
Y+20
X+30 X-30
ERHS Math Geometry
Group Work
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 1
Mr. Chin-Sung Lin
ABCD is a parallelogram, calculate the perimeter of ABCD
A B
D C
2y-10
x+30
2x-10
y+10
ERHS Math Geometry
Question 1
Mr. Chin-Sung Lin
ABCD is a parallelogram, calculate the perimeter of ABCD
perimeter = 200 A B
D C
2y-10
x+30
2x-10
y+10
ERHS Math Geometry
Question 2
Mr. Chin-Sung Lin
ABCD is a parallelogram, solve for x
A B
D C
O
X+30
2XX+10
X-10
ERHS Math Geometry
Question 2
Mr. Chin-Sung Lin
ABCD is a parallelogram, solve for x
x = 30 A B
D C
O
X+30
2XX+10
X-10
ERHS Math Geometry
Question 3
Mr. Chin-Sung Lin
Given: ABCD is a parallelogramProve: XO YO
A B
D C
O
Y
X
ERHS Math Geometry
Question 4
Mr. Chin-Sung Lin
Given: ABCD is a parallelogram, BO ODProve: EO OF
A B
D C
E
O
F
ERHS Math Geometry
Question 5
Mr. Chin-Sung Lin
Given: ABCD is a parallelogram, AF || CEProve: FAB ECD
A B
D C
E
F
ERHS Math Geometry
Review: Theorems of Parallelogram
Mr. Chin-Sung Lin
Theorem of Dividing Diagonals
Theorem of Opposite Sides
Theorem of Opposite Angles
Theorem of Bisecting Diagonals
Theorem of Consecutive Angles
ERHS Math Geometry
Prove Quadrilaterals are Parallelograms
Mr. Chin-Sung Lin
ERHS Math Geometry
Criteria for Proving Parallelograms
Mr. Chin-Sung Lin
Parallel opposite sides
Congruent opposite sides
Congruent & parallel opposite sides
Congruent opposite angles
Supplementary consecutive angles
Bisecting diagonals
ERHS Math Geometry
Parallel Opposite Sides
Mr. Chin-Sung Lin
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
A B
D C
ERHS Math Geometry
Parallel Opposite Sides
Mr. Chin-Sung Lin
Statements Reasons
1. AB || CD and BC || DA 1. Given
2. ABCD is a parallelogram 2. Definition of parallelogram
A B
D C
ERHS Math Geometry
Application Example 1
Mr. Chin-Sung Lin
If m1 = m2 = m3, then ABCD is a parallelogram
A B
D C
1
2
3
ERHS Math Geometry
Application Example 2
Mr. Chin-Sung Lin
ABCD is a quadrilateral as shown below, solve for x
A B
D C2x+10
3x-20
50o
50o
60o
60o
ERHS Math Geometry
Application Example 2
Mr. Chin-Sung Lin
ABCD is a quadrilateral as shown below, solve for x
x = 30 A B
D C2x+10
3x-20
50o
50o
60o
60o
ERHS Math Geometry
Congruent Opposite Sides
Mr. Chin-Sung Lin
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
A B
D C
ERHS Math Geometry
Congruent Opposite Sides
Mr. Chin-Sung Lin
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
A B
D C
ERHS Math Geometry
Congruent Opposite Sides
Mr. Chin-Sung Lin
Statements Reasons
1. Connect BD 1. Form two triangles2. AB CD and BC DA 2. Given 3. BD BD 3. Reflexive property4. ∆ ABD ∆ CDB 4. SSS postulate5. 1 2 and 3 4 5. CPCTC6. AB || DC and AD || BC 6. Converse of alternate interior
angles theorem7. ABCD is a parallelogram 7. Definition of
parallelogram
1
2
3
4
A B
D C
ERHS Math Geometry
Application Example 3
Mr. Chin-Sung Lin
ABCD is a quadrilateral, solve for x
A B
D C
10
15
10
15
X+50
2x-30
ERHS Math Geometry
Application Example 3
Mr. Chin-Sung Lin
ABCD is a quadrilateral, solve for x
x = 80 A B
D C
10
15
10
15
X+50
2x-30
ERHS Math Geometry
Application Example 4
Mr. Chin-Sung Lin
ABCD is a parallelogram, if DF = BE, then AECF is also a parallelogram
A B
D C
E
F
ERHS Math Geometry
Congruent & Parallel Opposite Sides
Mr. Chin-Sung Lin
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
A B
D C
ERHS Math Geometry
Congruent & Parallel Opposite Sides
Mr. Chin-Sung Lin
Statements Reasons
