answers to exercises - cerritos collegeweb.cerritos.edu/imccance/sitepages/worksheets and...

8 ANSWERS TO EXERCISES

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Answers to Exercises21. AB��, AC�� 22. PM��, PN��

23. XY��, XZ�� 24.

25. 26.

27.

28. 10

29–31.

32. Yes, P�(6, 2), Q�(5, �2), R�(4, �6); the slope

between any two of the points is �41

�.

33. possible answer:

34.

35. Answers will vary. Possible answers:

8 cm

11 cm

8 cm

11 cm

T

A

GY

R

y

x

N

P

Q

x

y

P

R

Q

R´

P´

Q´

y

xA (4, 0)

M

O

C

D

B

–5

–5

5

C

A BD

M

NX

Y

BA

CHAPTER 1 • CHAPTER CHAPTER 1 • CHAPTER

LESSON 1.1

1. sample answers: point: balls, where lines cross;

segment: lines of the parachute, court markings;

collinear: points along a line of the court structure;

coplanar: each boy’s hand, navel, and kneecap,

points along two strings

2. PT��, TP��3. any two of the following: AR��, RA��, AT��, TA��, RT��, TR��

4. any two of the following: MA��,MS��,AS��,AM��,SA��,SM��

5.6. 7.

8. AC� or CA� 9. PQ� or QP�

10. TR� or RT�, RI� or IR�, and TI� or IT�

11. 12.

13. mAB�� 14.3 cm 14. mCD�� 6.7 cm

15–17. Check each length to see if it is correct.

Refer to text for measurements.

18. R is the midpoint of PQ�. X is the midpoint of

WY��. Y is the midpoint of XZ�. No midpoints are

shown in �ABC.

19. possible answers:

20. P

Q

SM

T

orA C EDB

�� AC � CE

�� BC � CD

yS

R

x

11

BA

y

xD

E

–3

3

–3

3

K

L

BA

1

ANSWERS TO EXERCISES 9

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USING YOUR ALGEBRA SKILLS 1

1. (3, 4)

2. (�9, 1.5)

3. (5.5, 5.5)

4. (�6, 44)

5. Yes. The coordinates of the midpoint of a

segment with endpoints (a, b) and (c, d) are found

by taking the average of the x-coordinates, �a �

2c

�,

and the average of the y-coordinates, �b �

2d

�. Thus the

midpoint is ��a �2

c�, �

b �2

d��.

6. (3, 2) and (6, 4). To get the first point of

trisection, sum the coordinates of points A and B

to get (9, 6), then multiply those coordinates by �13

� to

get (3, 2). To get the second point of trisection, sum

the coordinates of points A and B to get (9, 6), then

multiply those coordinates by �23

� to get (6, 4). This

works because the coordinates of the first point

are (0, 0).

7. Find the midpoint, then find the midpoint of

each half.

8a. Midpoints for Figure 1 are (5.5, 6.5); for

Figure 2, (16, 6.75); and for Figure 3, (29.75, 5.5).

8b. For these figures the midpoints of the two

diagonals are the same point.


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LESSON 1.2

1. �TEN, �NET, �E; �FOU, �UOF, �1;

�ROU, �UOR, �2

2.

3.

4.

5. �S, �P, �R, �Q; none in the second figure

6. possible answer:

A

7. 90°

8. 120°

9. 45°

10. 135°

11. 45°

12. 135°

13. 30°

14. 90°

15. Yes; m�XQA � m�XQY � 45° � 90° � 135°,

which equals m�AQY.

16. 69°

17. 110°

18. 40°

19. 125°

20. 55°

21. �SML has the greater measure because

m�SML � 30° and m�BIG � 20°.

22.

23.

B90�

A44�

M

L

S

G

I

B

A T

N

24.

25.

26. no

27.

28.

29.

30. One possibility is 4:00.

31.

32.

33. G T

A

NI

R

T H

S A

T

H

O

90�

6 8

A

N

C

YD

A22�

22�

D

67.5�67.5�

C

D E

135�

B

C

D


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34.

35. MY; CK; m�I 36. �SEU; �EUO; MO

37. x � 54° 38. y � 102°

39. z � 32° 40. 288°

41. 242° 42. They add to 360°.

