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CK-12 Geometry - Basic,Answer Key
CK-12 Foundation
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Printed: March 31, 2014
AUTHORCK-12 Foundation
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Contents www.ck12.org
Contents
1 Basics Geometry, Answer Key 11.1 Basic Geometry, Points, Lines, and Planes, Review Answers . . . . . . . . . . . . . . . . . . . 21.2 Basic Geometry, Segments and Distance, Review Answers . . . . . . . . . . . . . . . . . . . . 41.3 Basic Geometry, Angles and Measurement, Review Answers . . . . . . . . . . . . . . . . . . . 51.4 Basic Geometry, Midpoints and Bisectors, Review Answers . . . . . . . . . . . . . . . . . . . . 71.5 Basic Geometry, Angle Pairs, Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.6 Basic Geometry, Classifying Polygons, Review Answers . . . . . . . . . . . . . . . . . . . . . 91.7 Chapter Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2 Reasoning and Proof, Answer Key 112.1 Basic Geometry, Inductive Reasoning, Review Answers . . . . . . . . . . . . . . . . . . . . . . 122.2 Basic Geometry, Conditional Statements, Review Answers . . . . . . . . . . . . . . . . . . . . 132.3 Basic Geometry, Deductive Reasoning, Review Answers . . . . . . . . . . . . . . . . . . . . . 152.4 Basic Geometry, Algebraic and Congruence Properties, Review Answers . . . . . . . . . . . . . 162.5 Basic Geometry, Proofs about Angle Pairs and Segments, Review Answers . . . . . . . . . . . . 182.6 Chapter Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 Parallel and Perpendicular Lines, Answer Key 223.1 Basic Geometry, Lines and Angles, Review Answers . . . . . . . . . . . . . . . . . . . . . . . 233.2 Basic Geometry, Properties of Parallel Lines, Review Answers . . . . . . . . . . . . . . . . . . 243.3 Basic Geometry, Proving Lines Parallel, Review Answers . . . . . . . . . . . . . . . . . . . . . 263.4 Basic Geometry, Properties of Perpendicular Lines, Review Answers . . . . . . . . . . . . . . . 283.5 Basic Geometry, Parallel and Perpendicular Lines in the Coordinate Plane, Review Answers . . . 293.6 Basic Geometry, The Distance Formula, Review Answers . . . . . . . . . . . . . . . . . . . . . 333.7 Chapter Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4 Triangles and Congruence, Answer Key 354.1 Basic Geometry, Triangle Sums, Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Basic Geometry, Congruent Figures, Review Answers . . . . . . . . . . . . . . . . . . . . . . . 384.3 Basic Geometry, Triangle Congruence using SSS and SAS, Review Answers . . . . . . . . . . . 394.4 Basic Geometry, Triangle Congruence using ASA, AAS, and HL, Review Answers . . . . . . . 414.5 Basic Geometry, Isosceles and Equilateral Triangles, Review Answers . . . . . . . . . . . . . . 434.6 Chapter Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5 Relationships with Triangles, Answer Key 475.1 Basic Geometry, Midsegments, Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 Basic Geometry, Perpendicular Bisectors and Angle Bisectors in Triangles, Review Answers . . 505.3 Basic Geometry, Medians and Altitudes in Triangles, Review Answers . . . . . . . . . . . . . . 525.4 Basic Geometry, Inequalities in Triangles, Review Answers . . . . . . . . . . . . . . . . . . . . 545.5 Basic Geometry, Extension: Indirect Proof, Review Answers . . . . . . . . . . . . . . . . . . . 565.6 Chapter Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
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www.ck12.org Contents
6 Polygons and Quadrilaterals, Answer Key 586.1 Basic Geometry, Angles in Polygons, Review Answers . . . . . . . . . . . . . . . . . . . . . . 596.2 Basic Geometry, Properties of Parallelograms, Review Answers . . . . . . . . . . . . . . . . . . 606.3 Basic Geometry, Proving Quadrilaterals are Parallelograms, Review Answers . . . . . . . . . . 626.4 Basic Geometry, Rectangles, Rhombuses and Squares, Review Answers . . . . . . . . . . . . . 646.5 Basic Geometry, Trapezoids and Kites, Review Answers . . . . . . . . . . . . . . . . . . . . . 666.6 Chapter Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
7 Similarity, Answer Key 707.1 Basic Geometry, Ratios and Proportions, Review Answers . . . . . . . . . . . . . . . . . . . . . 717.2 Basic Geometry, Similar Polygons, Review Answers . . . . . . . . . . . . . . . . . . . . . . . . 727.3 Basic Geometry, Similarity by AA, Review Answers . . . . . . . . . . . . . . . . . . . . . . . 737.4 Basic Geometry, Similarity by SSS and SAS, Review Answers . . . . . . . . . . . . . . . . . . 747.5 Basic Geometry, Proportionality Relationships, Review Answers . . . . . . . . . . . . . . . . . 757.6 Basic Geometry, Similarity Transformations, Review Answers . . . . . . . . . . . . . . . . . . 767.7 Basic Geometry, Extension: Self-Similarity, Review Answers . . . . . . . . . . . . . . . . . . . 787.8 Chapter Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