1. Connect BD 1. Form two triangles2. AB CD and AB || CD 2. Given 3. BD BD 3. Reflexive property4. 1 2 4. Alternate interior angles5. ∆ ABD ∆ CDB 5. SAS postulate6. 3 4 6. CPCTC7. AD || BC 7. Converse of alternate interior
angles theorem8. ABCD is a parallelogram 8. Definition of parallelogram
1
2
3
4
A B
D C
ERHS Math Geometry
Application Example 5
Mr. Chin-Sung Lin
ABCD is a quadrilateral, solve for x and y
A B
D C
10
30o
10
y+50
2y-20
30o
X+5
ERHS Math Geometry
Application Example 5
Mr. Chin-Sung Lin
ABCD is a quadrilateral, solve for x and y
x = 5y = 70o
A B
D C
10
30o
10
y+50o
2y-20o
30o
X+5
ERHS Math Geometry
Application Example 6
Mr. Chin-Sung Lin
ABCD is a parallelogram, if m1 = m2, then AECF is also a parallelogram
A B
D C
E
F
1
2
ERHS Math Geometry
Congruent Opposite Angles
Mr. Chin-Sung Lin
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
A B
D C
ERHS Math Geometry
Congruent Opposite Angles
Mr. Chin-Sung Lin
Statements Reasons
1. Connect BD 1. Form two triangles2. m1 +m4 + mA 180 2. Triangle angle-sum
theorem m2 +m3 + mB 1803. m1 +m4 + mA + 3. Addition property m2 +m3 + mC 3604. m1 +m3 = mB 4. Partition property m4 +m2 = mD5. mA +mB + mC + mD 5. Substitution property = 360
1
2
3
4
A B
D C
ERHS Math Geometry
Congruent Opposite Angles
Mr. Chin-Sung Lin
Statements Reasons
6. A C and B D 6. Given7. 2mA + 2mB = 360 7. Substitution property 2mA + 2mD = 3608. mA + mB = 180 8. Division property mA + mD = 1809. AD || BC, AB || DC 9. Converse of same-side
interior angles10. ABCD is a parallelogram 10. Definition of parallelogram
1
2
3
4
A B
D C
ERHS Math Geometry
Application Example 7
Mr. Chin-Sung Lin
ABCD is a quadrilateral, solve for x
A B
D C
50o
2x-40
130o
130o
50o
X+30
ERHS Math Geometry
Application Example 7
Mr. Chin-Sung Lin
ABCD is a quadrilateral, solve for x
x = 70 A B
D C
50o
2x-40
130o
130o
50o
X+30
ERHS Math Geometry
Application Example 8
Mr. Chin-Sung Lin
if m1 = m2, m3 = m4, then ABCD is a parallelogram
A B
DC
1
2
3
4
ERHS Math Geometry
Bisecting Diagonals
Mr. Chin-Sung Lin
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
D C
O
ERHS Math Geometry
Bisecting Diagonals
Mr. Chin-Sung Lin
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 angles4. ∆AOB ∆COD, ∆AOD ∆COB 4. SAS postulate5. 1 2 and 3 4 5. CPCTC6. AB || DC and AD || BC 6. Converse of alternate interior
angles theorem7. ABCD is a parallelogram 7. Definition of
parallelogram
1
2
3
4
A B
D C
O
ERHS Math Geometry
Application Example 9
Mr. Chin-Sung Lin
∆ AOB ∆ COD, then ABCD is a parallelogram
A B
D C
O
ERHS Math Geometry
Supplementary Consecutive Angles
Mr. Chin-Sung Lin
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 B
D C
ERHS Math Geometry
Supplementary Consecutive Angles
Mr. Chin-Sung Lin
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
A B
D C
ERHS Math Geometry
Application Example 10
Mr. Chin-Sung Lin
ABCD is a quadrilateral, solve for x
A B
D C
2x+80
2(x+45)-10
100-2x
3x
ERHS Math Geometry
Application Example 10
Mr. Chin-Sung Lin
ABCD is a quadrilateral, solve for x
x = 20
A B
D C
2x+80
2(x+45)-10
100-2x
3x
ERHS Math Geometry
Review: Proving Parallelograms
Mr. Chin-Sung Lin
Parallel opposite sides
Congruent opposite sides
Congruent & parallel opposite sides
Congruent opposite angles
Supplementary consecutive angles
Bisecting diagonals
ERHS Math Geometry
Rectangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Rectangles
Mr. Chin-Sung Lin
A rectangle is a parallelogram containing one right angle
A B
CD
ERHS Math Geometry
All Angles Are Right Angles
Mr. Chin-Sung Lin