43. no

44. You will miss the target because the incom-

ing angle is too big.

Target

Mirror

Laser lightsource

W

B OT

I

45. 180 km. The towns can be represented by

three collinear points, P, S, and G. Because S is

between P and G, PS � SG � PG.

46.

47.

48. MS � DG means that the distance between

M and S equals the distance between D and G.

MS � DG means that segment MS is congruent

to segment DG. The first statement equates two

numbers. The second statement concerns the

congruence between two geometric figures.

However, they convey the same information and

are marked the same way on a diagram.

49. Answers will vary. Possible answer.

70�40�

D E6 cm

F

70�40�

4 cmA B

C

4.36 cm

2.18 cm 2.18 cm

12 cm

6 cm 6 cm


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LESSON 1.3

1–3. possible answers:

1.

2.

3.

4.

5.

6.

7.

8.

9. B is a Zoid. A Zoid is a creature that has in its

interior a small triangle with a large black dot

taking up most of its center.

10. A good definition places an object in a class

and also differentiates it from other objects in that

class. A good definition has no counterexamples.

D

40�

C

140�

A

40�

B

50�

EC

A D

B

P ER

A

M

S

D

G

I G

EB

T E

R

O G

D

45�

11. The measures of complementary angles sum

to 90°, whereas the measures of supplementary

angles sum to 180°.

12. No, supplementary angles can be uncon-

nected, while a linear pair must share a vertex and

a common side.

13a. an angle; measures less than 90°

13b. angles; have measures that add to 90°

13c. a point; divides a segment into two congruent

segments

13d. a geometry tool; is used to measure the sizes

of angles in degrees

14.

If AB�� and CD�� intersect at point P so that P is

between A and B, and P is between C and D, then

�APC and �BPD are a pair of vertical angles.

15. true

16. true

17. true

18. false

19. false

P

BC

D

A


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20. false

21. true

22. false; Though the converse is true, a

counterexample is

23. false

24. false

25. possible answers: (�3, 0), (0, 2), or (6, 6)

26. possible answers: (2, �3) or (5, �1)

27. The reflected ray and the ray that passes

through (called the refracted ray) are mirror

images of each other. Or they form congruent

angles with the mirror.

B

A

Mirror

Reflectedsegment

C T

A

40�D

C140�

C T

A

50� 80�

50�

28. One possible answer is A(�8, 8), B(�4.5, 6.5),

C(�11, 1).

29. 12 cm

30. 36°


32.

33.

34a. �13

26

00

°°

� � �13

�, �23

� left

34b. �36600°°

� � �16

�, �16

� missing

34c. �36

90°� � 40°

1 pt.

4 pt.

5 pt. 6 pt.

2 pt.

3 pt.

Infinitelymany points

0 pt.

2 pt.

3 pt.

1 pt.

y

x–12 –6

A

B

C

T

S

R

6


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LESSON 1.4

1.


3.

4. octagon

5. hexagon

6. heptagon

7. pentagon

8–10. One possible answer for each is shown.

8. pentagon FIVER

9. quadrilateral FOUR

10. equilateral quadrilateral BLOC

11a. a polygon; has eight sides

11b. a polygon; has at least one diagonal outside

of the polygon

11c. a polygon; has 20 sides

11d. a polygon; has all sides of equal length

12. One possibility is �C and �Y are consecutive

angles; CY� and YN� are consecutive sides.

13. 9; possible answer:

14. AC�, AD��, BD�, BE�, CE�

15. �TIN

16. �WEN

17a. a � 44, b � 58, c � 34

17b. m�T � 87° and m�I � 165°

18. PA�� FI� and �IVE � �ANC



21. 84 in.

22. 5.25 cm

23. AB � 14 m, CD � 25 m

24. complementary angles: �AOS and �SOC;

vertical angles: �OCT and �ECR or �TCE and

�RCO

25.


27. All are possible except two points.

0 pts.

3 pts. 4 pts.

1 pt.

5 pts. 6 pts.

20

5

E

A B

C DF

130� 130�

5 cm

130�


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LESSON 1.5

1. D 2. A

3. C 4. B

5. 6.

