Chapter 10-1 The General Quadrilateral Section Quiz...

1. If ABCD is a parallelogram, which statement mustbe true?

(1)

(2)

(3) �A and �C are supplementary.

(4) �A � �D

2. If ABCD is a parallelogram, which statement mustbe true?

(1)

(2) bisects .

(3) bisects �C.

(4) �

3. If m�CDA � x � 32 and m�BCD � x � 20, findm�CDA.

(1) 26 (3) 56

(2) 45 (4) 60✔

52

12

BDAC

AC

BDAC✔

AC ' BD

AD > BC✔

AC > BD

4. In parallelogram ABCD, if m�A � 50, find m�C.

(1) 25 (3) 50

(2) 40 (4) 130

5. In parallelogram ABCD, diagonals intersect at E. If EC � 31, EB � 3x, and AE � 4x � 5, what is the value of BD?

(1) 9 (3) 36

(2) 27 (4) 54

6. Which statement is not always true for aparallelogram?

(1) Consecutive angles are supplementary.

(2) The diagonals are perpendicular.

(3) The opposite sides are congruent.

(4) The opposite angles are congruent.

✔

✔

AC and BD

✔

Chapter 10 Quadrilaterals 205

Name __________________________________________________________ Class ______________ Date ______________

Copyright © Amsco School Publications, Inc.

Chapter 10-1 The General Quadrilateral Section Quiz [20 points]Chapter 10-2 The Parallelogram

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12]

PART II

Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps,including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numericalanswer with no work shown will receive only 1 credit. [8]

7. Answer both questions.

a. The average of the degree measures of the angles of a quadrilateral is how much greater than the average ofthe degree measures of a triangle?

Answer: 30° more

Solution:The sum of the measures of the angles of a quadrilateral is 360°.

Thus, the average degree measure is .

The sum of the measures of the angles of a triangle is 180°.

Thus, the average degree measure is .1803 5 608

3604 5 908

206 Chapter 10 Quadrilaterals

Name __________________________________________________________ Class ______________ Date ______________


b. In parallelogram ABCD, m�A � 5x and m�C � 3x � 30. Find m�A and m�B.

Answer: m�A � 75, m�B � 105

Solution:

Therefore, m�A � 5(15) � 75, m�C � 180 � 75 � 105.

8. In parallelogram ABCD, m�D � 108 and bisects �BAD. Find m�C and m�AGC.

Answer: m�C � 72, m�AGC � 144

Solution:

m�GAD � 722 5 36

m/C 5 m/A 5 72

m/AGC 5 144m/A 5 72

36 1 108 1 72 1 m/AGC 5 360m/A 5 180 2 108

m/AGD 1 m/D 1 m/C 1 m/AGC 5 360m/A 5 180 2 m/D

D

C

A

B G

108°

GA

x 5 15

2x 5 30

5x 5 3x 1 30

1. A quadrilateral is a parallelogram if two of itsopposite sides are

(1) parallel and the other two sides are congruent.

(2) parallel only.

(3) congruent and parallel.

(4) congruent only.

2. In quadrilateral ABCD, AB � CD and AD � BC. Itmust necessarily follow that the diagonals AC andBD

(1) are equal.

(2) are perpendicular.

(3) bisect each other.

(4) bisect the angles of the quadrilateral.

3. A quadrilateral is a parallelogram if

(1) two adjacent sides are congruent.

(2) three sides are congruent.

(3) the diagonals form 45° angles with each other.

(4) the diagonals bisect each other✔

✔

✔

For 4 and 5, use the figure below.

4. Which of the following statements is not sufficient toshow that ABCD is a parallelogram?

(1) �1 � �3 and �2 � �4

(2) �1 � �3

(3)

(4) �AOB � �DOC

5. Which of the following statements is not sufficient toshow that ABCD is a parallelogram?

(1) �1 � �2 and �5 � �7

(2) �1 � �4 and

(3) �1 � �5, �2 � �4, and �4 � �7

(4) �6 � �7 and

6. Given: The vertices of ROSA are R(0, 4), O(6, 8),S(12, 0), and A(0, �2).

Which best describes ROSA?

