Chapter 10 Quadrilaterals Name __________________________________________________________

Chapter 10-1 The General Quadrilateral Chapter 10-2 The Parallelogram

Class ______________

205

Date ______________

Section Quiz [20 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. If ABCD is a parallelogram, which statement must be true?

4. In parallelogram ABCD, if mA  50, find mC.

(1) AC > BD

(1) 25

✔ (2) AD > BC

(2) 40

(3) A and C are supplementary. (4) A  D 2. If ABCD is a parallelogram, which statement must be true? ✔ (2) AC bisects BD.

(1) 9

(3) 36 ✔ (4) 54

6. Which statement is not always true for a parallelogram?

(3) AC bisects C.

(1) Consecutive angles are supplementary.

(4) AC  BD 3. If mCDA  mCDA.

(4) 130

5. In parallelogram ABCD, diagonals AC and BD intersect at E. If EC  31, EB  3x, and AE  4x  5, what is the value of BD? (2) 27

(1) AC ' BD

✔ (3) 50

✔ (2) The diagonals are perpendicular. 1 2x

 32 and mBCD 

5 2x

 20, find

(3) The opposite sides are congruent. (4) The opposite angles are congruent.

(1) 26

(3) 56

(2) 45

✔ (4) 60

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 numerical answer 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 of the 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 360 4 5 908. The sum of the measures of the angles of a triangle is 180°. Thus, the average degree measure is 180 3 5 608. Copyright © Amsco School Publications, Inc.

206

Chapter 10 Quadrilaterals

Name __________________________________________________________

Class ______________

Date ______________

b. In parallelogram ABCD, mA  5x and mC  3x  30. Find mA and mB. Answer: mA  75, mB  105 Solution: 5x 5 3x 1 30 2x 5 30 x 5 15 Therefore, mA  5(15)  75, mC  180  75  105.

8. In parallelogram ABCD, mD  108 and GA bisects BAD. Find mC and mAGC. B

A

G

C

108° D

Answer: mC  72, mAGC  144 Solution: m/A 5 180 2 m/D m/A 5 180 2 108 m/A 5 72

m/AGD 1 m/D 1 m/C 1 m/AGC 5 360 36 1 108 1 72 1 m/AGC 5 360 m/AGC 5 144

m/C 5 m/A 5 72 mGAD  72 2 5 36

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Chapter 10 Quadrilaterals Name __________________________________________________________

Class ______________

Chapter 10-3 Proving that a Quadrilateral Is a Parallelogram

207

Date ______________

Section Quiz [20 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. A quadrilateral is a parallelogram if two of its opposite sides are

For 4 and 5, use the figure below. D

(1) parallel and the other two sides are congruent. (2) parallel only. 2

(4) congruent only.

1

A

(1) 1  3 and 2  4

(2) are perpendicular.

(2) AB > DC and 1  3

(4) bisect the angles of the quadrilateral. 3. A quadrilateral is a parallelogram if

5

6

B

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

(1) are equal. ✔ (3) bisect each other.

4

O

✔ (3) congruent and parallel.

2. In quadrilateral ABCD, AB  CD and AD  BC. It must necessarily follow that the diagonals AC and BD

C 3

7 8

✔ (3) AB > BD (4) AOB  DOC

(1) two adjacent sides are congruent.

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

(2) three sides are congruent.

✔ (1) 1  2 and 5  7

(3) the diagonals form 45° angles with each other. ✔ (4) the diagonals bisect each other

(2) 1  4 and AD > DC > AB (3) 1  5, 2  4, and 4  7 (4) 6  7 and AO > OC 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

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208

Chapter 10 Quadrilaterals

Name __________________________________________________________

Class ______________

Date ______________

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 numerical answer with no work shown will receive only 1 credit. [8] 7. Complete the proof by filling in the missing reasons.

B

X

R

D

C

Given: Parallelogram ABCD and AR > CX Prove: BXDR is a parallelogram. Proof:

A

Statements

Reasons

1. Parallelogram ABCD

1. Given.

2. AD y BC

2. Definition of a parallelogram.

3. RD y BX

3. Segments of parallel lines are .

4. AD > BC

4. Opposite sides of a parallelogram are .

5. AR > CX

5. Given.

6. AD 2 AR > BC 2 CX

6. Subtraction postulate.

or RD > BX 7. BXDR is a parallelogram.