8 Right Triangle Trigonometry, Answer Key 808.1 Basic Geometry, The Pythagorean Theorem, Review Answers . . . . . . . . . . . . . . . . . . . 818.2 Basic Geometry, Converse of the Pythagorean Theorem, Review Answers . . . . . . . . . . . . 828.3 Basic Geometry, Using Similar Right Triangles, Review Answers . . . . . . . . . . . . . . . . . 838.4 Basic Geometry, Special Right Triangles, Review Answers . . . . . . . . . . . . . . . . . . . . 848.5 Basic Geometry, Tangent, Sine and Cosine, Review Answers . . . . . . . . . . . . . . . . . . . 858.6 Basic Geometry, Inverse Trigonometric Ratios, Review Answers . . . . . . . . . . . . . . . . . 868.7 Chapter Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
9 Circles, Answer Key 889.1 Basic Geometry, Parts of Circles and Tangent Lines, Review Answers . . . . . . . . . . . . . . 899.2 Basic Geometry, Properties of Arcs, Review Answers . . . . . . . . . . . . . . . . . . . . . . . 929.3 Basic Geometry, Properties of Chords, Review Answers . . . . . . . . . . . . . . . . . . . . . . 939.4 Basic Geometry, Inscribed Angles, Review Answers . . . . . . . . . . . . . . . . . . . . . . . . 949.5 Basic Geometry, Angles of Chords, Secants, and Tangents, Review Answers . . . . . . . . . . . 969.6 Basic Geometry, Segments of Chords, Secants, and Tangents, Review Answers . . . . . . . . . . 989.7 Basic Geometry, Extension: Writing and Graphing the Equations of Circles, Review Answers . . 1009.8 Chapter Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
10 Perimeter and Area, Answer Key 10210.1 Basic Geometry, Triangles and Parallelograms, Review Answers . . . . . . . . . . . . . . . . . 10310.2 Basic Geometry, Trapezoids, Rhombi, and Kites, Review Answers . . . . . . . . . . . . . . . . 10510.3 Basic Geometry, Areas of Similar Polygons, Review Answers . . . . . . . . . . . . . . . . . . . 10610.4 Basic Geometry, Circumference and Arc Length, Review Answers . . . . . . . . . . . . . . . . 10710.5 Basic Geometry, Areas of Circles and Sectors, Review Answers . . . . . . . . . . . . . . . . . . 10810.6 Chapter Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
11 Surface Area and Volume, Answer Key 11011.1 Basic Geometry, Exploring Solids, Review Answers . . . . . . . . . . . . . . . . . . . . . . . . 11111.2 Basic Geometry, Surface Area of Prisms and Cylinders, Review Answers . . . . . . . . . . . . . 11311.3 Basic Geometry, Surface Area of Pyramids and Cones, Review Answers . . . . . . . . . . . . . 11411.4 Basic Geometry, Volume of Prisms and Cylinders, Review Answers . . . . . . . . . . . . . . . 11511.5 Basic Geometry, Volume of Pyramids and Cones, Review Answers . . . . . . . . . . . . . . . . 11611.6 Basic Geometry, Surface Area and Volume of Spheres, Review Answers . . . . . . . . . . . . . 117
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Contents www.ck12.org
11.7 Basic Geometry, Extension: Exploring Similar Solids, Review Answers . . . . . . . . . . . . . 11811.8 Chapter Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
12 Rigid Transformations, Answer Key 12012.1 Basic Geometry, Exploring Symmetry, Review Answers . . . . . . . . . . . . . . . . . . . . . . 12112.2 Basic Geometry, Translations, Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . 12312.3 Basic Geometry, Reflections, Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . 12412.4 Basic Geometry, Rotations, Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12612.5 Basic Geometry, Composition of Transformations, Review Answers . . . . . . . . . . . . . . . 13112.6 Basic Geometry, Extension: Tessellating Polygons, Review Answers . . . . . . . . . . . . . . . 13212.7 Chapter Review Answers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
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www.ck12.org Chapter 1. Basics Geometry, Answer Key
CHAPTER 1 Basics Geometry, AnswerKey
Chapter Outline1.1 BASIC GEOMETRY, POINTS, LINES, AND PLANES, REVIEW ANSWERS
1.2 BASIC GEOMETRY, SEGMENTS AND DISTANCE, REVIEW ANSWERS
1.3 BASIC GEOMETRY, ANGLES AND MEASUREMENT, REVIEW ANSWERS
1.4 BASIC GEOMETRY, MIDPOINTS AND BISECTORS, REVIEW ANSWERS
1.5 BASIC GEOMETRY, ANGLE PAIRS, REVIEW ANSWERS
1.6 BASIC GEOMETRY, CLASSIFYING POLYGONS, REVIEW ANSWERS
1.7 CHAPTER REVIEW ANSWERS
1
1.1. Basic Geometry, Points, Lines, and Planes, Review Answers www.ck12.org
1.1 Basic Geometry, Points, Lines, and Planes,Review Answers
For 1-5, answers will vary. One possible answer for each is included.
1.
2.
3.
4.
5.6.←→WX ,
←→YW , line m,
←→XY and
←→WY .
7.−→T S,−→T R, and
←−RT .
8. Plane V or plane RST .9. A soccer field is like a plane since it is a flat two-dimensional surface.
10. Sun rays, laser beam, the hands on a clock are a few examples.11. A line and a plane intersect at a point.
2
www.ck12.org Chapter 1. Basics Geometry, Answer Key
12. A postulate is assumed true and a theorem must be proven true.13.−→PQ intersects
←→RS at point Q.
14.−→AC and AB are coplanar and point D is not.
15. Points E and H are coplanar, but their rays,−→EF and
−→GH are non-coplanar.
16.−→IJ ,−→IK,−→IL, and
−→IM with common endpoint I and J,K,L and M are non-collinear.