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
CD
ERHS Math Geometry
All Angles Are Right Angles
Mr. Chin-Sung Lin
Statements Reasons
1. ABCD is a rectangle & A = 90o 1. Given
2. C = 90o 2. Opposite angles
3. mA + mD = 180 3. Consecutive angles mA + mB = 1804. 90 + mD = 180 4. Substitution 90 + mB = 1805. mB = 90, mD = 90 5. Subtraction6. B = 90o, D = 90o 6. Def. of measurement of angles
ERHS Math Geometry
A B
CD
All Angles Are Right Angles
Mr. Chin-Sung Lin
The diagonals of a rectangle are congruent
Given: ABCD is a rectangle
Prove: AC BD
ERHS Math Geometry
A B
CD
All Angles Are Right Angles
Mr. Chin-Sung Lin
Statements Reasons
1. ABCD is a rectangle 1. Given
2. C = 90o, D = 90o 2. All angles are right angles
3. C D 3. Substitution4. DC DC 4. Reflexive5. AD BC 5. Opposite sides6. ∆ADC ∆BCD 6. SAS postulate7. AC BD 7. CPCTC
ERHS Math Geometry
A B
CD
Properties of Rectangle
Mr. Chin-Sung Lin
The properties of a rectangle
All the properties of a parallelogram
Four right angles (equiangular)
Congruent diagonals A B
CD
ERHS Math Geometry
Proving Rectangles
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Rectangles
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Rectangles
Mr. Chin-Sung Lin
If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle
Given: ABCD is a parallelogram and mA = 90Prove: ABCD is a rectangle
A B
CD
ERHS Math Geometry
Proving Rectangles
Mr. Chin-Sung Lin
If a quadrilateral is equiangular, it is a rectangle
Given: ABCD is a quadrangular &
mA = mB = mC = mDProve: ABCD is a rectangle
A B
CD
ERHS Math Geometry
Proving Rectangles
Mr. Chin-Sung Lin
The diagonals of a parallelogram are congruent
Given: AC BD
Prove: ABCD is a rectangle
A B
CD
O
ERHS Math Geometry
Application Example
ABCD is a parallelogram, mA = 6x - 30 and mC = 4x + 10. Show that ABCD is a rectangle
A B
CD
ERHS Math Geometry
Mr. Chin-Sung Lin
Application Example
ABCD is a parallelogram, mA = 6x - 30 and mC = 4x + 10. Show that ABCD is a rectangle
x =20
mA = 90
ABCD is a rectangle
A B
CD
ERHS Math Geometry
Mr. Chin-Sung Lin
Rhombuses
Mr. Chin-Sung Lin
ERHS Math Geometry
Rhombus
Mr. Chin-Sung Lin
A rhombus is a parallelogram that has two congruent consecutive sides
A
B
C
D
ERHS Math Geometry
All Sides Are Congruent
Mr. Chin-Sung Lin
All sides of a rhombus are congruent
Given: ABCD is a rhombus with AB DA
Prove: AB BC CD DA
ERHS Math Geometry
A
B
C
D
All Sides Are Congruent
Mr. Chin-Sung Lin
Statements Reasons
1. ABCD is a rhombus w. AB DA1. Given
2. AB DC, AD BC 2. Opposite sides are congruent
3. AB BC CD DA 3. Transitive
ERHS Math Geometry
A
B
C
D
Perpendicular Diagonals
Mr. Chin-Sung Lin
The diagonals of a rhombus are perpendicular to each other
Given: ABCD is a rhombus
Prove: AC BD
ERHS Math Geometry
A
B
C
DO
Perpendicular Diagonals
Statements Reasons
1. ABCD is a rhombus 1. Given
2. AO AO 2. Reflexive
3. AD AB 3. Congruent sides4. BO DO 4. Bisecting diagonals 5. ∆AOD ∆AOB 5. SSS postulate 6. AOD AOB 6. CPCTC7. mAOD + mAOB = 180 7. Supplementary angles8. 2mAOD = 180 8. Substitution 9. AOD = 90o 9. Division pustulate10. AC BD 10. Definition of
perpendicular
ERHS Math Geometry A
B
C
DO
Diagonals Bisecting Angles
Mr. Chin-Sung Lin
The diagonals of a rhombus bisect its angles
Given: ABCD is a rhombus
Prove: AC bisects DAB and DCB
DB bisects CDA and CBA
ERHS Math Geometry
A
B
C
D
Diagonals Bisecting Angles
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 postulate4. ∆ACD ∆ACB, ∆BAD ∆BCD 4. SSS postulate 5. DAC BAC, DCA BCA 5. CPCTC ADB CDB, ABD CBD6. AC bisects DAB and DCB 6. Definition of angle bisector DB bisects CDA and CBA