(1) a quadrilateral with no diagonals bisected

(2) a quadrilateral with one diagonal bisected

(3) a parallelogram with congruent diagonals

(4) a parallelogram with perpendicular diagonals

✔

AO > OC

AD > DC > AB

✔

AB > BD✔

AB > DC and

D C

A B

O

12

3 4

5 6

7 8


Name __________________________________________________________ Class ______________ Date ______________


Chapter 10-3 Proving that a Quadrilateral Is a Parallelogram Section Quiz [20 points]

PART I


PART II


7. Complete the proof by filling in the missing reasons.

Given: Parallelogram ABCD and

Prove: BXDR is a parallelogram.

Proof:

Statements Reasons

1. Parallelogram ABCD 1. Given.

2. 2.

3. 3.

4. 4.

5. 5. Given.

6. 6.

or

7. BXDR is a parallelogram. 7.

RD > BX

AD 2 AR > BC 2 CX

AR > CX

AD > BC

RD y BX

AD y BC

AR > CX


Name __________________________________________________________ Class ______________ Date ______________


D

C

A

B

R

X

Opposite sides of a parallelogram are �.

then the quadrilateral is a parallelogram.

If one pair of opposite sides is both � and �,

Subtraction postulate.

Segments of parallel lines are �.

Definition of a parallelogram.

8. Complete the proof by filling in the blanks.

Given:

Prove: Quadrilateral ABCD is a parallelogram.

Proof:

AB > DC, AD > BC


Name __________________________________________________________ Class ______________ Date ______________


AC > AC

�ABC � �CDA

�1 � �2 and �3 � �4

Reflexive property

Corresponding parts of congruent

If alternate interior angles are �,

then the two lines are �.

triangles are �.

Definition of a parallelogram

D

C

A

B

14

32

Statements Reasons

1. 1. Given.

2. 2.

3. 3. SSS.

4. 4.

5. 5.

6. ABCD is a parallelogram. 6.

AB y DC and AD y BC

AB > DC, AD > BC

For 1 and 2, use the rectangle given below.

1. If �CAD � 42°, what is the measure of �CED?

(1) 42 (3) 96

(2) 84 (4) 138

2. If AB � x and BC � 4x, what percent of theperimeter of the rectangle is the sum AB � BC � CD?

(1) 50%

(2) 60%

(3) %

(4) 90%

3. Which of the following is not a property of allrectangles?

(1) The diagonals bisect each other.

(2) The diagonals are perpendicular to each other.

(3) The diagonals are congruent.

(4) The angles are congruent.

✔

83.3

✔

✔

D

C

A

B

E

4. The perimeter of rectangle ABCD is 16. If the lengthof the rectangle is greater than 7, which of thefollowing is a possible value for the width?

(1) 0.5 (3) 1.5

(2) 1 (4) 2

5. The area of a rectangle is 54 square inches and theperimeter is 30 inches. If the length and width areintegers, what is the absolute value of the differencebetween the length and the width?

(1) 2 in. (3) 12 in.

(2) 3 in. (4) 24 in.

6.

Which of the following statements is not sufficient toshow that DEFG is a rectangle?

(1) �2 � �8, �4 � �5, and

(2) �2 � �5, , and

(3) �6 � �7 and m�GDE � 90

(4) , , and �7 � �3DG y EFDE y FG

✔

DP > PFDG > EF

GP > PE

E

F

D

G

P

12

34

56

78

✔

✔


Name __________________________________________________________ Class ______________ Date ______________


Chapter 10-4 The Rectangle Section Quiz [20 points]

PART I


PART II


7. a. In rectangle ABCD, E is the midpoint of diagonal . Find the measureof �AED.