17. True18. False19. False20. False21. False22. True23. False24. False25. True
3
1.2. Basic Geometry, Segments and Distance, Review Answers www.ck12.org
1.2 Basic Geometry, Segments and Distance,Review Answers
1. 1.625 in2. 2.875 in3. 0.875 in4. 3.875 in5. 3.7 cm6. 8.2 cm7. 1.5 cm8. 4.8 cm9. 2.75 in
10. 4.9 cm11. 4.625 in12. 8.7 cm
13.14. O would be halfway between L and T , so that LO = OT = 8 cm
1.2. TA+AQ = T Q3. T Q = 15 in
1.2. HM+MA = HA3. AM = 11 cm
1.2. MI + IT = MT3. MI = 19 cm
18. BC = 8 cm, BD = 25 cm, and CD = 17 cm
19. FE = 8 in, HG = 13 in, and FG = 17 in
20. x = 3, HJ = 21, JK = 12, HK = 33
21. x = 11, HJ = 52, JK = 79, HK = 131
22. 1323. 524. 925. 5
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www.ck12.org Chapter 1. Basics Geometry, Answer Key
1.3 Basic Geometry, Angles and Measurement,Review Answers
1. False, two angles could be 5◦ and 30◦.2. False, it is a straight angle.3. True4. True5. False, you use a compass.6. False, B is the vertex.7. True8. True9. True
10. False, it is equal to the sum of the smaller angles within it.11. Acute
12. Obtuse
13. Obtuse
14. Acute
15. Obtuse
16. Acute
5
1.3. Basic Geometry, Angles and Measurement, Review Answers www.ck12.org
17 & 18: Drawings should look exactly like 12 and 16, but with the appropriate arc marks.
19. 40◦
20. 122◦
21. 18◦
22. 87◦
23. m6 EAC = m 6 ABC, AE ∼=CD, ED∼=CB, ED⊥ DB
24.25. m6 QOP = 100◦
26. m6 QOT = 130◦
27. m6 ROQ = 30◦
28. m6 SOP = 70◦
29. x = 10◦
30. x = 22◦
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www.ck12.org Chapter 1. Basics Geometry, Answer Key
1.4 Basic Geometry, Midpoints and Bisectors,Review Answers
1.2. 12 in3. 5 in4. 5 in5. 13 in6. 90◦
7. 10 in8. 24 in9. 90◦
10. 8 triangles11. PS12. QT ,V S13. 90◦
14. 45◦
15. bisector16. bisector17. 45◦
18. PU is a segment bisector of QT19. x = 20◦
20. x = 14◦
21. x = 1222. d = 13◦
23. 55◦ each24. Each half should be 3.5 cm each.25. (3, 5)26. (1.5, 6)27. (5, 5)28. (-4.5, 2)29. (7, 10)30. (6, 9)
7
1.5. Basic Geometry, Angle Pairs, Review Answers www.ck12.org
1.5 Basic Geometry, Angle Pairs, Review An-swers
1. 45◦
2. 8◦
3. 81◦
4. (90− z)◦
1. 135◦
2. 62◦
3. 148◦
4. (180− x)◦
3. 6 JNI and 6 MNL (or 6 INM and 6 JNL)4. 6 INM and 6 MNL (or 6 INK and 6 KNL)5. 6 INJ and 6 JNK6. 6 INM and 6 MNL (or 6 INK and 6 KNL)
1. 180◦
2. 90◦
1. 117◦
2. 27◦
3. 63◦
4. 117◦
9. True10. False11. False12. True13. True14. False15. False16. True17. x = 7◦
18. x = 34◦
19. y = 13◦
20. x = 17◦
21. x = 15◦
22. y = 9◦
23. x = 10.5◦
24. x = 4◦
25. y = 3◦
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www.ck12.org Chapter 1. Basics Geometry, Answer Key
1.6 Basic Geometry, Classifying Polygons,Review Answers
1. Acute scalene triangle2. Equilateral and equiangular triangle3. Right isosceles triangle.4. Obtuse scalene triangle5. Acute isosceles triangle6. Obtuse isosceles triangle7. No, there would be more than 180◦ in the triangle, which is impossible.8. No, same reason as #7.9. Concave pentagon
10. Convex octagon11. Convex 17-gon12. Convex decagon13. Concave quadrilateral14. Concave hexagon15. A is not a polygon because the two sides do not meet at a vertex; B is not a polygon because one side is curved;
C is not a polygon because it is not closed.16. 2 diagonals17. 5 diagonals18. A dodecagon has twelve sides, so you can draw nine diagonals from one vertex.19. An octagon has 20 diagonals, a nonagon has 27 diagonals, a decagon has 35 diagonals, an undecagon has 44
diagonals, and a dodecagon has 54 diagonals20. True21. True22. False23. False24. True25. False26. True
9
1.7. Chapter Review Answers www.ck12.org
1.7 Chapter Review Answers
1. E2. B3. L4. A5. H6. M7. F8. P9. J
10. G11. I12. K13. D14. C15. N
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www.ck12.org Chapter 2. Reasoning and Proof, Answer Key
CHAPTER 2 Reasoning and Proof,Answer Key
Chapter Outline2.1 BASIC GEOMETRY, INDUCTIVE REASONING, REVIEW ANSWERS
2.2 BASIC GEOMETRY, CONDITIONAL STATEMENTS, REVIEW ANSWERS
2.3 BASIC GEOMETRY, DEDUCTIVE REASONING, REVIEW ANSWERS
2.4 BASIC GEOMETRY, ALGEBRAIC AND CONGRUENCE PROPERTIES, REVIEW AN-SWERS
2.5 BASIC GEOMETRY, PROOFS ABOUT ANGLE PAIRS AND SEGMENTS, REVIEW
ANSWERS
2.6 CHAPTER REVIEW ANSWERS
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2.1. Basic Geometry, Inductive Reasoning, Review Answers www.ck12.org
2.1 Basic Geometry, Inductive Reasoning, Re-view Answers
1. 9, 212. 20, 1103. 13, 37
4. 1.2. there are two more points in each star than its figure number.
5. 10
1.2. 48
6. 20, 23, 267. -19, -24, -298. 64, 128, 2569. 12, 1, -10
10. -12, 0, -16 OR -16, 0, -3211. 6
7 ,78 ,
89
12. 1223 ,
1427 ,
1631
13. 21, -25, 2914. 38, 57; the amount that is added is increasing by two with each term.15. 48, 67; the amount that is added is increasing by two with each term.16. 216, 343; the term number cubed, n3.17. 8, 13; add the previous two terms together to get the current term.18. n = 119. 2 is a prime number that is not odd20. 21, 51, 81, . . .21. 4
3 or any improper fraction.22. A triangle is a counterexample.