ERHS Math Geometry A
B
C
D
Mr. Chin-Sung Lin
Properties of Rhombus
Mr. Chin-Sung Lin
The properties of a rhombus
All the properties of a parallelogram
Four congruent sides (equilateral)
Perpendicular diagonals
Diagonals that bisect opposite pairs of angles
A
B
C
D
ERHS Math Geometry
Proving Rhombus
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Rhombus
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Rhombus
Mr. Chin-Sung Lin
If a parallelogram has two congruent consecutive sides, then the parallelogram is a rhombus
Given: ABCD is a parallelogram and AB DAProve: ABCD is a rhombus
ERHS Math Geometry A
B
C
D
Proving Rhombus
Mr. Chin-Sung Lin
If a quadrilateral is equilateral, it is a rhombus
Given: ABCD is a parallelogram and
AB BC CD DAProve: ABCD is a rhombus
ERHS Math Geometry A
B
C
D
Proving Rhombus
Mr. Chin-Sung Lin
The diagonals of a parallelogram are perpendicular
Given: AC BD
Prove: ABCD is a rhombus
ERHS Math Geometry
A
B
C
D
Proving Rhombus
Mr. Chin-Sung Lin
Each diagonal of a rhombus bisects two angles of the rhombus
Given: AC bisects DAB and DCBProve: ABCD is a rhombus
A
B
C
D
1 2
3 4
ERHS Math Geometry
Application Example
Mr. Chin-Sung Lin
ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13
Prove: ABCD is a rhombusA B
D C
2x+1
3x-11
x+13
ERHS Math Geometry
Application Example
Mr. Chin-Sung Lin
ABCD is a parallelogram. AB = 2x + 1, DC = 3x - 11, AD = x + 13
Prove: ABCD is a rhombus
x = 12AB = AD = 25ABCD is a rhombus
A B
D C
2x+1
3x-11
x+13
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
B
C
D
ERHS Math Geometry
Mr. Chin-Sung Lin
Application Example
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
A
B
C
D
ERHS Math Geometry
Mr. Chin-Sung Lin
Squares
Mr. Chin-Sung Lin
ERHS Math Geometry
Squares
Mr. Chin-Sung Lin
A square is a rectangle that has two congruent consecutive sides
A B
CD
ERHS Math Geometry
Squares
Mr. Chin-Sung Lin
A square is a rectangle with four congruent sides (an equilateral rectangle)
ERHS Math Geometry
A B
CD
Squares
Mr. Chin-Sung Lin
A square is a rhombus with four right angles (an equiangular rhombus)
ERHS Math Geometry
A B
CD
Squares
Mr. Chin-Sung Lin
A square is an equilateral quadrilateral
A square is an equiangular quadrilateral
ERHS Math Geometry
A B
CD
Squares
Mr. Chin-Sung Lin
A square is a rhombus
A square is a rectangle
ERHS Math Geometry
A B
CD
Properties of Square
Mr. Chin-Sung Lin
The properties of a square
All the properties of a parallelogram
All the properties of a rectangle
All the properties of a rhombus
A B
CD
ERHS Math Geometry
Proving Squares
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Squares
Mr. Chin-Sung Lin
If a rectangle has two congruent consecutive sides, then the
rectangle is a square
Given: ABCD is a rectangle and AB DAProve: ABCD is a square
ERHS Math Geometry
A B
CD
Proving Squares
Mr. Chin-Sung Lin
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
ERHS Math Geometry
A B
CD
Proving Squares
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Application Example
ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y
A B
CD
ERHS Math Geometry
Mr. Chin-Sung Lin
Application Example
ABCD is a square, mA = 4x - 30, AB = 3x + 10 and BC = 4y. Solve x and y
4x – 30 = 90
x = 30
y = 25
A B
CD
ERHS Math Geometry
Mr. Chin-Sung Lin
Review Questions
Mr. Chin-Sung Lin
ERHS Math Geometry
Question 1
A parallelogram where all angles are right angles (90o) is a _________?
