32 ANSWERS TO EXERCISES Answers to Exercises LESSON 3.1 1. 2. 3. 4. 5. possible answer: 6. m3 m1 m2; possible answer: 7. possible answer: 8. A B C AC BC AB 1 2 3 1 2 copy E G L AB 2EF CD EF EF AB CD AB AB CD CD E F C D A B 9. For Exercise 7, trace the triangle. For Exercise 8, trace the segment onto three separate pieces of patty paper. Lay them on top of each other, and slide them around until the segments join at the endpoints and form a triangle. 10. One method: Draw DU . Copy Q and construct COY QUD. Duplicate DUA at point O. Construct OYP UDA. 11. One construction method is to create congruent circles that pass through each other’s center. One side of the triangle is between the centers of the circles; the other sides meet where the circles intersect. 12. a 50°, b 130°, c 50°, d 130°, e 50°, f 50°, g 130°, h 130°, k 155°, l 90°, m 115°, n 65° 13. west 14. An isosceles triangle is a triangle that has at least one line of reflectional symmetry.Yes, all equilateral triangles are isosceles. 15. 16. new coordinates: A(0, 0), Y(4, 0), D(0, 2) 17. Methods will vary. It isn’t possible to draw a second triangle with the same side lengths that is not congruent to the first. 11 cm 8 cm 10 cm y x D A Y –6 –6 6 6 A' D' Y' U A D Q O P Y C Answers to Exercises CHAPTER 3 • CHAPTER CHAPTER 3 • CHAPTER 3
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32 ANSWERS TO EXERCISES
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LESSON 3.1
1.
2.
3.
4.
5. possible answer:
6. m�3 � m�1 � m�2; possible answer:
7. possible answer:
8.
A B
C
AC BC
AB
1 2 3�1
�2
copy
EG
L
AB � 2EF � CD
EFEFAB
CD
AB
AB � CD
CD
EF
CD
AB
9. For Exercise 7, trace the triangle. For Exercise 8,
trace the segment onto three separate pieces of
patty paper. Lay them on top of each other, and
slide them around until the segments join at the
endpoints and form a triangle.
10. One method: Draw DU��. Copy �Q and
construct �COY � �QUD. Duplicate �DUA
at point O. Construct �OYP � �UDA.
11. One construction method is to create congruent
circles that pass through each other’s center. One
side of the triangle is between the centers of the
circles; the other sides meet where the circles
intersect.
12. a � 50°, b � 130°, c � 50°, d � 130°, e � 50°,
f � 50°, g � 130°, h � 130°, k � 155°, l � 90°,
m � 115°, n � 65°
13. west
14. An isosceles triangle is a triangle that has at
least one line of reflectional symmetry.Yes, all
equilateral triangles are isosceles.
15.
16. new coordinates: A�(0, 0), Y�(4, 0), D�(0, 2)
17. Methods will vary. It isn’t possible to draw a
second triangle with the same side lengths that is
not congruent to the first.
11 cm 8 cm
10 cm
y
xD A
Y
–6
–6 6
6
A'
D'
Y'
UA
DQ
OP
YC
Answers to Exercises
CHAPTER 3 • CHAPTER CHAPTER 3 • CHAPTER3
ANSWERS TO EXERCISES 33
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LESSON 3.2
1.
2.
3.
4.
5.
6. Exercises 1–5 with patty paper:
Exercise 1 This is the same as Investigation 1.
Exercise 2Step 1 Draw a segment on patty paper. Label it QD��.
Step 2 Fold your patty paper so that endpoints Q
and D coincide. Crease along the fold.
Step 3 Unfold and draw a line in the crease.
Step 4 Label the point of intersection A.
Step 5 Fold your patty paper so that endpoints Q
and A coincide. Crease along the fold.
Step 6 Unfold and draw a line in the crease.
Step 7 Label the point of intersection B.
Step 8 Fold your patty paper so that endpoints A
and D coincide. Crease along the fold.
Step 9 Unfold and draw a line in the crease.
Step 10 Label the point of intersection C.
AB CD
M N
12
MN = (AB + CD)
C D
CD12
12
CD12
CD12
AB AB
2AB – CD
Edge of the paper
Original segment
Q D
A B
Exercise 3 This is the same as Investigation 1.
Exercise 4Step 1 Do Investigation 1 to get �
12
�CD.
Step 2 On a second piece of patty paper, trace AB�two times so that the two segments form a segment
of length 2AB.
Step 3 Lay the first piece of patty paper on top of
the second so that the endpoints coincide and the
shorter segment is on top of the longer segment.
Step 4 Trace the rest of the longer segment with a
different colored writing utensil. That will be the
answer.
Exercise 5Step 1 Trace segments AB and CD so that the two
segments form a segment of length AB � CD.
Step 2 Fold your patty paper so that points A and
D coincide. Crease along the fold.
Step 3 Unfold and draw a line in the crease.
7. The perpendicular bisectors all intersect in one
point.
8. The medians all intersect in one point.
9. GH�� appears to be parallel to EF�, and its length
is half the length of EF�.