23.Any two angles in this rectangle are congruent, supplementary and NOT a linear pair.
24. No, anyone can buy a used car. It is a coincidence.25. Answers will vary.
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www.ck12.org Chapter 2. Reasoning and Proof, Answer Key
2.2 Basic Geometry, Conditional Statements,Review Answers
1. Hypothesis: 5 divides evenly into x. Conclusion: x ends in 0 or 5.2. Hypothesis: A triangle has three congruent sides. Conclusion: It is an equilateral triangle.3. Hypothesis: Three points lie in the same plane. Conclusion: The three points are coplanar.4. Hypothesis: x = 3. Conclusion: x2 = 9.5. Hypothesis: You take yoga. Conclusion: You are relaxed.6. Hypothesis: You are a baseball player. Conclusion: You wear a hat.7. Converse: If x ends in 0 or 5, then 5 divides evenly into x. True. Inverse: If 5 does not divide evenly into x,
then x does not end in 0 or 5. True. Contrapositive: If x does not end in 0 or 5, then 5 does not divide evenlyinto it. True
8. Converse: If you are relaxed, then you take yoga. False. You could have gone to a spa. Inverse: If you do nottake yoga, then you are not relaxed. False. You can be relaxed without having had taking yoga. You couldhave gone to a spa. Contrapositive: If you are not relaxed, then you did not take yoga. True
9. Converse: If you wear a hat, then you are a baseball player. False. You could be a cowboy or anyone else whowears a hat. Inverse: If you are not a baseball player, then you do not wear a hat. False. Again, you could bea cowboy. Contrapositive: If you do not wear a hat, then you are not a baseball player. True
10. If a triangle is equilateral, then it has three congruent sides. True. A triangle has three congruent sides if andonly if it is equilateral.
11. If three points are coplanar, then they lie in the same plane. True. Three points lie in the same plane if andonly if they are coplanar.
12. If x2 = 9, then x = 3. False. x could also be -3.13. Not necessary, A,B, and C need to be collinear in order for B to be a midpoint.14. If B is the midpoint of AC, then AB = 5 and BC = 5. This could be true, but we don’t know AC. AB = BC, but
we don’t know the specific values of AB and BC.15. If AB 6= 5 and BC 6= 5, then B is not the midpoint of AC. Again, this could be true, but we don’t know AC.
Also, A,B and C might not be collinear.16. If AB 6= 5 and BC 6= 5, then B is not the midpoint of AC. It is the same as #15.17. If an angle is less than 90◦, then it is acute. True. An angle is acute if and only if it is less than 90◦.18. If you are sun burnt, then you are at the beach. False.19. If x+3 > 7, then x > 4. True. x+3 > 7 if and only if x > 4.20. If a U.S. citizen can vote, then he or she is 18 or more years old. If a U.S. citizen is 18 or more years old, then
he or she can vote.21. If a whole number is prime, then its factors are 1 and itself. If a whole number’s factors are only 1 and itself,
then it is prime.22. If 2x = 18, then x = 9. If x = 9, then 2x = 18.
1. Yes.2. No, he could be 20.3. No, he could be 20.4. Yes.
1. Yes2. No, it could also end in 0.3. No, it could end in 0.4. Yes.
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2.2. Basic Geometry, Conditional Statements, Review Answers www.ck12.org
1. Yes.2. No, x could equal -4.3. No, again x could equal -4.4. Yes.
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www.ck12.org Chapter 2. Reasoning and Proof, Answer Key
2.3 Basic Geometry, Deductive Reasoning,Review Answers
1. I am a smart person. Law of Detachment2. No conclusion3. 4ABC is equilateral. Law of Detachment4. If North wins, then East loses. Law of Syllogism.5. If z > 5, then y > 7. Law of Syllogism.6. I am not cold. Law of Contrapositive.7. No conclusion, Inverse Error.8. If a shape is a circle, then we don’t need to study it. Law of Syllogism.9. You don’t text while driving. Law of Contrapositive.
10. It is sunny outside. Law of Detachment.11. You are not wearing sunglasses. Law of Contrapositive.12. My mom didn’t ask me to. Law of Contrapositive.13. This is a logical argument, but it doesn’t make sense because we know that circles exist.
p→ q
q→ r
r→ s
s→ t
∴ p→ t
p→ q
p
∴ q
p→ q
∼ q
∴∼ p
16. Inductive; a pattern of weather was observed.17. Deductive; Beth used a fact to determine what her sister would eat.18. Deductive; Jeff used a fact about Nolan Ryan.19. Either reasoning. Inductive; surfers observed patterns of weather and waves to determine when the best time
to surf is. Deductive; surfers could take the given statement as a fact and use that to determine when the besttime to surf is.
20. Inductive; observed a pattern.21. True22. False23. True24. True25. False
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2.4. Basic Geometry, Algebraic and Congruence Properties, Review Answers www.ck12.org
2.4 Basic Geometry, Algebraic and Congru-ence Properties, Review Answers
1.
3x+11 =−16
3x =−27 Subtraction PoE
x =−9 Division PoE
2.
7x−3 = 3x−35
4x−3 =−35 Subtraction PoE
4x =−32 Addition PoE
x =−8 Division PoE
3.
23
g+1 = 19
23
g = 18 Subtraction PoE
g = 27 Multiplication PoE
4.
12
MN = 5
MN = 10 Multiplication PoE
5.
5m6 ABC = 540◦
m6 ABC = 108◦ Division PoE
6.