7. If one pair of opposite sides is both  and , then the quadrilateral is a parallelogram.

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Chapter 10 Quadrilaterals Name __________________________________________________________

Class ______________

8. Complete the proof by filling in the blanks.

Date ______________

C

B 3

Given: AB > DC, AD > BC Prove: Quadrilateral ABCD is a parallelogram.

1

Proof:

4

A

Statements

Reasons

1. AB > DC, AD > BC

1. Given.

2. AC > AC

2. Reflexive property

3. ABC  CDA

3. SSS.

4. 1  2 and 3  4

4. Corresponding parts of congruent triangles are .

5. AB y DC and AD y BC

5. If alternate interior angles are , then the two lines are .

6. ABCD is a parallelogram.

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6. Definition of a parallelogram

209

D

2

210

Chapter 10 Quadrilaterals

Name __________________________________________________________

Class ______________

Chapter 10-4 The Rectangle

Date ______________

Section Quiz [20 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] For 1 and 2, use the rectangle given below. B

C

4. The perimeter of rectangle ABCD is 16. If the length of the rectangle is greater than 7, which of the following is a possible value for the width? ✔ (1) 0.5

E

(3) 1.5

(2) 1 A

D

1. If CAD  42°, what is the measure of CED? (1) 42 (2) 84

✔ (3) 96 (4) 138

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

5. The area of a rectangle is 54 square inches and the perimeter is 30 inches. If the length and width are integers, what is the absolute value of the difference between the length and the width? (1) 2 in.

(3) 12 in.

✔ (2) 3 in.

(4) 24 in.

6. G 8

F 7

(1) 50% ✔ (2) 60% (3) 83.3%

(4) 2

3

4

P 2 D

1

6

5 E

(4) 90% 3. Which of the following is not a property of all rectangles? (1) The diagonals bisect each other. ✔ (2) The diagonals are perpendicular to each other. (3) The diagonals are congruent. (4) The angles are congruent.

Which of the following statements is not sufficient to show that DEFG is a rectangle? (1) 2  8, 4  5, and GP > PE (2) 2  5, DG > EF, and DP > PF ✔ (3) 6  7 and mGDE  90 (4) DE y FG, DG y EF, and 7  3

Copyright © Amsco School Publications, Inc.

Chapter 10 Quadrilaterals Name __________________________________________________________

Class ______________

211

Date ______________

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 numerical answer with no work shown will receive only 1 credit. [8] 7. a. In rectangle ABCD, E is the midpoint of diagonal AC. Find the measure of AED.

C

B

60°

Answer: mAED  120

E

Solution: mCAD  180  60  90  30 The diagonals are congruent and bisect each other. Thus, AE  EC  ED, and so:

A

D

mEDA  mCAD  30 mAED  180  30  30  120 b. In rectangle ABCD, AC  3x  1 and DE  x  13. Find the length of AE. Answer: AE  40 Solution: AC 5 2DE 3x 2 1 5 2(x 1 13) 3x 2 1 5 2x 1 26 x 5 27 Therefore, AE 5 DE 5 27 1 13 5 40. 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: 22 24 Slope of ST 5 22 21 2 3 5 24 5 1

20 22 Slope of SN 5 22 21 1 3 5 2 5 21

0 4 4 22 Slope of AN 5 41 2 Slope of TA 5 23 2 13 5 4 5 1 2 1 5 22 5 21 The slopes of opposite sides are negative reciprocals of each other. Therefore, the angles of STAN are all right angles, and STAN is a rectangle.

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212

Chapter 10 Quadrilaterals

Name __________________________________________________________

Class ______________

Chapter 10-5 The Rhombus

Date ______________

Section Quiz [20 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1.

5.

C

B

D

20°

6

C 7

E 5 A

D

4 3 E 2

1

A

In rhombus ABCD, diagonals AC and BD intersect at E. If mBCE  20, find mADC. (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

B

Which statement is not sufficient to show that parallelogram ABCD is a rhombus? (1) AB > AD ✔ (2) 1  2 (3) 1  5 and 6  7 (4) 3  4 6.

B

3. Which of the following statements is not always true for a rhombus? ✔ (1) The diagonals are congruent.

A

120° C

(2) The diagonals are perpendicular. (3) The diagonals bisect each other. (4) The diagonals bisect opposite angles. 4. In rhombus PQRS, PQ  3x  3, PS  5x  1, and RS  10x  11. What is the perimeter of the rhombus?

