8 ANSWERS TO EXERCISES Answers to Exercises Answers to Exercises 21. 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 4 1 . 33. possible answer: 34. 35. Answers will vary. Possible answers: 8 cm 11 cm 8 cm 11 cm T A G Y R y x N P Q x y P R Q R´ P´ Q´ y x A (4, 0) M O C D B –5 –5 5 C A B D M N X Y B A 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 S M T or A C E D B AC CE BC CD y S R x 11 B A y x D E –3 3 –3 3 K L B A 1
15
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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.
10 ANSWERS TO EXERCISES
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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
ANSWERS TO EXERCISES 11
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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
12 ANSWERS TO EXERCISES
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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
ANSWERS TO EXERCISES 13
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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°
31. possible answer:
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
14 ANSWERS TO EXERCISES
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LESSON 1.4
1.
2. possible answers:
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
19. possible answer:
20. possible answer:
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.
26. possible answer:
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�
ANSWERS TO EXERCISES 15
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LESSON 1.5
1. D 2. A
3. C 4. B
5. 6.
7.
8. possible answer:
9. possible answer:
10. possible answer:
11. 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)
15. possible answers:
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
16. possible answers:
17. true
18. true
19. False, a diagonal connects nonconsecutive
vertices.
20. False, an angle bisector divides an angle into