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 1 Answer
A parallelogram where all angles are right angles (90o) is a _________?
ERHS Math Geometry
Mr. Chin-Sung Lin
Rectangle
Question 2
A parallelogram where all sides are congruent is a _________?
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 2 Answer
A parallelogram where all sides are congruent is a _________?
Rhombus
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 3
A rectangle with four congruent sides is a _________?
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 3 Answer
A rectangle with four congruent sides is a _________?
Square
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 4
A rhombus with four right angles is a _________?
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 4 Answer
A rhombus with four right angles is a _________?
Square
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 5
A parallelogram with congruent diagonals is a _________?
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 5 Answer
A parallelogram with congruent diagonals is a _________?
Rectangle
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 6
A parallelogram where all angles are right angles and all sides are congruent is a _________?
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 6 Answer
A parallelogram where all angles are right angles and all sides are congruent is a _________?
Square
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 7
A parallelogram with perpendicular diagonals is a _________?
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 7 Answer
A parallelogram with perpendicular diagonals is a _________?
Rhombus
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 8
A parallelogram whose diagonals bisect opposite pairs of angles is a ______?
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 8 Answer
A parallelogram whose diagonals bisect opposite pairs of angles is a ______?
Rhombus
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 9
A quadrilateral which is both rectangle and rhombus is a _________?
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 9 Answer
A quadrilateral which is both rectangle and rhombus is a _________?
Square
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 10
Choose the right answer(s):
1. A parallelogram is a rhombus2. A rectangle is a square3. A rhombus is a parallelogram
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 10 Answer
Choose the right answer(s):
1. A parallelogram is a rhombus2. A rectangle is a square3. A rhombus is a parallelogram
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 11
Choose the right answer(s):
1. A quadrilateral is a parallelogram2. A square is a rhombus3. A rectangle is a rhombus
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 11 Answer
Choose the right answer(s):
1. A quadrilateral is a parallelogram2. A square is a rhombus3. A rectangle is a rhombus
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 12
Choose the right answer(s):
1. A rectangle is a parallelogram2. A square is a rectangle3. A rhombus is a square
ERHS Math Geometry
Mr. Chin-Sung Lin
Question 12 Answer
Choose the right answer(s):
1. A rectangle is a parallelogram2. A square is a rectangle3. A rhombus is a square
ERHS Math Geometry
Mr. Chin-Sung Lin
Trapezoids
Mr. Chin-Sung Lin
ERHS Math Geometry
Definitions of Trapezoids
Mr. Chin-Sung Lin
ERHS Math Geometry
Trapezoids
Mr. Chin-Sung Lin
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 B
CD
Upper base
Lower base
LegLeg
ERHS Math Geometry
Isosceles Trapezoids
Mr. Chin-Sung Lin
A trapezoid whose nonparallel sides are congruent is called an isosceles trapezoid
ERHS Math Geometry
A B
CD
Upper base
Lower base
LegLeg
Median of a Trapezoid
Mr. Chin-Sung Lin
The median of a trapezoid is the line segment connecting the midpoints of the nonparallel sides
A B
CD
Upper base
Lower base
Median
ERHS Math Geometry
Examples of Trapezoids
Mr. Chin-Sung Lin
C
A
B
D
100o
80o
80o
100o
110o
70o
120o
60o
D C
BA
110o
70o
45o
135o
D C
BA
120o
60o90o
90o
D C
BA
ERHS Math Geometry
Exercise - Trapezoids
Mr. Chin-Sung Lin
110o
75o
45o
130o
D C
BA
105o
75o
D
C
B
A
Which one is a trapezoid? Why?