7.

8. possible answer:

9. possible answer:



12. possible answers: (�1, �1), (�1, 0), or (�4, 3)

13. possible answers: (3, �3), (3, 4), (�11, 4),

(�11, �3), (�0.5, 0.5), or (�7.5, 0.5)

14. possible answers: (�3, 1), (�1, �9), (2, 2),

(4, �8), (0, �1), or (1, �6)


9 cm45�

7 cm

BA

C

9 cm

12 cm

B

C

A45�

120�

4 cm

4 cm

ZP

A

80�4 cm

4 cm

BA

C

C

BA

C

BA6 cm 6 cm

40� 40�

b

2a2a

b – 2a

3a3a

R

A

C

S M L

C

A T

C


17. true

18. true

19. False, a diagonal connects nonconsecutive

vertices.

20. False, an angle bisector divides an angle into

two congruent angles.

21. true

22. (�4,1) → (�3,�2) (1,1) → (2, �2) (2, 4) →(3, 1) (�3, 5) → (�2, 2) Yes, the quadrilaterals are

congruent.

23. Find the midpoint of each rod. All the

midpoints lie on the same line; place the edge

of a ruler under this line.

24–26. Sample answers for 24–26

24. 25.

26. Q U

D A

P

AE

N T

P

AE

N T

y

(2, 4)

(1, 1)

(–3, 5)

(–4, 1)x

10 cm

40�

10 cm

E

F

A

10 cm

40�

14 cm

B

C

A


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LESSON 1.6

1. first figure: quadrilateral ABCD with two

congruent sides and one right angle; second

figure kite EFGH; third figure: trapezoid IJKL

with two right angles; fourth figure parallelogram

MNPQ

2. B 3. D

4. F 5. C

6. A, D, F

7.

8.

9.

10.

11.

12. square

13. possible answers: (2, 6), (�2, 4); (6, �2),

(2, �4); (3, �1), (1, 3)

14. 90 cm

15. (5, 6)

16. S(9, 0), I(4, �2)

17. S(�3, 0), I(�1, �5)

18. A(7, 6), N(5, 9) or A(�5, �2), N(�7, 1)

19a. Possible answer: Flip one triangle and align it

so that a congruent pair of sides forms a diagonal

of a kite.

T

R

H

G

I

E U

L

Q

A

N

BF

E

D I

Z O

19b. Possible answer: Rotate one triangle so that

a congruent pair of sides forms a diagonal of a

parallelogram.

20. There are three possibilities: a rhombus, a

concave kite, or a parallelogram.

21a. right triangles

21b. isosceles right triangles

22.

23.

24.

25. no

y

x

C (0, 5)

B (4, 4)

A (5, 0)

72�

72�

72�

72�

72�

5 cm5 cm


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LESSON 1.7

1. Answers will vary. Sample answers:The green

area in the irrigation photo is a circle,the water is a

radius,and a path on the far side of the circle appears

to be tangent to the circle.The wood bridge is an arc of

a circle,and the railings are arcs of concentric circles.

The horizontal support beam under the bridge is

a chord.

2. three of the following: AB�, BD�, EC�, EF�

3. EC�

4. AP�, EP�, FP�, BP�, CP�

5. five of the following: EF�, AE�, AB�, BC�, CD�, DF�,

EB�, ED�, FC�, AC�, DB�, AF�, AD�, BF�

6. EDC� or EFC�, EBC� or EAC�

7. two of the following: ECD�, EDF�, FEC�, DEC�, . . .

8. FG��, HB��9. either F or B

10. possible answers: cars, trains, motorcycles;

washing machines, dishwashers, vacuum cleaners;

compact disc players, record players, car racing,

Ferris wheel

11. mPQ� � 110°; mPRQ� � 250°

12.

13. possible answers: concentric rings on cross

sections of trees (annual rings),bull’s-eye or target,

ripples from a rock falling into a pond.

14.

15.

P Q

65�

215�

16. Equilateral quadrilateral. (The figure is

actually a rhombus, but students have not yet

learned the properties needed to conclude that the

sides are parallel.)

17. equilateral; 3 to 1

18.

