48 ANSWERS TO EXERCISES Answers to Exercises Answers to Exercises angles P and D can be drawn at each endpoint using the protractor. 17. The third angles of the triangles also have the same measures; are equal in measure 18. You know from the Triangle Sum Conjecture that mA mB mC 180°, and mD mE mF 180°. By the transitive property, mA mB mC mD mE mF. You also know that mA mD, and mB mE. You can substitute for mD and mE in the longer equation to get mA mB mC mA mB mF. Subtracting equal terms from both sides, you are left with mC mF. 19. For any triangle, the sum of the angle measures is 180°, by the Triangle Sum Conjecture. Since the triangle is equiangular, each angle has the same measure, say x. So x x x 180°, and x 60°. 20. false 21. false 22. false 23. false 24. true 25. eight; 100 855540P D 7 cm Q CHAPTER 4 • CHAPTER CHAPTER 4 • CHAPTER LESSON 4.1 1. The angle measures change, but the sum remains 180°. 2. 73° 3. 60° 4. 110° 5. 24° 6. 3 360° 180° 900° 7. 3 180° 180° 360° 8. 69°; 47°; 116°; 93°; 86° 9. 30°; 50°; 82°; 28°; 32°; 78°; 118°; 50° 10. 11. 12. First construct E, using the method used in Exercise 10. 13. 14. From the Triangle Sum Conjecture mA mS mM 180°. Because M is a right angle, mM 90°. By substitution, mA mS 90° 180°. By subtraction, mA mS 90°. So two wrongs make a right! 15. Answers will vary. See the proof on page 202. To prove the Triangle Sum Conjecture, the Linear Pair Conjecture and the Alternate Interior Angles Conjecture must be accepted as true. 16. It is easier to draw PDQ if the Triangle Sum Conjecture is used to find that the measure of D is 85°. Then PD can be drawn to be 7 cm, and E R A L G Fold M R A E R A L G M A R 4
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48 ANSWERS TO EXERCISES
An
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Answers to Exercisesangles P and D can be drawn at each endpoint
using the protractor.
17. The third angles of the triangles also have the
same measures; are equal in measure
18. You know from the Triangle Sum Conjecture
that m�A � m�B � m�C � 180°, and m�D �m�E � m�F � 180°. By the transitive property,
m�A � m�B � m�C � m�D � m�E � m�F.
You also know that m�A � m�D, and m�B �m�E. You can substitute for m�D and m�E in the
longer equation to get m�A � m�B � m�C �m�A � m�B � m�F. Subtracting equal terms
from both sides, you are left with m�C � m�F.
19. For any triangle, the sum of the angle measures
is 180°, by the Triangle Sum Conjecture. Since the
triangle is equiangular, each angle has the same
measure, say x. So x � x � x � 180°, and x � 60°.
20. false
21. false
22. false
23. false
24. true
25. eight; 100
85�
55�
40�P D7 cm
Q
CHAPTER 4 • CHAPTER CHAPTER 4 • CHAPTER
LESSON 4.1
1. The angle measures change, but the sum
remains 180°.
2. 73°
3. 60°
4. 110°
5. 24°
6. 3 � 360° � 180° � 900°
7. 3 � 180° � 180° � 360°
8. 69°; 47°; 116°; 93°; 86°
9. 30°; 50°; 82°; 28°; 32°; 78°; 118°; 50°
10.
11.
12. First construct �E, using the method used in
Exercise 10.
13.
14. From the Triangle Sum Conjecture
m�A � m�S � m�M � 180°. Because �M is a
right angle, m�M � 90°. By substitution,
m�A � m�S � 90° � 180°. By subtraction,
m�A � m�S � 90°. So two wrongs make a right!
15. Answers will vary. See the proof on page 202.
To prove the Triangle Sum Conjecture, the Linear
Pair Conjecture and the Alternate Interior Angles
Conjecture must be accepted as true.
16. It is easier to draw �PDQ if the Triangle Sum
Conjecture is used to find that the measure of
�D is 85°. Then PD� can be drawn to be 7 cm, and
E
RA�L
�G
Fold
�M
�R
�A
E
RA
�L
�G
�M �A
�R
4
LESSON 4.2
1. 79°
2. 54°
3. 107.5°
4. 44°; 35 cm
5. 76°; 3.5 cm
6. 72°; 36°; 8.6 cm
7. 78°; 93 cm
8. 75 m; 81°
9. 160 in.; 126°
10. a � 124°, b � 56°, c � 56°, d � 38°, e � 38°,
f � 76°, g � 66°, h � 104°, k � 76°, n � 86°,
p � 38°; Possible explanation: The angles with
measures 66° and d form a triangle with the angle
with measure e and its adjacent angle. Because d,
e, and the adjacent angle are all congruent,
3d � 66° � 180°. Solve to get d � 38°. This is
also the measure of one of the base angles of the
isosceles triangle with vertex angle measure h.
Using the Isosceles Triangle Conjecture, the other
base angle measures d, so 2d � h � 180°, or
76° � h � 180°. Therefore, h � 104°.
11. a � 36°, b � 36°, c � 72°, d � 108°, e � 36°;
none
12a. Yes. Two sides are radii of a circle. Radii must
be congruent; therefore, each triangle must be
isosceles.
12b. 60°
13. NCA
14. IEC
15.
16.
17. possible answer:
18. perpendicular
19. parallel
20. parallel
21. neither
22. parallelogram
23. 40
24. New: (6, �3), (2, �5), (3, 0). Triangles are
congruent.