Answer: m�AED � 120

Solution:

m�CAD � 180 � 60 � 90 � 30

The diagonals are congruent and bisect each other. Thus, AE � EC � ED,and so:

m�EDA � m�CAD � 30

m�AED � 180 � 30 � 30 � 120

b. In rectangle ABCD, AC � 3x � 1 and DE � x � 13. Find the length of .

Answer: AE � 40

Solution:

Therefore, .

8. The vertices of quadrilateral STAN are S(�1, �2), T(3, 2), A(1, 4), and N(�3, 0). Show that STAN is a rectangle.

Proof:

Slope of Slope of

Slope of Slope of

The slopes of opposite sides are negative reciprocals of each other. Therefore, the angles of STAN are all rightangles, and STAN is a rectangle.

TA 5 2 2 43 2 1 5 22

22 5 21AN 5 4 2 01 1 3 5 4

4 5 1

SN 5 22 2 021 1 3 5 22

2 5 21ST 5 22 2 221 2 3 5 24

24 5 1

AE 5 DE 5 27 1 13 5 40

x 5 27

3x 2 1 5 2x 1 26

3x 2 1 5 2(x 1 13)

AC 5 2DE

AE

AC


Name __________________________________________________________ Class ______________ Date ______________


D

C

A

B

E60°

1.

In rhombus ABCD, diagonals intersectat E. If m�BCE � 20, find m�ADC.

(1) 20 (3) 110

(2) 70 (4) 140

2. In rhombus PQRS, PQ � 5x � 15 and QR � 2x � 45, what is RS?

(1) 10 (3) 65

(2) 20 (4) 85

3. Which of the following statements is not always truefor a rhombus?




(4) The diagonals bisect opposite angles.

4. In rhombus PQRS, PQ � 3x � 3, PS � 5x � 1, andRS � 10x � 11. What is the perimeter of therhombus?

(1) 8

(2) 9

(3) 32

(4) 36✔

✔

✔

✔

AC and BD

D

C

A

B

E

20°5.

Which statement is not sufficient to show thatparallelogram ABCD is a rhombus?

(1)

(2) �1 � �2

(3) �1 � �5 and �6 � �7

(4) �3 � �4

6.

In rhombus ABCD with diagonal , if

m�C � 120, what is m�ADE?

(1) 30 (3) 120

(2) 60 (4) 150✔

BDEg

D

CA

B

E

120°

✔

AB > AD

D C

A B

E

15

34

6 7

2


Name __________________________________________________________ Class ______________ Date ______________


Chapter 10-5 The Rhombus Section Quiz [20 points]

PART I


PART II


7. The vertices of quadrilateral PQRS are P(�3, 0), Q(1, �3), R(1, 2), and S(�3, 5).

a. Prove that PQRS is a parallelogram.

Proof:and are both vertical segments and SP � RQ � 5.

Therefore, PQRS has a pair of opposite sides that are congruent and parallel, and so PQRS is aparallelogram.

b. Prove that PQRS is a rhombus.

Proof:Slope of

Slope of

The slopes of the diagonals and are negative reciprocals of each other, and so .A parallelogram with perpendicular diagonals is a rhombus. Therefore, PQRS is a rhombus.

8. Complete the proof by filling in the missing reasons.

Given: Parallelogram ABCD, at E.

Prove: ABCD is a rhombus.

Statements Reasons

1. Parallelogram ABCD 1. Given.

2. 2.

3. at E 3. Given.

4. �BEA � �AED 4.

5. 5.

6. �AEB � �AED 6.

7. 7.

8. ABCD is a rhombus. 8.

AB > AD

AE > AE

BD ' AC

BE > ED

BD ' AC

PR ' SQSQPR

SQ 5 5 1 323 2 1 5 8

24 5 22

PR 5 0 2 223 2 1 5 22

24 5 12

RQSP


Name __________________________________________________________ Class ______________ Date ______________


D

C

A

B

E

The diagonals of a parallelogram bisect

each other.