10b−2(b+3) = 5b
10b−2b+6 = 5b Distributive Property
8b+6 = 5b Combine like terms
6 =−3b Subtraction PoE
−2 = b Division PoE
b =−2 Symmetric PoE
7.
14
y+56=
13
3y+10 = 4 Multiplication PoE (multiplied everything by 12)
3y =−6 Subtraction PoE
y =−2 Division PoE
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www.ck12.org Chapter 2. Reasoning and Proof, Answer Key
8.
14
AB+13
AB = 12+12
AB
3AB+4AB = 144+6AB Multiplication PoE (multiplied everything by 12)
7AB = 144+6AB Combine like terms
AB = 144 Subtraction PoE
9. 3 = x10. 12x−3211. x = 1212. y+ z = x+ y13. CD = 514. y−7 = z+415. Yes, they are collinear. 16+7 = 2316. No, they are not collinear, 9+9 6= 16. I cannot be the midpoint.17. 6 NOP must be an obtuse angle because it is supplementary with 56◦, meaning that m6 NOP is 180◦−56◦ =
124◦. 90◦ < 124◦ < 180◦, so by definition 6 NOP is an obtuse angle.18. ∼= 6 s have = measures; m6 ABC+m6 GHI = m6 DEF +m6 GHI; Substitution19. M is the midpoint of AN, N is the midpoint MB;AM = MN,MN = NB; Transitive20. 6 BFE or 6 BFG21.←→EF⊥←→BF
22. Yes, EG = FH because EF = GH and EF + FG = EG and FG + GH = FH by the Segment AdditionPostulate. FG = FG by the Reflexive Property and with substitution EF +FG = EG and FG+EF = FH.Therefore, EG = FH by the Transitive Property.
23. Not enough information.24. m6 EBF,m6 HCG25. BC,BC26. No27. No28. Yes29. Yes30. No31. No32. 633. 334. 9
17
2.5. Basic Geometry, Proofs about Angle Pairs and Segments, Review Answers www.ck12.org
2.5 Basic Geometry, Proofs about Angle Pairsand Segments, Review Answers
TABLE 2.1:
Statement Reason1. AC⊥BD, 6 1∼= 6 4 Given2. m6 1 = m6 4 ∼= angles have = measures3. 6 ACB and 6 ACD are right angles ⊥ lines create right angles4. m6 ACB = 90◦
m6 ACD = 90◦Definition of right angles
5. m 6 1+m 6 2 = m6 ACBm6 3+m6 4 = m 6 ACD
Angle Addition Postulate
6. m6 1+m 6 2 = 90◦
m6 3+m6 4 = 90◦Substitution
7. m6 1+m 6 2 = m6 3+m6 4 Substitution8. m6 1+m 6 2 = m6 3+m6 1 Substitution9. m6 2 = m6 3 Subtraction PoE10. 6 2∼= 6 3 ∼= angles have = measures
TABLE 2.2:
Statement Reason1. 6 MLN ∼= 6 OLP Given2. m6 MLN = m 6 OLP ∼= angles have = measures3. m6 MLO = m6 MLN +m6 NLOm6 NLP = m6 NLO+m6 OLP
Angle Addition Postulate
4. m6 NLP = m 6 NLO+m6 MLN Substitution5. m6 MLO = m6 NLP Substitution6. 6 NLP∼= 6 MLO ∼= angles have = measures
TABLE 2.3:
Statement Reason1. AE⊥EC,BE⊥ED Given2. 6 BED is a right angle6 AEC is a right angle
⊥ lines create right angles
3. m6 BED = 90◦
m6 AEC = 90◦Definition of a right angle
4. m6 BED = m6 2+m6 3m6 AEC = m 6 1+m6 3
Angle Addition Postulate
5. 90◦ = m6 2+m 6 390◦ = m 6 1+m6 3
Substitution
6. m6 2+m 6 3 = m6 1+m6 3 Substitution7. m6 2 = m6 1 Subtraction PoE8. 6 2∼= 6 1 ∼= angles have = measures
18
www.ck12.org Chapter 2. Reasoning and Proof, Answer Key
TABLE 2.4:
Statement Reason1. 6 L is supplementary to 6 M6 P is supplementary to 6 O6 L∼= 6 O
Given
2. m6 L = m6 O ∼= angles have = measures3. m6 L+m6 M = 180◦
m6 P+m6 O = 180◦Definition of supplementary angles
4. m6 L+m6 M = m 6 P+m 6 O Substitution5. m6 L+m6 M = m 6 P+m 6 L Substitution6. m6 M = m6 P Subtraction PoE7. 6 M ∼= 6 P ∼= angles have = measures
TABLE 2.5:
Statement Reason1. 6 1∼= 6 4 Given2. m6 1 = m6 4 ∼= angles have = measures3. 6 1 and 6 2 are a linear pair6 3 and 6 4 are a linear pair
Given (by looking at the picture) couldalso be Definition of a Linear Pair
4. 6 1 and 6 2 are supplementary6 3 and 6 4 are supplementary
Linear Pair Postulate
5. m6 1+m 6 2 = 180◦
m6 3+m6 4 = 180◦Definition of supplementary angles
6. m6 1+m 6 2 = m6 3+m6 4 Substitution7. m6 1+m 6 2 = m6 3+m6 1 Substitution8. m6 2 = m6 3 Subtraction PoE9. 6 2∼= 6 3 ∼= angles have = measures
TABLE 2.6:
Statement Reason1. 6 C and 6 F are right angles Given2. m6 C = 90◦,m6 F = 90◦ Definition of a right angle3. 90◦+90◦ = 180◦ Addition of real numbers4. m 6 C+m6 F = 180◦ Substitution
TABLE 2.7:
Statement Reason1. l⊥m Given2. 6 1 and 6 2 are right angles ⊥ lines create right angles.3. 6 1∼= 6 2 Right Angles Theorem
TABLE 2.8:
Statement Reason1. m6 1 = 90◦ Given
19
2.5. Basic Geometry, Proofs about Angle Pairs and Segments, Review Answers www.ck12.org