D E g

In rhombus ABCD with diagonal BDE, if

(1) 8

mC  120, what is mADE?

(2) 9

(1) 30

(3) 120

(3) 32

(2) 60

✔ (4) 150

✔ (4) 36

Copyright © Amsco School Publications, Inc.

Chapter 10 Quadrilaterals Name __________________________________________________________

Class ______________

Date ______________

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 numerical answer with no work shown will receive only 1 credit. [8] 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: SP and RQ 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 a parallelogram. b. Prove that PQRS is a rhombus. Proof: 022 22 1 Slope of PR 5 23 2 1 5 24 5 2 513 8 Slope of SQ 5 23 2 1 5 24 5 22 The slopes of the diagonals PR and SQ are negative reciprocals of each other, and so PR ' SQ. A parallelogram with perpendicular diagonals is a rhombus. Therefore, PQRS is a rhombus.

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

B

C

Given: Parallelogram ABCD, BD ' AC at E. E

Prove: ABCD is a rhombus. A

D

Statements

Reasons

1. Parallelogram ABCD

1. Given. 2. The diagonals of a parallelogram bisect

2. BE > ED

each other. 3. BD ' AC at E 4. BEA  AED

3. Given. Right angles are congruent. 4.

5. AE > AE

5.

6. AEB  AED

6. SAS.

7. AB > AD

7.

8. ABCD is a rhombus.

8.

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Reflexive property.

Definition of a rhombus.

213

214

Chapter 10 Quadrilaterals

Name __________________________________________________________

Class ______________

Chapter 10-6 The Square

Date ______________

Section Quiz [20 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. Parallelogram RSTV must be a square if its

5. D

C 4

(1) opposite angles and opposite sides are congruent. 1

✔ (2) sides and angles are congruent. (3) diagonals bisect each other and are perpendicular to each other.

2 E 3

A

B

(4) diagonals are congruent. 2. If (3, 2), (1, 5), and (4, 1) are consecutive vertices of a square, which of the following represents the coordinates of the fourth vertex? (1) (2, 0)

(3) (1, 2)

✔ (2) (0, 2)

(4) (1, 3)

3. Equilateral ABC and square PQRS have the same perimeters. If a side of the triangle is 3x  3 and a side of the square is 2x  5, what is the length of the side of the square? (1) 29 ✔ (2) 63

Parallelogram ABCD is a square when which of the following is true? (1) AD > AB ✔ (2) AE > DE and 1  2 (3) DB > AC (4) 3  4 6. Of all rectangles with a given perimeter, the square has maximum area. What is the maximum area of a rectangle with perimeter 24?

(3) 84

(1) 24 sq units

(4) 252

(2) 25 sq units

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

✔ (3) 36 sq units (4) 576 sq units

i (A)  C (1) rBD

(2) AC bisects BAD (3) AC ' BD ✔ (4) AB > AC

Copyright © Amsco School Publications, Inc.

Chapter 10 Quadrilaterals Name __________________________________________________________

Class ______________

Date ______________

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 numerical answer with no work shown will receive only 1 credit. [8] Q

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

R

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

A

Solution: 5x 2 3 5 2(4x 2 2) 5x 2 3 5 8x 2 12 9 5 3x x53 Therefore, PR  QS  5(3)  3  12 and QA  6.

8. If mPAQ  3x  y and mPSR  x  2y, what are the values of x and y? Answer: x  18 and y  36 Solution: 3x 2 y 5 90 S y 5 3x 2 90 Substituting y into the equation x  2y  90 yields: x 2 2(3x 2 90) 5 90 x 2 6x 2 180 5 90 25x 5 290 x 5 18 Therefore, y  3(18)  90  36.

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P

215

S

216

Chapter 10 Quadrilaterals

Name __________________________________________________________

Class ______________

Chapter 10-7 The Trapezoid

Date ______________

Section Quiz [20 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. The length of one base of a trapezoid is three times the length of the other base. If the length of the median of the trapezoid is 10, the length of the other base is (1) 7.5 (2) 12

5.

D

✔ (3) 15 (4) 20

40° A

E

C 130°

50° B

(1) one triangle and one parallelogram.

In the given figure, AE bisects DAC. If mEAB  40, mABC  50, and mBCD  130, what is mADC?

(2) two trapezoids of equal area.