Right angles are congruent.

Reflexive property.

SAS.

Definition of a rhombus.

1. Parallelogram RSTV must be a square if its

(1) opposite angles and opposite sides arecongruent.

(2) sides and angles are congruent.

(3) diagonals bisect each other and areperpendicular to each other.

(4) diagonals are congruent.

2. If (�3, 2), (1, 5), and (4, 1) are consecutive vertices ofa square, which of the following represents thecoordinates of the fourth vertex?

(1) (�2, 0) (3) (1, �2)

(2) (0, �2) (4) (1, �3)

3. Equilateral �ABC and square PQRS have the sameperimeters. If a side of the triangle is 3x � 3 and aside of the square is 2x � 5, what is the length of theside of the square?

(1) 29 (3) 84

(2) 63 (4) 252

4. Given any square ABCD, which of the followingstatements is not true?

(1) � C

(2) bisects �BAD

(3)

(4) AB > AC✔

AC ' BD

AC

rBDi (A)

✔

✔

✔

5.

Parallelogram ABCD is a square when which of thefollowing is true?

(1)

(2) and �1 � �2

(3)

(4) �3 � �4

6. Of all rectangles with a given perimeter, the squarehas maximum area. What is the maximum area of arectangle with perimeter 24?

(1) 24 sq units

(2) 25 sq units

(3) 36 sq units

(4) 576 sq units

✔

DB > AC

AE > DE✔

AD > AB

D C

A B

E1

2

4

3


Name __________________________________________________________ Class ______________ Date ______________


Chapter 10-6 The Square Section Quiz [20 points]

PART I


PART II


In 7 and 8, use the square on the right.

7. If PR � 5x � 3 and QA � 4x � 6, what are the values of x, PR, QS, and QA?

Answer: x � 3, PR � QS � 12, QA � 6

Solution:

Therefore, PR � QS � 5(3) � 3 � 12 and QA � 6.

8. If m�PAQ � 3x � y and m�PSR � x � 2y, what are the values of x and y?

Answer: x � 18 and y � �36

Solution:

Substituting y into the equation x � 2y � 90 yields:

Therefore, y � 3(18) � 90 � �36.

x 5 18

25x 5 290

x 2 6x 2 180 5 90

x 2 2(3x 2 90) 5 90

3x 2 y 5 90 S y 5 3x 2 90

x 5 3

9 5 3x

5x 2 3 5 8x 2 12

5x 2 3 5 2(4x 2 2)


Name __________________________________________________________ Class ______________ Date ______________


S

R

P

Q

A

1. The length of one base of a trapezoid is three timesthe length of the other base. If the length of themedian of the trapezoid is 10, the length of the otherbase is

(1) 7.5 (3) 15

(2) 12 (4) 20

2. The median of an isosceles trapezoid always dividesthe trapezoid into

(1) one triangle and one parallelogram.

(2) two trapezoids of equal area.

(3) two isosceles trapezoids.

(4) two congruent trapezoids.

3. PQRS is a trapezoid with . Which additionalpiece of information would guarantee that thetrapezoid is isosceles?

(1) �P and �Q are supplementary.

(2) �P � �R

(3)

(4) bisect each other.

4. The opposite angles of an isosceles trapezoid arealways

(1) acute

(2) congruent

(3) supplementary

(4) complementary

✔

PR and QS

PR > QS✔

PQ y SR

✔

✔

5.

In the given figure, bisects �DAC. If m�EAB � 40, m�ABC � 50, and m�BCD � 130,what is m�ADC?

(1) 40

(2) 80

(3) 100

(4) 140

6. The area of a trapezoid is 100 square inches, itsaltitude is 10 inches, and the length of one of its bases is 5 inches. What is the length of the otherbase?

(1) 5 in.

(2) 10 in.

(3) 15 in.

(4) 20 in.