25. New: (3, �3), (�3, �1), (�1, �5). Triangles
are congruent.
0 8 16 24 32 40
Fold 2
Fold 1
Fold 4
Fold 3
105� 60�
45�
M N
G
K
H
P
ED
BA
F
C
ANSWERS TO EXERCISES 49
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50 ANSWERS TO EXERCISES
USING YOUR ALGEBRA SKILLS 4
1. false 2. true
3. not a solution 4. solution
5. not a solution 6. x � 7
7. y � �4 8. x � �8
9. x � 4.2 10. n � ��12
�
11. x � 2 12. t � 18
13. n � �25
� 14a. x � �49
�
14b. x � �49
�; The two methods produce identical
results. Multiplying by the lowest common
denominator (which is comprised of the factors of
both denominators) and then reducing common
factors (which clears the denominators on either
side) is the same as simply multiplying each numerator
by the opposite denominator (or cross multiplying).
Algebraically you could show that the two methods
are equivalent as follows:
�a�b
� � �dc
�
bd��ab
�� � bd��dc
���ab
bd
� � �b
dcd�
ad � bc
The method of “clearing fractions” results in the
method of “cross multiplying.”
15. You get an equation that is always false, so
there is no solution to the equation.
16. Camella is not correct. Because the equation
0 � 0 is always true, the truth of the equation does
not depend on the value of x. Therefore, x can be
any real number. Camella’s answer, x � 0, is only
one of infinitely many solutions.
17.
If x equals the measure of the vertex angle, then
the base angles each measure 2x. Applying the
Triangle Sum Conjecture results in the following
equation:
x � 2x � 2x � 180°
5x � 180°
x � 36°
The measure of the vertex angle is 36° and the
measure of each base angle is 72°.
2x
2x
x
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LESSON 4.3
1. yes
2. no
3. no
4. yes
5. a, b, c
6. c, b, a
7. b, a, c
8. a, c, b
9. a, b, c
10. v, z, y, w, x
11. 6 � length � 102
12. By the Triangle Inequality Conjecture, the
sum of 11 cm and 25 cm should be greater than
48 cm.
13. b � 55°, but 55° � 130° � 180°, which is
impossible by the Triangle Sum Conjecture.
5 6
12
4 5
9
14. 135°
15. 72°
16. 72°
17. a � b � c � 180° and x � c � 180°. Subtract c
from both sides of both equations to get x � 180 � c
and a � b � 180 � c. Substitute a � b for 180 � c
in the first equation to get x � a � b.
18. 45°
19. a � 52°, b � 38°, c � 110°, d � 35°
20. a � 90°, b � 68°, c � 112°, d � 112°, e � 68°,
f � 56°, g � 124°, h � 124°
21. By the Triangle Sum Conjecture, the third
angle must measure 36° in the small triangle, but it
measures 32° in the large triangle. These are the
same angle, so they can’t have different measures.
22. ABE
23. FNK
24. cannot be determined
ANSWERS TO EXERCISES 51
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52 ANSWERS TO EXERCISES
LESSON 4.4
1. Answers will vary. Possible answer: If three
sides of one triangle are congruent to three sides of
another triangle, then the triangles are congruent
(all corresponding angles are also congruent).
2. Answers will vary. Possible answer: The picture
statement means that if two sides of one triangle
are congruent to two sides of another triangle, and
the angles between those sides are also congruent,
then the two triangles are congruent.
If you know this: then you also know this:
3. Answers will vary. Possible answer:
4. SAS
5. SSS
6. cannot be determined
7. SSS
8. SAS
9. SSS (and the Converse of the Isosceles Triangle
Conjecture)
10. yes, �ABC � �ADE by SAS
11. Possible answer: Boards nailed diagonally in
the corners of the gate form triangles in those
corners. Triangles are rigid, so the triangles in the
gate’s corners will increase the stability of those
corners and keep them from changing shape.
12. FLE by SSS
13. Cannot be determined. SSA is not a congruence
conjecture.
14. AIN by SSS or SAS
15. Cannot be determined. Parts do not correspond.
16. SAO by SAS
17. Cannot be determined. Parts do not correspond.
18. RAY by SAS
19. The midpoint of SD� and PR� is (0, 0).
Therefore, �DRO � �SPO by SAS.
20. Because the LEV is marking out two triangles
that are congruent by SAS, measuring the length
of the segment leading to the finish will also
approximate the distance across the crater.
21. 22.
23. a � 37°, b � 143°, c � 37°, d � 58°,
e � 37°, f � 53°, g � 48°, h � 84°, k � 96°,
m � 26°, p � 69°, r � 111°, s � 69°; Possible
explanation: The angle with measure h is the vertex
angle of an isosceles triangle with base angles
measuring 48°, so h � 2(48) � 180°, and h � 84°.
The angle with measure s and the angle with
measure p are corresponding angles formed by
parallel lines, so s � p � 69°.
24. 3 cm � third side � 19 cm
25. See table below.
26a. y � 6 b. y � �133� c. y � ��
34
�x � 2
27. (�5, �3)
An
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Side length 1 2 3 4 5 … n … 20
Elbows 4 4 4 4 4 4 4
T’s 0 4 8 12 16 76
Crosses 0 1 4 9 16 361
25. (Lesson 4.4)
4n � 4 (n � 1)2
LESSON 4.5
1. If two angles and the included side of one triangle
are congruent to the corresponding side and angles
of another triangle, then the triangles are congruent.
2. If two angles and a non-included side of one
triangle are congruent to the corresponding side
and angles of another triangle, then the triangles