(1) 40

2. The median of an isosceles trapezoid always divides the trapezoid into

✔ (3) two isosceles trapezoids. (4) two congruent trapezoids. 3. PQRS is a trapezoid with PQ y SR. Which additional piece of information would guarantee that the trapezoid is isosceles? (1) P and Q are supplementary. (2) P  R ✔ (3) PR > QS (4) PR and QS bisect each other. 4. The opposite angles of an isosceles trapezoid are always

(2) 80 ✔ (3) 100 (4) 140 6. The area of a trapezoid is 100 square inches, its altitude is 10 inches, and the length of one of its bases is 5 inches. What is the length of the other base? (1) 5 in. (2) 10 in. ✔ (3) 15 in. (4) 20 in.

(1) acute (2) congruent ✔ (3) supplementary (4) complementary

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Chapter 10 Quadrilaterals Name __________________________________________________________

Class ______________

217

Date ______________

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 numerical answer with no work shown will receive only 1 credit. [8] 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 the area, in square feet, of the trapezoid? Area 5 12 (6)(9 1 17) 5 3(26) 5 78 ft2 Answer b. In isosceles trapezoid DEFG, mD is three times mF. Find mF. Answer: mF  45 Solution: In an isosceles trapezoid, any two opposite angles are supplementary. m/D 1 m/F 5 180 3m/F 1 m/F 5 180 4m/F 5 180 m/F 5 45 B

8. Given: Isosceles Trapezoid ABCD with AB > DC and AD y BC. Diagonal BD bisects CDA.

C

Prove: AB > BC Proof:

A

D

Statements

Reasons

1. AD y BC

1. Given.

2. ADB  CBD

2. Alternate interior angles are congruent in  lines.

3. Diagonal BD bisects CDA.

3. Given.

4. ADB  CDB

4. Definition of angle bisector.

5. CBD  CDB

5. Transitive property.

6. BC > DC

6. Converse of isosceles triangle theorem.

7. AB > DC

7. Given.

8. AB > BC

8. Transitive property.

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218

Chapter 10 Quadrilaterals

Name __________________________________________________________

Class ______________

Chapter 10-8 Areas of Polygons

Date ______________

Section Quiz [20 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. If the coordinates of the vertices of parallelogram ABCD are A(0, 2), B(2, 5), C(10, 5), and D(8, 2), the area of the parallelogram is: (1) 12 sq units

4. If the diagonals of a rhombus have lengths of 6 and 12, the area of the rhombus is: (1) 72 sq units ✔ (2) 36 sq units

✔ (2) 24 sq units

(3) 30 sq units

(3) 36 sq units

(4) 18 sq units

(4) 48 sq units 2. The area of a square whose perimeter is 8k is:

5. Q

A

R

(1) 4k!2 sq units (2) 4k2 !2 sq units

✔ (3) 4k2 sq units (4) 8k2 sq units 3.

P

Q

P

A

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

S

In rhombus PQRS, diagonals QS and PR intersect at A. If QS  14 and PR  12, what is the area of PRS? (1) 21 sq units ✔ (2) 42 sq units

S

(1) 14

✔ (3) 12

(2) 31

(4) 12

6. The perimeter of a rectangle is 6x. If one side has length x2, what is the area of the rectangle? 2 (1) x4 sq units 2 ✔ (2) 5x 4 sq units 2 (3) 5x 2 sq units

(4) 3x2 sq units

(3) 84 sq units (4) 126 sq units

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Chapter 10 Quadrilaterals Name __________________________________________________________

Class ______________

219

Date ______________

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 numerical answer with no work shown will receive only 1 credit. [8] 7. a. If the perimeter of rectangle ABCD is 58 and the length of AD is 16, what is the area of ABCD? Answer: 208 sq units Solution: Let x  AB  DC and y  AD  BC  16. Then: x 1 y 5 29 y 5 16 x 5 29 2 16 5 13 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 of the rhombus? Solution: The diagonals partition the rhombus into four congruent right triangles with hypotenuses that are 5 centimeters long. Since the diagonals have integer lengths, the only possible lengths are 3 and 4 centimeters. (The triangles are 3-4-5 right triangles.) Thus, the area is: 4 C12 (3)(4) D 5 24 cm2 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 the smaller 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. 72 5 12 (8)(x 1 2x) 72 5 4(3x) 65x 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 the rectangle. Answer: 180 in.2 Solution: Let x and 5x represent the width and height of the rectangle, respectively. 2(x 1 5x) 5 72 6x 5 36 x56 Thus, the width is 6 and the height is 30. The area is 180 square inches. Copyright © Amsco School Publications, Inc.