✔

✔

AE

B

E CD

A40° 50°

130°


Name __________________________________________________________ Class ______________ Date ______________


Chapter 10-7 The Trapezoid Section Quiz [20 points]

PART I


PART II


7. a. The lengths of the bases of a trapezoid are 9 feet and 17 feet. If the height of the trapezoid is 6 feet, what is thearea, in square feet, of the trapezoid?

Answer

b. In isosceles trapezoid DEFG, m�D is three times m�F. Find m�F.

Answer: m�F � 45

Solution:In an isosceles trapezoid, any two opposite angles are supplementary.

8. Given: Isosceles Trapezoid ABCD with and .Diagonal bisects �CDA.

Prove:

Proof:

Statements Reasons

1. 1. Given.

2. �ADB � �CBD 2. Alternate interior angles are congruent in � lines.

3. Diagonal bisects �CDA. 3. Given.

4. �ADB � �CDB 4. Definition of angle bisector.

5. �CBD � �CDB 5. Transitive property.

6. 6. Converse of isosceles triangle theorem.

7. 7. Given.

8. 8. Transitive property.AB > BC

AB > DC

BC > DC

BD

AD y BC

AB > BC

BDAD y BCAB > DC

m/F 5 45

4m/F 5 180

3m/F 1 m/F 5 180

m/D 1 m/F 5 180

5 78 ft2

5 3(26)

Area 5 12(6)(9 1 17)


Name __________________________________________________________ Class ______________ Date ______________


B C

DA

1. If the coordinates of the vertices of parallelogramABCD are A(0, 2), B(2, 5), C(10, 5), and D(8, 2), thearea of the parallelogram is:

(1) 12 sq units

(2) 24 sq units

(3) 36 sq units

(4) 48 sq units

2. The area of a square whose perimeter is 8k is:

(1) sq units

(2) sq units

(3) 4k2 sq units

(4) 8k2 sq units

3.

In rhombus PQRS, diagonals intersect atA. If QS � 14 and PR � 12, what is the area of�PRS?

(1) 21 sq units

(2) 42 sq units

(3) 84 sq units

(4) 126 sq units

✔

QS and PR

P

Q

RA

S

✔

4k2!2

4k!2

✔

4. If the diagonals of a rhombus have lengths of 6 and12, the area of the rhombus is:

(1) 72 sq units

(2) 36 sq units

(3) 30 sq units

(4) 18 sq units

5.

In the given figure, what is the ratio of the area of�PAS to the area of square PQRS?

(1) (3)

(2) (4)

6. The perimeter of a rectangle is 6x. If one side haslength , what is the area of the rectangle?

(1) sq units

(2) sq units

(3) sq units

(4) 3x2 sq units

5x2

2

5x2

4✔

x2

4

x2

21

13

12✔1

4

P

Q RA

S

✔


Name __________________________________________________________ Class ______________ Date ______________


Chapter 10-8 Areas of Polygons Section Quiz [20 points]

PART I


PART II


7. a. If the perimeter of rectangle ABCD is 58 and the length of is 16, what is the area of ABCD?

Answer: 208 sq units

Solution:Let x � AB � DC and y � AD � BC � 16.Then:

Therefore, the area of ABCD is (16)(13) � 208 square units.

b. The length of the side of a rhombus is 5 centimeters. If the diagonals have integer lengths, what is the area ofthe rhombus?

Solution:The diagonals partition the rhombus into four congruent right triangles with hypotenuses that are5 centimeters long. Since the diagonals have integer lengths, the only possible lengths are 3 and 4centimeters. (The triangles are 3-4-5 right triangles.) Thus, the area is:

8. a. If the area of a trapezoid is 72 square units, the altitude is 8, and the length of the larger base is twice thesmaller base, what are the lengths of the bases?

Answer: 6 and 12

Solution:Let x � the length of the smaller base.Then 2x � the length of the larger base.

Therefore, the length of the bases are 6 and 6(2) � 12.

b. The width and height of a rectangle are in the ratio 5 : 1 and the perimeter is 72 inches. Find the area of therectangle.