TABLE 2.8: (continued)
Statement Reason2. 6 1 and 6 2 are a linear pair Definition of a linear pair3. 6 1 and 6 2 are supplementary Linear Pair Postulate4. m6 1+m 6 2 = 180◦ Definition of supplementary angles5. 90◦+m6 2 = 180◦ Substitution6. m6 2 = 90◦ Subtraction PoE
TABLE 2.9:
Statement Reason1. l⊥m Given2. 6 1 and 6 2 make a right angle ⊥ lines create right angles3. m6 1+m 6 2 = 90◦ Definition of a right angle OR
Substitution PoE4. 6 1 and 6 2 are complementary Definition of complementary angles
TABLE 2.10:
Statement Reason1. l⊥m, 6 2∼= 6 6 Given2. m6 2 = m6 6 ∼= angles have = measures3. 6 5∼= 6 2 Vertical Angles Theorem4. m6 5 = m6 2 ∼= angles have = measures5. m6 5 = m6 6 Transitive
11. 6 AHM, 6 PHE12. AM ∼= MG,CP∼= PE,AH ∼= HE,MH ∼= HP,GH ∼= HC13. 6 AMH, 6 HMG and 6 CPH, 6 HPE14. 6 AHC15. 6 MAH, 6 HAC and 6 MGH, 6 HGE16. GC17. AE,GC18. 6 AHM, 6 MHG19. 6 AGH ∼= 6 HE20. 90◦
21. 90◦
22. 26◦
23. 154◦
24. 26◦
25. 64◦
20
www.ck12.org Chapter 2. Reasoning and Proof, Answer Key
2.6 Chapter Review Answers
1. D2. F3. H4. B5. I6. C7. G8. A9. J
10. E
21
www.ck12.org
CHAPTER 3 Parallel and PerpendicularLines, Answer Key
Chapter Outline3.1 BASIC GEOMETRY, LINES AND ANGLES, REVIEW ANSWERS
3.2 BASIC GEOMETRY, PROPERTIES OF PARALLEL LINES, REVIEW ANSWERS
3.3 BASIC GEOMETRY, PROVING LINES PARALLEL, REVIEW ANSWERS
3.4 BASIC GEOMETRY, PROPERTIES OF PERPENDICULAR LINES, REVIEW AN-SWERS
3.5 BASIC GEOMETRY, PARALLEL AND PERPENDICULAR LINES IN THE COORDI-NATE PLANE, REVIEW ANSWERS
3.6 BASIC GEOMETRY, THE DISTANCE FORMULA, REVIEW ANSWERS
3.7 CHAPTER REVIEW ANSWERS
22
www.ck12.org Chapter 3. Parallel and Perpendicular Lines, Answer Key
3.1 Basic Geometry, Lines and Angles, ReviewAnswers
1. Railroad Tracks2. Adjacent Sides of a Picture Frame3. Northbound Freeway and an Eastbound Overpass4. The only difference between the Parallel Line Postulate and the Perpendicular Line Postulate are the words
“parallel” and “perpendicular.” They are similar because both postulates state that there is only one line that isparallel (or perpendicular) to a given line, through a point.
5. AB and EZ,XY and BW , among others6. AB||VW , among others7. BC ⊥ BW , among others8. one, AV9. one, CD
10. 6 611. 6 312. 6 213. 6 114. 6 815. 6 816. 6 517. m6 3 = 55◦ (vertical angles), m 6 1 = 125◦ (linear pair), m6 4 = 125◦ (linear pair)18. m6 8 = 123◦ (vertical angles), m6 6 = 57◦ (linear pair), m6 7 = 57◦ (linear pair)19. No, we do not know anything about line m.20. No, even though they look parallel, we cannot assume it.
21.22. -123. -224. 325. −1
326. y =−x−127. y =−2x+128. y = 3x−1329. Yes, the lines are parallel because they have the same slope.30. No, the lines are not perpendicular because they have the same slope.
23
3.2. Basic Geometry, Properties of Parallel Lines, Review Answers www.ck12.org
3.2 Basic Geometry, Properties of ParallelLines, Review Answers
1. Supplementary2. Congruent3. Congruent4. Supplementary5. Congruent6. Supplementary7. Supplementary8. Alternate Exterior9. Alternate Interior
10. None11. Same Side Interior12. Vertical Angles13. Corresponding Angles14. Alternate Exterior15. None16. 6 1, 6 3, 6 6, 6 9, 6 11, 6 14, and 6 1617. x = 70◦, y = 90◦
18. x = 15◦, y = 40◦
19. x = 9◦, y = 22◦
20. x = 21◦, y = 17◦
21. x = 25◦
22. y = 18◦
23. x = 20◦
24. x = 21◦
25. y = 12◦
TABLE 3.1:
Statement Reason1. l||m Given2. 6 1∼= 6 5 Corresponding Angles Postulate3. m6 1 = m6 5 ∼= angles have = measures4. 6 1 and 6 3 are supplementary Linear Pair Postulate5. m6 1+m 6 3 = 180◦ Definition of Supplementary Angles6. m 6 3+m 6 5 = 180◦ Substitution PoE7. 6 3 and 6 5 are supplementary Definition of Supplementary Angles
TABLE 3.2:
Statement Reason1. l||m Given2. 6 1∼= 6 5 Corresponding Angles Postulate3. 6 5∼= 6 8 Vertical Angles Theorem4. 6 1∼= 6 8 Transitive PoC
24
www.ck12.org Chapter 3. Parallel and Perpendicular Lines, Answer Key
TABLE 3.3:
Statement Reason1. l||m,s||t Given2. 6 2∼= 6 13 Alternate Exterior Angles Theorem3. 6 13∼= 6 15 Corresponding Angles Postulate4. 6 2∼= 6 15 Transitive PoC.