D EH
G
F
C
B
N
M
L
A
LA
I
10. The quadrilateral appears to be a rhombus.
11.
Any point on the perpendicular bisector of the
segment connecting the two offices would be
equidistant from the two post offices. Therefore,
any point on one side of the perpendicular bisector
would be closer to the post office on that side.
12. It is a parallelogram.
13. The triangles are not necessarily congruent,
but their areas are equal. A cardboard triangle
would balance on its median.
F L
AT
Ness Station
Umsar Station
V
R
I
C
E
D
O
S
14. One way to balance it is along the median. The
two halves weigh the same.
sample figure:
15. F 16. E
17. B 18. A
19. D 20. C
21. B, C, D, E, H, I, O, X (K in some fonts, though
not this one)
22. Methods will vary.
It is not possible to draw a second triangle with the
same angle and side measures that is not congruent
to the first.
10 cm
40� 70�A B
C
Area CDB = 158 in2
Area DAB = 158 in2
RulerA
C
D
B
34 ANSWERS TO EXERCISES
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ANSWERS TO EXERCISES 35
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LESSON 3.3
1.
The answer depends on the angle drawn and where
P is placed.
2.
3.
Two altitudes fall outside the triangle, and one falls
inside.
4. From the point, swing arcs on the line to
construct a segment whose midpoint is that point.
Then construct the perpendicular bisector of the
segment.
5. Construct perpendiculars from point Q and
point R. Mark off QS� and RE� congruent to QR��.
Connect points S and E.
ES
RQ
B
O TU
C
AD
B
G
PB
I
6. The two folds are parallel.
7. Fold the patty paper through the point so that
two perpendiculars coincide to see the side closest
to the point. Fold again using the perpendicular
of the side closest to the point and the third
perpendicular; compare those sides.
8. Draw a line. Mark two points on it, and label
them A and C. Construct a perpendicular at C.
Mark off CB� congruent to CA�. The altitude CD�� is
also the angle bisector, median, and perpendicular
bisector.
9.
10.
11.
12.
Complement of �A
�A
A B
E L
T
O B
U
C
A
D
B
P
Q
13. See table below.
14.
15.
16.
17.
C D
E
F
A
B
Y
F
I T
18. not congruent
19. possible answer:
20. possible answer:
5 cm
9 cm
5 cm
9 cm
40�
8 cm
6 cm
40�
8 cm
6 cm
36 ANSWERS TO EXERCISES
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Rectangle 1 2 3 4 5 6 … n … 35
Number of shaded 2 9 20 35 54 77 … … 2484triangles
Rectangular pattern with triangles
13. (Lesson 3.3)
(2n � 1)(n � 1)
ANSWERS TO EXERCISES 37
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LESSON 3.4
1. D
2. F
3. A
4. C
5. E
6.
7.
8.
9a, b.
9c.
10.AnglebisectorAltitude
Median
45�135�
45�45�
90�
M S
U
O
E
M
M
E
S
M
A
A
B
P
PR
M
R AR P
z
z
z
11.
12. The angle bisectors are perpendicular. The
sum of the measures of the linear pair is 180°. The
sum of half of each must be 90°.
13. If a point is equally distant from the sides of
an angle, then it is on the bisector of an angle. This
is true for points in the same plane as the angle.
Mosaic answers: Square pattern constructions:
perpendiculars, equal segments, and midpoints;
The triangles are not identical, as the downward
ones have longer bases.
14. y � 110°
15. m�R � 46°
16. Angle A is the largest; m�A � 66°,
m�B � 64°, m�C � 50°.
17. STOP
18.G
T
IN
SA
OL
19.
20.
21. No, the triangles don’t look the same.
8 cm 8 cm40� 40�60�
60�
6.5 cmB C
A
3.5 cm
5.6 cm
130�A
B
C
22a. A web of lines fills most of the plane, except
a U-shaped region and a V-shaped region. (The
U-shaped region is actually bounded by a section
of a parabola and straight lines. If AB� were
extended to AB���, the U would be a complete
parabola.)
22b. a line segment parallel to AB� and half the
length (The segment is actually the midsegment
of �ABD.)
A
B
D
C Parabola
38 ANSWERS TO EXERCISES
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ANSWERS TO EXERCISES 39
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LESSON 3.5
1. 2.
3. Construct a segment with length z. Bisect the
segment to get length �2z
�. Bisect again to get a
segment with length �4z
�. Construct a square with
each side of length �4z
�.
4.
5. sample construction:
6. Draw a line and construct ML�� perpendicular to
it. Swing an arc from point M to point G so that
MG � RA. From point G, swing an arc to construct
RG�. Finish the parallelogram by swinging an arc of
length RA from R and swinging an arc of length GR
from M. There is only one possible parallelogram.
7. �1 � �S, �2 � �U
RL
M A
G
R
AP
T
x
x
x
x
x
A
A
z12
z14
z
8. �1 � �S and �2 � �U by of the Alternate
Interior Angles Conjecture
9. The ratios appear to be the same.
10. �1 � �3 and �2 � �4 by the
Corresponding Angles Conjecture
11. A parallelogram
12. Using the Converse of the Parallel Lines
Conjecture, the angle bisectors are parallel:
�DAB � �ABC, so AD��� � BC���.