Answer: 180 in.2

Solution:Let x and 5x represent the width and height of the rectangle, respectively.

Thus, the width is 6 and the height is 30. The area is 180 square inches.

x 5 6

6x 5 36

2(x 1 5x) 5 72

6 5 x

72 5 4(3x)

72 5 12(8)(x 1 2x)

4C12(3)(4) D 5 24 cm2

x 5 29 2 16 5 13

y 5 16

x 1 y 5 29

AD


Name __________________________________________________________ Class ______________ Date ______________


1. Which is always true of the diagonals of aparallelogram?




(4) The diagonals bisect the angles of theparallelogram.

2. Which statement is false?

(1) A parallelogram is a quadrilateral.

(2) A rectangle is a parallelogram.

(3) A square is a rhombus.

(4) A rectangle is a square.

3. In parallelogram ABCD, if m�B exceeds m�A by56°, what is m�B?

(1) 62

(2) 112

(3) 118

(4) 124

4. In quadrilateral ABCD, .Which statement must be true?

(1) The diagonals bisect the angles of thequadrilateral.


(3) The diagonals are equal in measure.


✔

AB > CD and AB y CD

✔

✔

✔

5. In a trapezoid, the length of the median is 14 and thelength of one base is 10. The length of the other baseis:

(1) 4

(2) 6

(3) 12

(4) 18

6. The coordinates of rectangle ABCD are A(1, 4),B(1, 1), C(7, 1), and D(7, 4). Which of the following isthe point of intersection of the diagonals?

(1) (1, 2.5)

(2) (4, 2.5)

(3) (7, 2.5)

(4) (8, 5)

7. A bag is filled with an isosceles trapezoid, aparallelogram, a rhombus, a rectangle, and a square.If one of these quadrilaterals is picked at random,what is the probability that the diagonals of thechosen figure bisect each other?

(1) 0 (3)

(2) (4) 1

8. A parallelogram must be a rectangle if the oppositeangles

(1) are complementary.

(2) are supplementary.

(3) are congruent.

(4) sum to 120°.

✔

15

45✔

✔

✔


Name __________________________________________________________ Class ______________ Date ______________


Chapter 10 Quadrilaterals Chapter Review [40 points]

PART I


PART II


9. In parallelogram MRST, m�M � 3x � 40 and m�S � 7x � 100. Find:

a. the value of x.

Answer

b. the measure of �R.

Answer

10. If the perimeter of a rhombus is 40 and the length of one of the diagonals is 16, what is the length of the otherdiagonal?

Answer: 12

Solution:The diagonals bisect each other and form congruent right triangles.

Let y � one-half the length of the other diagonal.

Then:

Therefore, the length of the other diagonal is 2(6) � 12.

y 5 6

y2 5 36

82 1 y2 5 102

5 175

5 180 2 (45 2 40)

5 180 2 f3(15) 2 40g

m/R 5 180 2 m/M

x 5 15

60 5 4x

3x 2 40 5 7x 2 100


Name __________________________________________________________ Class ______________ Date ______________


PART III


11. The length of a rectangle is twice its width. The perimeter of the rectangle is 84 inches.

a. What are the dimensions of the rectangle?

Answer: Length � 28, width � 14

Solution:Let w � the width of the rectangle.

Then 2w � the length of the rectangle.

Therefore, the length is 2(14) � 28.

b. What is the area of the rectangle?

Answer

12. Given: ABCD is a parallelogram, AB � 3x � 1, BC � x � 23, and CD � 2x � 11.

Show that ABCD is a rhombus.

Proof:

Since ABCD is a parallelogram, opposite sides are congruent.