ERHS Math Geometry
Exercise - Trapezoids
Mr. Chin-Sung Lin
110o
75o
45o
130o
D C
BA
105o
75o
D
C
B
A
Which one is a trapezoid? Why?
ERHS Math Geometry
Exercise - Trapezoids
Mr. Chin-Sung Lin
110o
65o
120o
65o
D C
BA
Which one is a trapezoid?
C
A
B
D
90o
80o
90o
100o
ERHS Math Geometry
Exercise - Trapezoids
Mr. Chin-Sung Lin
110o
65o
120o
65o
D C
BA
Which one is a trapezoid?
C
A
B
D
90o
80o
90o
100o
ERHS Math Geometry
Properties of Isosceles Trapezoids
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Isosceles Trapezoids
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Congruent Base Angles
Mr. Chin-Sung Lin
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
A B
CD
ERHS Math Geometry
Congruent Base Angles
Mr. Chin-Sung Lin
Given: Isosceles trapezoid ABCD
AB || CD and AD BC
Prove: A B; C D
E
A B
CD
A B
CD
ERHS Math Geometry
Congruent Diagonals
Mr. Chin-Sung Lin
The diagonals of an isosceles trapezoid are congruent
Given: Isosceles trapezoid ABCD
AB || CD and AD BC
Prove: AC BD
A B
CD
ERHS Math Geometry
Congruent Diagonals
Mr. Chin-Sung Lin
Given: Isosceles trapezoid ABCD
AB || CD and AD BC
Prove: AC BD
A B
CD
ERHS Math Geometry
Parallel and Average Median
Mr. Chin-Sung Lin
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 B
CD
E F
ERHS Math Geometry
Parallel and Average Median
Mr. Chin-Sung Lin
Given: Isosceles trapezoid ABCD
AB || CD and median EF
Prove: AB || EF , CD || EF and
EF = (1/2)(AB + CD)
A B
CD H
FE G
ERHS Math Geometry
Proving Trapezoids
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Trapezoids
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Isosceles Trapezoids
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Application Examples
Mr. Chin-Sung Lin
ERHS Math Geometry
Numeric Example of Trapezoids
Mr. Chin-Sung Lin
Isosceles Trapezoid ABCD, AB || CD and AD BC
Solve for x and yA B
CD
2xo
xo 3yo
ERHS Math Geometry
Numeric Example of Trapezoids
Mr. Chin-Sung Lin
Isosceles Trapezoid ABCD, AB || CD and AD BC
Solve for x and y
x = 60
y = 20
A B
CD
2xo
xo 3yo
ERHS Math Geometry
Numeric Example of Trapezoids
Mr. Chin-Sung Lin
Trapezoid ABCD, AB || CD and median EF
Solve for x
A B
CD
E F
2x
2x + 4
3x + 2
ERHS Math Geometry
Numeric Example of Trapezoids
Mr. Chin-Sung Lin
Trapezoid ABCD, AB || CD and median EF
Solve for x
x = 6
A B
CD
E F
2x
2x + 4
3x + 2
ERHS Math Geometry
Proving Isosceles Trapezoids
Mr. Chin-Sung Lin
Given: Trapezoid ABCD and A B
Prove: ABCD is an isosceles trapezoid
A B
CD
ERHS Math Geometry
Proving Isosceles Trapezoids
Mr. Chin-Sung Lin
Given: Trapezoid ABCD and AC BD
Prove: ABCD is an isosceles trapezoid
A B
CD
O
ERHS Math Geometry
Proving Isosceles Trapezoids
Mr. Chin-Sung Lin
Given: Trapezoid ABCD, AB || CD and AE BE
Prove: ABCD is an isosceles trapezoid
A B
CD
E
ERHS Math Geometry
Summary of Quadrilaterals
Mr. Chin-Sung Lin
ERHS Math Geometry
Properties of Quadrilaterals - 1
Mr. Chin-Sung Lin
Properties
Cong. Oppo. Sides (1 P)
Cong. Oppo. Sides (2 P)
Cong. Four Sides
Parallel Oppo. Sides (1P)
Parallel Oppo. Sides (2P)
ERHS Math Geometry
Properties of Quadrilaterals - 1
Mr. Chin-Sung Lin
Properties
Cong. Oppo. Sides (1 P)
Cong. Oppo. Sides (2 P)
Cong. Four Sides
Parallel Oppo. Sides (1P)
Parallel Oppo. Sides (2P)
ERHS Math Geometry
Properties of Quadrilaterals - 1
Mr. Chin-Sung Lin
Properties
Cong. Oppo. Sides (1 P)
Cong. Oppo. Sides (2 P)
Cong. Four Sides
Parallel Oppo. Sides (1P)
Parallel Oppo. Sides (2P)
ERHS Math Geometry