TABLE 3.4:
Statement Reason1. l||m,s||t Given2. 6 6∼= 6 9 Alternate Interior Angles Theorem3. 6 4∼= 6 7 Vertical Angles Theorem4. 6 6 and 6 7 are supplementary Same Side Interior Angles5. 6 9 an 6 4 are supplementary Same Angle Supplements Theorem
30. m6 1 = 102◦, m6 2 = 78◦, m6 3 = 102◦, m6 4 = 78◦, m6 5 = 22◦, m6 6 = 78◦, m 6 7 = 102◦
25
3.3. Basic Geometry, Proving Lines Parallel, Review Answers www.ck12.org
3.3 Basic Geometry, Proving Lines Parallel,Review Answers
1. Yes. If alternate interior angles are congruent, then the lines are parallel.2. No. Since alternate exterior angles are NOT congruent the lines are NOT parallel.3. No. Since alternate interior angles are NOT congruent, the lines are NOT parallel.4. Yes. If corresponding angles are congruent, then the lines are parallel.5. Yes. If exterior angles on the same side of the transversal are supplementary, then the lines are parallel.6. a = 90◦
7. b = 40◦
8. c = 140◦
9. d = 50◦
10. e = 90◦
11. f = 140◦
12. g = 130◦
13. h = 40◦
14. Start by copying the same angle as in Investigation 3-1, but place the copy where the alternate interior anglewould be.
15. The Corresponding Angles Postulate is used when two lines are parallel and we need to prove that two anglesare congruent. The Converse is used when we know the two angles are congruent and need to prove that thetwo lines are parallel.
16. Given, 6 1∼= 6 3, Given, 6 2∼= 6 3, Corresponding Angles Theorem, Transitive Property17. Given, 6 1∼= 6 3, Given, 6 2∼= 6 3, l||m18. Give, Converse of the Alternate Interior Angles Theorem, Given, Converse of the Alternate Interior Angles
Theorem, Parallel Lines Property
TABLE 3.5:
Statement Reason1. m⊥ l, n⊥ l Given2. m6 1 = 90◦,m6 2 = 90◦ Definition of Perpendicular Lines3. m6 1 = m6 2 Transitive Property4. m||n Converse of Corresponding Angles Theorem
TABLE 3.6:
Statement Reason1. 6 1∼= 6 3 Given
26
www.ck12.org Chapter 3. Parallel and Perpendicular Lines, Answer Key
TABLE 3.6: (continued)
Statement Reason2. m||n Converse of Alternate Interior Angles Theorem3. m 6 3+m 6 4 = 180◦ Linear Pair Postulate4. m 6 1+m 6 4 = 180◦ Substitution5. 6 1 and 6 4 are supplementary Definition of Supplementary Angles
TABLE 3.7:
Statement Reason1. 6 2∼= 6 4 Given2. m||n Converse of Corresponding Angles Theorem3. 6 1∼= 6 3 Alternate Interior Angles Theorem
TABLE 3.8:
Statement Reason1. 6 2∼= 6 3 Given2. m||n Converse of Corresponding Angles Theorem3. 6 1∼= 6 4 Alternate Exterior Angles Theorem
23.←→CG||←→HK
24.−→IB||−→AM
25. none26.←→CG||←→HK
27.−→IB||−→AM
28. none29.−→IB||−→AM
30. 58◦
31. 73◦
32. 107◦
33. 58◦
34. 49◦
35. 107◦
36. 49◦
37. x = 30◦
38. x = 15◦
39. x = 12◦
40. x = 26◦
27
3.4. Basic Geometry, Properties of Perpendicular Lines, Review Answers www.ck12.org
3.4 Basic Geometry, Properties of Perpendicu-lar Lines, Review Answers
1. 90◦
2. 90◦
3. 45◦
4. 16◦
5. 72◦
6. 84◦
7. 41◦
8. 24◦
9. 78◦
10. 90◦
11. 126◦
12. 54◦
13. 180◦
14. ⊥15. not ⊥16. not ⊥17. ⊥18. 90◦
19. 34◦
20. 56◦
21. 90◦
22. 56◦
23. 134◦
24. 134◦
25. 34◦
TABLE 3.9:
Statement Reason1. l ⊥ m, l ⊥ n Given2. 6 1 and 6 2 are right angles Definition of perpendicular lines3. m6 1 = 90◦, m6 2 = 90◦ Definition of right angles4. m6 1 = m6 2 Transitive PoE5. 6 1∼= 6 2 ∼= angles have = measures6. m||n Converse of the Corresponding Angles Postulate
27. x = 12◦
28. x = 9◦
29. x = 14.25◦
30. x = 8◦
28
www.ck12.org Chapter 3. Parallel and Perpendicular Lines, Answer Key
3.5 Basic Geometry, Parallel and Perpendicu-lar Lines in the Coordinate Plane, ReviewAnswers
1. 13
2. -13. 2
74. -25. 46. undefined7. Perpendicular
8. Parallel
9. Perpendicular
29
3.5. Basic Geometry, Parallel and Perpendicular Lines in the Coordinate Plane, Review Answers www.ck12.org
10. Neither
11. Perpendicular
12. Parallel
13. Neither
30
www.ck12.org Chapter 3. Parallel and Perpendicular Lines, Answer Key
14. Parallel
15. y =−5x−716. y = 2
3 x−517. y = 1
4 x+218. y =−3
2 x+119. y =−x−420. y =−1
3 x−421. y =−2
5 x+722. x =−123. Perpendicular
y =23
x+2
y =−32
x−4
24. Parallel
y =−15
x+7
y =−15
x−3
25. Perpendicular
y = x
y =−x
26. Neither
y =−2x+2
y = 2x−3
27.