Therefore, AB � CD � 3(12) � 1 � 35 and BC � AD � 12 � 23 � 35, and so the sides of the parallelogram are all congruent, and ABCD is a rhombus.

x 5 12

3x 2 1 5 2x 1 11

5 392 in.2

5 14(28)

Area

w 5 14

3w 5 42

2(2w 1 w) 5 84


Name __________________________________________________________ Class ______________ Date ______________


PART IV


13. Given: Rectangle ABCD with E, the midpoint of .

Prove: �EAB � �EBA

Proof:

Statements Reasons

1. Rectangle ABCD 1. Given.

2. 2. In a rectangle, opposite sides are �.

3. �D � �C 3. The angles of a rectangle are all right �’s.

4. E, the midpoint of . 4. Given.

5. 5. Definition of midpoint.

6. �AED � �BEC 6. SAS.

7. 7. Corresponding parts of congruent triangles are �.

8. �EAB � �EBA 8. Isosceles triangle theorem.

14. Given: at E and �CAD � �BCA.

Prove: ABCD is a parallelogram.

Proof:

Statements Reasons

1. bisects at E. 1. Given.

2. 2. Definition of bisector.

3. �CAD � �BCA 3. Given.

4. 4. If alternate interior angles are �, then the two lines are �.

5. �AED � �BEC 5. Vertical angles are congruent.

6. �AED � �CEB 6. ASA.

7. 7. Corresponding parts of congruent triangles are �.

8. ABCD is a parallelogram. 8. If two opposite sides are both � and �, then the quadrilateral is a parallelogram.

AD > BC

AD y BC

AE > EC

ACBD

BD bisects AC

AE > EB

DE > EC

DC

AD > BC

DC


Name __________________________________________________________ Class ______________ Date ______________


D

B

C

A

E

D

B

C

A

E

1. Which statement is logically equivalent to ~p → q?

(1) q → p

(2) ~q → p

(3) q → ~p

(4) ~q → ~p

2.

What is the measure of �ABC?

(1) 50°

(2) 55°

(3) 60°

(4) 65°

3. In �ABC, if AB � 6 and BC � 10, which of thefollowing statements must be true?

(1) AC � 4

(2) m�A � m�C

(3) 6 � AC � 10

(4) m�A � m�C � m�B

4. How many positive integers are in the solution set ofthe inequality 3x � 5 � 2?

(1) One

(2) Two

(3) Three

(4) Infinitely many

✔

✔

✔

D

B

CA

50° 140°35°

✔

5.

If is a line segment and AO � OB, what are thecoordinates of point A?

(1) (s, r)

(2) (�s, �r)

(3) (�r, s)

(4) (�r, �s)

6. If the sum of the interior angles of a regular polygonis 1,080°, how many sides does this polygon have?

(1) 3 (3) 6

(2) 4 (4) 8

7. Given parallelogram HOPE, which of the followingstatements may not be true?

(1) m�HOP � m�PEH

(2)

(3)

(4)

8. If a regular polygon has 12 sides, what is the measureof each exterior angle?

(1) 30°

(2) 36°

(3) 360°

(4) 1,800°

✔

HE > OP

HO y EP

HP > EO✔

✔

✔

AB

A

x

B(r, s)y

O


Name __________________________________________________________ Class ______________ Date ______________


Chapter 10 Quadrilaterals Cumulative Review [40 points]

PART I


PART II


9. The length of a rectangle is one more than twice its width. The area of the rectangle is 10 square units. Find thelength and width of the rectangle.

Answer: Length � 5, width � 2

Solution:

Let w � the width of the rectangle.

Then 2w � 1 � the length of the rectangle.

The width cannot have negative length, so the width is 2 and the length is 5.

10. In right triangle ABC, m�C � 90. If the measure of an exterior angle at A is 140°, which side of the triangle is theshortest side?