Properties of Quadrilaterals - 1
Mr. Chin-Sung Lin
Properties
Cong. Oppo. Sides (1 P)
Cong. Oppo. Sides (2 P)
Cong. Four Sides
Parallel Oppo. Sides (1P)
Parallel Oppo. Sides (2P)
ERHS Math Geometry
Properties of Quadrilaterals - 1
Mr. Chin-Sung Lin
Properties
Cong. Oppo. Sides (1 P)
Cong. Oppo. Sides (2 P)
Cong. Four Sides
Parallel Oppo. Sides (1P)
Parallel Oppo. Sides (2P)
ERHS Math Geometry
Properties of Quadrilaterals - 1
Mr. Chin-Sung Lin
Properties
Cong. Oppo. Sides (1 P)
Cong. Oppo. Sides (2 P)
Cong. Four Sides
Parallel Oppo. Sides (1P)
Parallel Oppo. Sides (2P)
ERHS Math Geometry
Properties of Quadrilaterals - 2
Mr. Chin-Sung Lin
Properties
Cong. Diagonals
Bisecting Diagonals
Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles
ERHS Math Geometry
Properties of Quadrilaterals - 2
Mr. Chin-Sung Lin
Properties
Cong. Diagonals
Bisecting Diagonals
Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles
ERHS Math Geometry
Properties of Quadrilaterals - 2
Mr. Chin-Sung Lin
Properties
Cong. Diagonals
Bisecting Diagonals
Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles
ERHS Math Geometry
Properties of Quadrilaterals - 2
Mr. Chin-Sung Lin
Properties
Cong. Diagonals
Bisecting Diagonals
Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles
ERHS Math Geometry
Properties of Quadrilaterals - 2
Mr. Chin-Sung Lin
Properties
Cong. Diagonals
Bisecting Diagonals
Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles
ERHS Math Geometry
Properties of Quadrilaterals - 2
Mr. Chin-Sung Lin
Properties
Cong. Diagonals
Bisecting Diagonals
Perpendicular Diagonals
Cong. Opposite Angles
Supp. Opposite Angles
ERHS Math Geometry
Properties of Quadrilaterals - 3
Mr. Chin-Sung Lin
Properties
Cong. Adj. Angles (1 P)
Cong. Adj. Angles (2 P)
Cong. Four Right Angles
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
ERHS Math Geometry
Properties of Quadrilaterals - 3
Mr. Chin-Sung Lin
Properties
Cong. Adj. Angles (1 P)
Cong. Adj. Angles (2 P)
Cong. Four Right Angles
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
ERHS Math Geometry
Properties of Quadrilaterals - 3
Mr. Chin-Sung Lin
Properties
Cong. Adj. Angles (1 P)
Cong. Adj. Angles (2 P)
Cong. Four Right Angles
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
ERHS Math Geometry
Properties of Quadrilaterals - 3
Mr. Chin-Sung Lin
Properties
Cong. Adj. Angles (1 P)
Cong. Adj. Angles (2 P)
Cong. Four Right Angles
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
ERHS Math Geometry
Properties of Quadrilaterals - 3
Mr. Chin-Sung Lin
Properties
Cong. Adj. Angles (1 P)
Cong. Adj. Angles (2 P)
Cong. Four Right Angles
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
ERHS Math Geometry
Properties of Quadrilaterals - 3
Mr. Chin-Sung Lin
Properties
Cong. Adj. Angles (1 P)
Cong. Adj. Angles (2 P)
Cong. Four Right Angles
Diagonals Bisect Angles
Non-Parallel Oppo. Sides
ERHS Math Geometry
Quadrilaterals and Proofs
Mr. Chin-Sung Lin
ERHS Math Geometry
Quadrilaterals and Proofs
Mr. Chin-Sung Lin
Given: Isosceles trapezoid ABCD
AB || CD and AD BC
Prove: 1 2A B
CD 1 2
ERHS Math Geometry
Quadrilaterals and Proofs
Mr. Chin-Sung Lin
Given: Parallelogram ABCD and ABDE
Prove: EAD DBCA B
D CE
ERHS Math Geometry
Quadrilaterals and Proofs
Mr. Chin-Sung Lin
Given: ABC is a right , O is the midpoint of AC
Prove: 1 2A
CB
O
1 2
ERHS Math Geometry
Quadrilaterals and Proofs
Mr. Chin-Sung Lin
Given: ABCD is a rhombus, DBFE is an isosceles trapezoid