220

Chapter 10 Quadrilaterals

Name __________________________________________________________

Class ______________

Chapter 10 Quadrilaterals

Date ______________

Chapter Review [40 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. Which is always true of the diagonals of a parallelogram? (1) The diagonals are congruent. (2) The diagonals are perpendicular. ✔ (3) The diagonals bisect each other. (4) The diagonals bisect the angles of the parallelogram. 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 mB exceeds mA by 56°, what is mB? (1) 62 (2) 112 ✔ (3) 118 (4) 124 4. In quadrilateral ABCD, AB > CD and AB y CD. Which statement must be true? (1) The diagonals bisect the angles of the quadrilateral. ✔ (2) The diagonals bisect each other. (3) The diagonals are equal in measure. (4) The diagonals are perpendicular.

5. In a trapezoid, the length of the median is 14 and the length of one base is 10. The length of the other base is: (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 is the 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, a parallelogram, a rhombus, a rectangle, and a square. If one of these quadrilaterals is picked at random, what is the probability that the diagonals of the chosen figure bisect each other? (1) 0

✔ (3) 45

(2) 15

(4) 1

8. A parallelogram must be a rectangle if the opposite angles (1) are complementary. ✔ (2) are supplementary. (3) are congruent. (4) sum to 120°.

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Chapter 10 Quadrilaterals Name __________________________________________________________

Class ______________

221

Date ______________

PART II

Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [4] 9. In parallelogram MRST, mM  3x  40 and mS  7x  100. Find: a. the value of x. 3x 2 40 5 7x 2 100 60 5 4x x 5 15 Answer b. the measure of R. m/R 5 180 2 m/M 5 180 2 f3(15) 2 40g 5 180 2 (45 2 40) 5 175 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 other diagonal? Answer: 12 Solution: The diagonals bisect each other and form congruent right triangles. Let y  one-half the length of the other diagonal. Then: 82 1 y2 5 102 y2 5 36 y56 Therefore, the length of the other diagonal is 2(6)  12.

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222

Chapter 10 Quadrilaterals

Name __________________________________________________________

Class ______________

Date ______________

PART III

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 numerical answer with no work shown will receive only 1 credit. [8] 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. 2(2w 1 w) 5 84 3w 5 42 w 5 14 Therefore, the length is 2(14)  28. b. What is the area of the rectangle? Area 5 14(28) 5 392 in. 2 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. 3x 2 1 5 2x 1 11 x 5 12 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.

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Chapter 10 Quadrilaterals Name __________________________________________________________

Class ______________

Date ______________

PART IV

Answer all questions in this part. Each correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [12] E

D

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

C

Prove: EAB  EBA Proof:

A

B

Statements

Reasons

1. Rectangle ABCD

1. Given.

2. AD > BC

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 DC.

4. Given.

5. DE > EC

5. Definition of midpoint.

6. AED  BEC

6. SAS.

7. AE > EB

7. Corresponding parts of congruent triangles are .

8. EAB  EBA

8. Isosceles triangle theorem.

14. Given: BD bisects AC at E and CAD  BCA.

A

Prove: ABCD is a parallelogram.

B E

D

Proof:

C

Statements

Reasons

1. BD bisects AC at E.

1. Given.

2. AE > EC

2. Definition of bisector.

3. CAD  BCA

3. Given.

4. AD y BC

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. AD > BC

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.

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Name __________________________________________________________

Class ______________

Chapter 10 Quadrilaterals

Date ______________

Cumulative Review [40 points]

PART I

Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12] 1. Which statement is logically equivalent to ~p → q?

5.

y B(r, s)

(1) q → p ✔ (2) ~q → p (3) q → ~p

O

(4) ~q → ~p 2.

x

A

B

If AB is a line segment and AO  OB, what are the coordinates of point A? (1) (s, r) (2) (s, r) D

50°

35°

140°

A

C

What is the measure of ABC?

(3) (r, s) ✔ (4) (r, s) 6. If the sum of the interior angles of a regular polygon is 1,080°, how many sides does this polygon have?