13. Construct the perpendicular bisector of each
of the three segments connecting the fire stations.
Eliminate the rays beyond where the bisectors
intersect. A point within any region will be closest
to the fire station in that region.
14. 15.
16.
17. a � 72°, b � 108°, c � 108°, d � 108°, e � 72°,
f �108°, g � 108°, h � 72°, j � 90°, k � 18°, l � 90°,
m � 54°, n � 62°, p � 62°, q � 59°, r � 118°;
Explanations will vary. Sample explanation:
c is 108° because of the Corresponding Angles
Conjecture. Using the Vertical Angles Conjecture,
2m � 108°, so m � 54°. p � n because of the
Corresponding Angles Conjecture. Using the
Linear Pair Conjecture, n � 62°, so p � 62°.
Using the Linear Pair Conjecture, r � p � 180°.
Because p � 62°, r � 118°.
R E
W
RC = KE = 8 cmK C
MB
R O60�
Z
D
O
TR
I
A C
BD
USING YOUR ALGEBRA SKILLS 3
1. perpendicular
2. neither
3. perpendicular
4. parallel
5. possible answer: (2, 5) and (7, 11)
6. possible answer: (1, �5) and (�2, �12)
7. Ordinary; no two slopes are the same, so no
sides are parallel, although TE�� EM�� because their
slopes are opposite reciprocals.
8. Ordinary, for the same reason as in Exercise 7—
none of the sides are quite parallel.
9. trapezoid: KC� � RO�
10a. Slope HA�� � slope ND�� � �16
�;
slope HD�� � slope NA�� �6. Quadrilateral HAND
is a rectangle because opposite sides are parallel
and adjacent sides are perpendicular.
10b. Midpoint HN�� � midpoint AD�� � ��12
�, 3�. The
diagonals of a rectangle bisect each other.
11a. Yes, the diagonals are perpendicular.
Slope OE�� 1; slope VR�� �1.
11b. Midpoint VR�� midpoint OE�� (�2, 4).
The diagonals of OVER bisect each other.
11c. OVER appears to be a rhombus. Slope
OV�� slope RE�� ��15
� and slope OR�� slope
VE�� �5, so opposite sides are parallel. Also, all of
the sides appear to have the same length.
12a. Both slopes equal �12
�.
12b. The segments are not parallel because they
are coincident.
12c. distinct
13. (3, �6)
40 ANSWERS TO EXERCISES
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ANSWERS TO EXERCISES 41
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LESSON 3.6
1. Sample description: Construct one of the
segments, and mark arcs of the correct length from
the endpoints. Draw sides to where those arcs meet.
2.
Sample description: Construct �O. Mark off
distances OD and OT on the sides of the angle.
Connect D and T.
3.
Sample description: Construct IY�. Construct �I at
I and �Y at Y. Label the intersection of the rays
point G.
4. Yes, students’ constructions must be either
larger or smaller than the triangle in the book.
Sample description: Draw one side with a different
length than the lengths in the book. Duplicate an
I Y
Y
GY
I
I
O
D
T
TO
O D
O
M
A
M
S
S
A
M A
S
angle at each end of that segment congruent to one
of the angles in the book. Where they meet is the
third vertex of the triangle.
5.
Sample description: Construct �A and mark off
the distance AB. From B swing an arc of length BC
to intersect the other side of �A at two points.
Each gives a different triangle.
6.
Sample description: Mark the distance y, mark
back the distance x, and bisect the remaining
length of y � x. Using an arc of that length, mark
arcs on the ends of segment x. The point where
they intersect is the vertex angle of the triangle.
7.
Sample description: Draw an angle. Mark off equal
segments on the sides of the angle. Use a different
compass setting to draw intersecting arcs from the
ends of those segments.
8. Sample description: Draw an angle and mark
off unequal distances on the sides. At the endpoint
of the longer segment (not the angle vertex), swing
an arc with the length of the shorter segment. From
the endpoint of the shorter segment, swing an arc
the length of the longer segment. Connect the
endpoints of the segments to the intersection
points of the arcs to form a quadrilateral.
y
C
A
T
x
xy �x____
2
y �x____2
A
A
B
B
C
C
9.
Sample description: Draw a segment and draw an
angle at one end of the segment. Mark off a
distance equal to that segment on the other side of
the angle. Draw an angle at that point and mark off
the same distance. Connect that point to the other
end of the original segment.
10.
Sample description: Draw an angle and mark off
equal lengths on the two sides. Use that length to
determine another point that distance from the
points on the sides. Connect that point with the
two points on the side of the angle.
11. Answers will vary. The angle bisector lies
between the median and the altitude. The order of
the points is either M, R, S or S, R, M. One possible
conjecture: In a scalene obtuse triangle the angle
bisector is always between the median and the
altitude.
CA
Altitude
Median
Anglebisector
S MR
B
m�ABC = 111�
12. new coordinates: E�(4, �6), A�(7, 0), T�(1, 2)