Prove: CE CF
E
A
B
C
D
F
ERHS Math Geometry
Coordinate Geometry and Quadrilaterals
Mr. Chin-Sung Lin
ERHS Math Geometry
Proving Rectangles
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Rectangles
Mr. Chin-Sung Lin
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)
ERHS Math Geometry
Proving Rectangles
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Rectangle - Parallelogram with a Right Angle
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Rectangle - Parallelogram with Congruent Diagonals
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Rhombuses
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Rhombuses
Mr. Chin-Sung Lin
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)
ERHS Math Geometry
Proving Rhombuses
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Rhombus - Quadrilateral with Four Congruent Sides
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Rhombus - Parallelogram with Congruent Adjacent Sides
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Rhombus - Parallelogram with Perpendicular Diagonals
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Squares
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Squares
Mr. Chin-Sung Lin
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)
ERHS Math Geometry
Proving Squares
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Squares - Rhombus with a Right Angle
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Squares - Rectangle with Congruent Adjacent Sides
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Trapezoids
Mr. Chin-Sung Lin
To prove that a quadrilateral is a trapezoid, show that two sides are parallel and the other two sides are not parallel
ERHS Math Geometry
Proving Trapezoids
Mr. Chin-Sung Lin
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)
ERHS Math Geometry
Proving Trapezoids
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Trapezoids - Parallel Bases and Non-Parallel Legs
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Isosceles Trapezoids
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Isosceles Trapezoids
Mr. Chin-Sung Lin
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)
ERHS Math Geometry
Proving Isosceles Trapezoids
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Isosceles Trapezoids - Trapezoid with Congruent Legs
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Proving Isosceles Trapezoids - Trapezoid w. Congruent Diagonals
Mr. Chin-Sung Lin
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
ERHS Math Geometry
Application Example
Mr. Chin-Sung Lin
ERHS Math Geometry
Finding the Type of Quadrilateral
Mr. Chin-Sung Lin
Given ABCD is a quadrilateral,
where A (3, 6), B(7, 0), C(1, -4), D(-3, 2)
Find the type of quadrilateral ABCD
ERHS Math Geometry
Areas of Polygons
Mr. Chin-Sung Lin
ERHS Math Geometry
Areas of Polygons
Mr. Chin-Sung Lin
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 polygon’s interior
ERHS Math Geometry
Areas of Quadrilaterals
Mr. Chin-Sung Lin
The area of a quadrilateral is the product of the length of the base and the length of the altitude (height)
ERHS Math Geometry
A B
CD base
altitude
Areas of Parallelograms
Mr. Chin-Sung Lin
The area of a parallelogram is the product of the length of the base and the length of the altitude (height)
ERHS Math Geometry
A B
CD base
altitude
Q & A
Mr. Chin-Sung Lin
ERHS Math Geometry
The End
Mr. Chin-Sung Lin
ERHS Math Geometry