Answer:

Explanation:Since the exterior angle at A is 140°, �A is a 40° angle. Since �C is a right angle, �B is a 50° angle. Thus, �A isthe angle with the shortest measure, and the side opposite �A, , is the shortest side.BC

BC

w 5 2w 5 252

w 2 2 5 02w 1 5 5 0

(2w 1 5)(w 2 2) 5 0

2w2 1 w 2 10 5 0

w(2w 1 1) 5 10


Name __________________________________________________________ Class ______________ Date ______________


PART III


11. Given: �ABC with coordinates A(8, �4), B(2, 0), and C(6, �8).

Graph and state the coordinates of , the image of �ABC under the composition .

Answer: A(�6, �2), B(0, �6), C(�4, 2)

Under :

Under R180°:

12. Given: l � m and transversal r.Find the degree measures of the angles numbered 1 to 6.

Answer: m�1 � 50, m�2 � 60, m�3 � 10, m�4 � 60, m�5 � 110,m�6 � 70

Solution:�5 is supplement of the 70° angle. Thus, m�5 � 110.

�6 is the supplement of �5. Thus, m�6 � 70.

The 110° angle is an external angle of the triangle with the 50° angleand �4.

Thus, m�4 � 50 � 110 or m�4 � 60.

�4 � �3 and �6 are vertical angles. Thus, m�4 � m�3 � 70, so 60 � m�3 � 70 or m�3 � 10.

m�2 � m�3 � 110 � 180, so m�2 � 10 � 110 � 180 or m�2 � 60.

Similarly, m�1 � m�2 � 70 � 180, so m�1 � 60 � 70 � 180 or m�1 � 50.

(4, 22) S Cr(24, 2)

(0, 6) S Br(0, 26)

(6, 2) S Ar(26, 22)

C(6, 28) S (4, 22)

B(2, 0) S (0, 6)

A(8, 24) S (6, 2)

T22, 6

R1808 + T22,6nArBrCr


Name __________________________________________________________ Class ______________ Date ______________


O

y

x

A

B1 2 3 4 5 6 7�6�5�4�3�2�1

7654321

�1�2�3�4�5�6�7

�9�8�7

�8�9

89

8 9

C

A

B

C

l

1r

m

2

3

456

70°

50°

110°

PART IV


13. Given: Rectangle ABCD, , and .

Prove: a. �1 � �2

b. �3 � �4

c.

Proof:

Statements Reasons

a. 1. Rectangle ABCD 1. Given.

2. 2. Opposite sides of a rectangle are �.

3. �A � �D 3. The angles of a rectangle are all right �’s.

4. 4. Given.

5. 5. Addition postulate.

6. , 6. Partition postulate.

7. 7. Transitive property.

8. �ABG � �DCH 8. SAS.

9. �1 � �2 9. Corresponding parts of congruent triangles are �.

b. 10. �3 � �4 10. Corresponding parts of congruent triangles are �.

c. 11. 11. Corresponding parts of congruent triangles are �.

12. 12. Converse of isosceles triangle theorem.

13. 13. Subtraction postulate.

14. , 14. Partition postulate.

15. 15. Transitive property.BP > CP

CP > HC 2 HPBP > BG 2 PG

BG 2 PG > HC 2 HP

HP > PG

BG > HC

AG > HD

HG 1 GD > HDAH 1 HG > AG

AH 1 HG > HG 1 GD

AH > GD

AB > CD

BP > CP

AH > GDAHGD, BPG, CPH


Name __________________________________________________________ Class ______________ Date ______________


A

B

2

3

1

4

H G D

C

P

14. a. Write the equation of a line through (�1, 0) and perpendicular to the line 2x � y � 6.

The given line is y � 2x � 6, so the slope of a perpendicular line is .

Answer

b. Find the equation of a line through the point (3, 1) and parallel to the line x � 2y � 4.

The given line is y � , so the slope of a parallel line is .

Answery 5 212x 1 52

y 5 212x 1 32 1 1

y 2 1 5 212(x 2 3)

21221

2x 1 2

y 5 212x 2 12

y 2 0 5 212(x 1 1)

212


Name __________________________________________________________ Class ______________ Date ______________


Chapter 10-1 The General Quadrilateral Section Quiz...

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