Quadrilaterals

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


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


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


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


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


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


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


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


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


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


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


Mr. Chin-Sung Lin


x = 120o y = 60o

A B

D C

x

120o

y

60o

ERHS Math Geometry


Mr. Chin-Sung Lin


A B

D C

2x - 60

X+20

180 - y

y - 20

ERHS Math Geometry


Mr. Chin-Sung Lin


x = 80o y = 100o

A B

D C

2x - 60

X+20

180 - y

y - 20

ERHS Math Geometry


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


Mr. Chin-Sung Lin

Statements Reasons


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


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


Mr. Chin-Sung Lin

ABCD is a parallelogram, if AO = 3, BO = 4 AB = 6, AC + BD = ?

AC + BD = 14

A B

D C

O3 4

ERHS Math Geometry

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


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


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


Mr. Chin-Sung Lin

Statements Reasons


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


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


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


Mr. Chin-Sung Lin


A B

D C

Y+20

X+30 X-30

ERHS Math Geometry


Mr. Chin-Sung Lin


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






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


Mr. Chin-Sung Lin

If m1 = m2 = m3, then ABCD is a parallelogram

A B

D C

1

2

3

ERHS Math Geometry


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


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


A B

D C

ERHS Math Geometry


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


A B

D C

ERHS Math Geometry


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


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


Mr. Chin-Sung Lin


x = 80 A B

D C

10

15

10

15

X+50

2x-30

ERHS Math Geometry


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


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


Mr. Chin-Sung Lin

ABCD is a quadrilateral, solve for x and y

A B

D C

10

30o

10

y+50o

2y-20o

30o

X+5

ERHS Math Geometry


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


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


Mr. Chin-Sung Lin

Statements Reasons

1. Connect BD 1. Form two triangles2. m1 +m4 + mA 180 2. Triangle angle-sum

theorem m2 +m3 + mC 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


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


Mr. Chin-Sung Lin


A B

D C

50o

2x-40

130o

130o

50o

X+30

ERHS Math Geometry


Mr. Chin-Sung Lin


x = 70 A B

D C

50o

2x-40

130o

130o

50o

X+30

ERHS Math Geometry


Mr. Chin-Sung Lin

if m1 = m2, m3 = m4, then ABCD is a parallelogram

A B

CD

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


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


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


Mr. Chin-Sung Lin


A B

D C

2x+80

2(x+45)-10

100-2x

3x

ERHS Math Geometry


Mr. Chin-Sung Lin


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


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

Diagonals Are Congruent

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


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


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 postulate 10. 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


B

C

D

Mr. Chin-Sung Lin

Properties of Rhombus

Mr. Chin-Sung Lin

The properties of a rhombus


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


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


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

If each diagonal of a parallelogram bisects two opposite angles, then it is a rhombus

Given: AC bisects DAB and DCB BD bisects ADC and ABC


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


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 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


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


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


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


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


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


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


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


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


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


AB || CD and AD BC

Prove: AC BD

A B

CD

ERHS Math Geometry

Congruent Diagonals

Mr. Chin-Sung Lin


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


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


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


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


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


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


Mr. Chin-Sung Lin

Given: Trapezoid ABCD and A B

Prove: ABCD is an isosceles trapezoid

A B

CD

ERHS Math Geometry


Mr. Chin-Sung Lin

Given: Trapezoid ABCD and AC BD


A B

CD

O

ERHS Math Geometry


Mr. Chin-Sung Lin

Given: Trapezoid ABCD, AB || CD and AE BE


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. Four Sides

Parallel Oppo. Sides (1P)


ERHS Math Geometry


Mr. Chin-Sung Lin

Properties



Cong. Four Sides



ERHS Math Geometry


Mr. Chin-Sung Lin

Properties



Cong. Four Sides



ERHS Math Geometry


Mr. Chin-Sung Lin

Properties



Cong. Four Sides



ERHS Math Geometry


Mr. Chin-Sung Lin

Properties



Cong. Four Sides



ERHS Math Geometry


Mr. Chin-Sung Lin

Properties



Cong. Four Sides



ERHS Math Geometry


Mr. Chin-Sung Lin

Properties

Cong. Diagonals

Bisecting Diagonals


Cong. Opposite Angles

Supp. Opposite Angles

ERHS Math Geometry


Mr. Chin-Sung Lin

Properties

Cong. Diagonals

Bisecting Diagonals




ERHS Math Geometry


Mr. Chin-Sung Lin

Properties

Cong. Diagonals

Bisecting Diagonals




ERHS Math Geometry


Mr. Chin-Sung Lin

Properties

Cong. Diagonals

Bisecting Diagonals




ERHS Math Geometry


Mr. Chin-Sung Lin

Properties

Cong. Diagonals

Bisecting Diagonals




ERHS Math Geometry


Mr. Chin-Sung Lin

Properties

Cong. Diagonals

Bisecting Diagonals




ERHS Math Geometry


Mr. Chin-Sung Lin

Properties

Cong. Adj. Angles (1 P)


Cong. Four Right Angles

Diagonals Bisect Angles

Non-Parallel Oppo. Sides

ERHS Math Geometry


Mr. Chin-Sung Lin

Properties






ERHS Math Geometry


Mr. Chin-Sung Lin

Properties






ERHS Math Geometry


Mr. Chin-Sung Lin

Properties






ERHS Math Geometry


Mr. Chin-Sung Lin

Properties






ERHS Math Geometry


Mr. Chin-Sung Lin

Properties






ERHS Math Geometry

Quadrilaterals and Proofs

Mr. Chin-Sung Lin

ERHS Math Geometry


Mr. Chin-Sung Lin


AB || CD and AD BC

Prove: 1 2A B

CD 1 2

ERHS Math Geometry


Mr. Chin-Sung Lin

Given: Parallelogram ABCD and ABDE

Prove: EAD DBCA B

D CE

ERHS Math Geometry


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


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


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


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


Prove: A given quadrilateral is a rhombus



ERHS Math Geometry

Proving Rhombuses

Mr. Chin-Sung Lin


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


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


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


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


Prove: A given quadrilateral is a square



ERHS Math Geometry

Proving Squares

Mr. Chin-Sung Lin


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


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


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


Prove: A given quadrilateral is a trapezoid



ERHS Math Geometry

Proving Trapezoids

Mr. Chin-Sung Lin


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


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


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


Mr. Chin-Sung Lin


Prove: A given quadrilateral is an isosceles trapezoid



ERHS Math Geometry


Mr. Chin-Sung Lin


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


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


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

Documents

Quadrilaterals