19. yes; yes

20. yes; no

21. no; no

y

x8 16

8

y

x

5

–5

–5

5

y

x

5

–5

–5

5

s

A

QP

B


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22. 80°. The slices of pizza do not overlap, so

60° � x � 140°, where x is the measure in

degrees of the angle of the second slice.

23.

24. not possible

25.

26.

27. about 0.986°

28. 15°

29.

30.

T R

P A

F

T

A100�

C B

A

B

C A

31.

32. not possible

33.

34.

35.

36.

37. E

I

TK

120�

E2p

2p

2p 2p

I U

Q55�

50� 50�70� 70�

4a + 2b

T

R

G

120�120� 120�60�

120�60� 60�

60�


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LESSON 1.8

1. 2.

3. 4.

5. 6.

7. 8.

9. 60 boxes 10.

11. 12.

13. B, D 14. B

15. C 16. D

17. A

18. 19.

20. true 21. true

5

3 4

2

34

22. False. The two lines are not necessarily in the

same plane, so they might be skew.

23. true 24. true

25. true

26. False. They divide space into seven or eight

parts.

27. true 28. (�3, 1)

29. perimeter � 20.5 cm; m(largest angle) � 100°

30.

120�

13 cm

8 cm


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LESSON 1.9

1. Sample answer: Furniture movers might

visualize how to rotate a couch to get it up a

narrow staircase.

2. yes

3. W O M E N��; Nadine is ahead.

4. 28 posts

5. 28 days

6. 0 ft (The poles must be touching!)

7.

8.

9.

10.

11. no

12. C�(�2, 3), A�(0, 0), B�(0, 5), D�(�2, 1)

13. Y�(�4, 1), C�(3, �1), N�(0, 3)

14. (x, y) → (x � 3, y � 2)

15. AB�� CP��, EF�� GH��; i � k, j � k

16. perimeter � 34 cm

17. Left photo: Three points determine a plane.

Middle photo: Two intersecting lines determine a

plane. Right photo: A line and a point not on the

line determine a plane.

Polygons

Quadrilaterals

Trapezoids

Triangles

Obtuse Isosceles

BA

BA

3 cm

3 cm

3 cm

A

A

B

A B

18. pyramid with hexagonal base

19. prism with hexagonal base

20. pyramid with square base

21. x � 15, y � 27

22. x � 12, y � 4

23.

24.

25. point, line, plane

26. AB�� 27. AB�

28. vertex 29. AB��30. AB�� CD�� 31. protractor

32. �ABC 33. AB�� CD��

34. congruent to

35. The distance is two times the radius.

36. They bisect each other and are perpendicular.

QP

B

A

PQ = 2r

QP

rr


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CHAPTER 1 REVIEW

1. true

2. False; it is written as QP��.

3. true

4. False; the vertex is point D.

5. true

6. true

7. False; its measure is less than 90°.

8. false; two possible counterexamples:

9. true

10. true

11. False; they are supplementary.

12. true

13. true

14. true

15. False; it has five diagonals.

16. true

17. F

18. G

19. L

20. J

21. C

22. I

23. no match

24. A

25. no match

26.

27.

28.

N

A

G

R

I E

S

PT

5

7

3

O

E Y

T

K

C

AA

CD

D

BPP

�APD and �APCare a linear pair.

�APD and �APCare the same angle.

29.

30.

31.

32.

33.

34.

35.

36.

37.

40�

125�

2 in.

3 in.5 in.

DC

A

B

T

A

P

R T

A P

EY


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38. 114°

39. x � 2, y � 1

40. x � 12, y � 4

41. x � 4, y � 2.5

42. x � 10, y � 8

43. AB � 16 cm

44. 96°

45. 105°

46. 30°

47.

48. 66 inches

49. 3 feet

50. The path taken by the midpoint of the ladder is

an arc of a circle or a quarter-circle if the ladder

slides all the way from the vertical to the horizontal.

Shed

Path of flashlight

Quadrilateral

Rectangle Square Rhombus

Trapezoid

51. He will get home at 5:46 (assuming he goes

inside before he gets blown back again).

52. (2, 3)

53.

54.

55.

56.

57.

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