(1) 50°

(1) 3

(3) 6

✔ (2) 55°

(2) 4

✔ (4) 8

(3) 60° (4) 65° 3. In ABC, if AB  6 and BC  10, which of the following statements must be true? ✔ (1) AC  4 (2) mA  mC

7. Given parallelogram HOPE, which of the following statements may not be true? (1) mHOP  mPEH ✔ (2) HP > EO (3) HO y EP (4) HE > OP

(3) 6  AC  10 (4) mA  mC  mB 4. How many positive integers are in the solution set of the inequality 3x  5  2? (1) One ✔ (2) Two

8. If a regular polygon has 12 sides, what is the measure of each exterior angle? ✔ (1) 30° (2) 36° (3) 360° (4) 1,800°

(3) Three (4) Infinitely many

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Chapter 10 Quadrilaterals Name __________________________________________________________

Class ______________

225

Date ______________

PART II

Answer all questions in this part. Each correct answer will receive 2 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [4] 9. The length of a rectangle is one more than twice its width. The area of the rectangle is 10 square units. Find the length 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. w(2w 1 1) 5 10 2w2 1 w 2 10 5 0 (2w 1 5)(w 2 2) 5 0 2w 1 5 5 0 w 5 252

w22 5 0 w52

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

10. In right triangle ABC, mC  90. If the measure of an exterior angle at A is 140°, which side of the triangle is the shortest side? Answer: BC 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 is the angle with the shortest measure, and the side opposite A, BC, is the shortest side.

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Name __________________________________________________________

Class ______________

Date ______________

PART III

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 numerical answer with no work shown will receive only 1 credit. [8] 11. Given: ABC with coordinates A(8, 4), B(2, 0), and C(6, 8). Graph and state the coordinates of nArBrCr , the image of ABC under the composition R1808 + T22,6. Answer: A (6, 2), B (0, 6), C (4, 2)

C

9 8 7 6 5 4 3 2 1

98765432 1 1 2 A 3 4 5 6 7 8 9

Under T22, 6:

y

A(8, 24) S (6, 2) B(2, 0) S (0, 6) C(6, 28) S (4, 22) Under R180°: x

O B 1 2 3 4 5 6 7 8 9

A

(6, 2) S Ar(26, 22) (0, 6) S Br(0, 26) (4, 22) S Cr(24, 2)

B C

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

l m

Answer: m1  50, m2  60, m3  10, m4  60, m5  110, m6  70

50°

Solution: 5 is supplement of the 70° angle. Thus, m5  110. 6 is the supplement of 5. Thus, m6  70.

110° 5 4 6

2 3

1

70°

r

The 110° angle is an external angle of the triangle with the 50° angle and 4. Thus, m4  50  110 or m4  60. 4  3 and 6 are vertical angles. Thus, m4  m3  70, so 60  m3  70 or m3  10. m2  m3  110  180, so m2  10  110  180 or m2  60. Similarly, m1  m2  70  180, so m1  60  70  180 or m1  50.

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Chapter 10 Quadrilaterals Name __________________________________________________________

Class ______________

Date ______________

PART IV

Answer all questions in this part. Each correct answer will receive 6 credits. Clearly indicate the necessary steps, including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical answer with no work shown will receive only 1 credit. [12] 13. Given: Rectangle ABCD, AHGD, BPG, CPH, and AH > GD.

B

C

3

Prove: a. 1  2

4

P

b. 3  4

1

A

2

H

G

D

c. BP > CP Proof: Statements a.

Reasons

1. Rectangle ABCD

1. Given.

2. AB > CD

2. Opposite sides of a rectangle are .

3. A  D

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

4. AH > GD

4. Given.

5. AH 1 HG > HG 1 GD

5. Addition postulate.

6. AH 1 HG > AG, HG 1 GD > HD

6. Partition postulate.

7. AG > HD

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. BG > HC

11. Corresponding parts of congruent triangles are .

12. HP > PG

12. Converse of isosceles triangle theorem.

13. BG 2 PG > HC 2 HP

13. Subtraction postulate.

14. BP > BG 2 PG, CP > HC 2 HP

14. Partition postulate.

15. BP > CP

15. Transitive property.

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Date ______________

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 212. y 2 0 5 212 (x 1 1) y 5 212x 2 12 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  212x 1 2, so the slope of a parallel line is 212. y 2 1 5 212 (x 2 3) y 5 212x 1 32 1 1 y 5 212x 1 52 Answer

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