⊥: y =−43
x−1
||: y =34
x+514
28.
⊥: y = 3x−3
||: y =−13
x+7
31
3.5. Basic Geometry, Parallel and Perpendicular Lines in the Coordinate Plane, Review Answers www.ck12.org
29.
⊥: y = 7
||: x =−3
30.
⊥: y = x−4
||: y =−x+8
32
www.ck12.org Chapter 3. Parallel and Perpendicular Lines, Answer Key
3.6 Basic Geometry, The Distance Formula,Review Answers
1. 17.09 units2. 19.20 units3. 5 units4. 17.80 units5. 22.20 units6. 14.21 units7. 6.40 units8. 9.22 units9. 7 units
10. 10.44 units11. 8 units12. 11 units13. 12 units14. 7 units15. 4 units16. 10 units17. 12 units18. 9 units19. 8 units20. 19 units21. 122. -123. (0, 2)24. y =−x+225. (3, -1)26. 4.24 units27. 9.90 units28. 2.83 units29. 4.24 units30. 12.73 units
33
3.7. Chapter Review Answers www.ck12.org
3.7 Chapter Review Answers
amp;m 6 1 = 90◦ m 6 2 = 118◦ m6 3 = 90◦ m6 4 = 98◦
amp;m 6 5 = 28◦ m 6 6 = 118◦ m6 7 = 128◦ m6 8 = 52◦
amp;m 6 9 = 62◦
34
www.ck12.org Chapter 4. Triangles and Congruence, Answer Key
CHAPTER 4 Triangles and Congruence,Answer Key
Chapter Outline4.1 BASIC GEOMETRY, TRIANGLE SUMS, REVIEW ANSWERS
4.2 BASIC GEOMETRY, CONGRUENT FIGURES, REVIEW ANSWERS
4.3 BASIC GEOMETRY, TRIANGLE CONGRUENCE USING SSS AND SAS, REVIEW
ANSWERS
4.4 BASIC GEOMETRY, TRIANGLE CONGRUENCE USING ASA, AAS, AND HL,REVIEW ANSWERS
4.5 BASIC GEOMETRY, ISOSCELES AND EQUILATERAL TRIANGLES, REVIEW AN-SWERS
4.6 CHAPTER REVIEW ANSWERS
35
4.1. Basic Geometry, Triangle Sums, Review Answers www.ck12.org
4.1 Basic Geometry, Triangle Sums, ReviewAnswers
1. 43◦
2. 121◦
3. 41◦
4. 86◦
5. 61◦
6. 51◦
7. 13◦
8. 60◦
9. 70◦
10. 118◦
11. 68◦
12. 116◦
13. 161◦
14. 141◦
15. 135◦
16. a = 68◦, b = 68◦, c = 25◦, d = 155◦, e = 43.5◦, f = 111.5◦
1. 180◦
2. 360◦
3. 360◦
4. 6 4∼= 6 7, 6 5∼= 6 8, and 6 6∼= 6 9
TABLE 4.1:
Statement Reason1. Triangle with interior and exterior angles. Given2. m6 1+m 6 2+m6 3 = 180◦ Triangle Sum Theorem3. 6 3 and 6 4 are a linear pair, 6 2 and 6 5 are a linearpair, and 6 1 and 6 6 are a linear pair
Definition of a linear pair
4. 6 3 and 6 4 are supplementary, 6 2 and 6 5 aresupplementary, and 6 1 and 6 6 are supplementary
Linear Pair Postulate
5. m6 1+m 6 6 = 180◦, m6 2+m6 5 = 180◦
m6 3+m6 4 = 180◦Definition of supplementary angles
6. m6 1+m 6 6+m6 2+m6 5+m 6 3+m6 4 = 540◦ Combine the 3 equations from #5.7. m6 4+m 6 5+m6 6 = 360◦ Subtraction PoE
TABLE 4.2:
Statement Reason1. 4ABC with right angle B Given2. m6 B = 90◦ Definition of a right angle3. m6 A+m 6 B+m6 C = 180◦ Triangle Sum Theorem4. m6 A+90◦+m6 C = 180◦ Substitution5. m6 A+m 6 C = 90◦ Subtraction PoE6. 6 A and 6 C are complementary Definition of complementary angles
36
www.ck12.org Chapter 4. Triangles and Congruence, Answer Key
20. x = 14◦
21. x = 9◦
22. x = 22◦
23. x = 17◦
24. x = 12◦
25. x = 30◦
26. x = 25◦
27. x = 7◦
37
4.2. Basic Geometry, Congruent Figures, Review Answers www.ck12.org
4.2 Basic Geometry, Congruent Figures, Re-view Answers
1. 6 R∼= 6 U, 6 A∼= 6 G, 6 T ∼= 6 H, RA∼=UG, AT ∼= GH, RT ∼=UH2. 6 B∼= 6 T, 6 I ∼= 6 O, 6 G∼= 6 P, BI ∼= TO, IG∼= OP, BG∼= T P3. Third Angle Theorem4. 90◦, they are congruent supplements5. Reflexive, FG∼= FG6. Angle Bisector7. 4FGI ∼=4FGH8. 6 A∼= 6 E and 6 B∼= 6 D by Alternate Interior Angles Theorem9. Vertical Angles Theorem
10. No, we need to know if the other two sets of sides are congruent.11. AC ∼=CE and BC ∼=CD12. 4ABC ∼=4EDC13. Yes,4FGH ∼=4KLM14. Not enough information to show congruence using the definition of congruent triangles.15. Yes,4ABE ∼=4DCE16. No17. 4BCD∼=4ZY X18. CPCTC19. m6 A = m6 X = 86◦, m6 B = m 6 Z = 52◦, m6 C = m 6 Y = 42◦
20. m6 A = m6 C = m6 Y = m6 Z = 35◦, m6 B = m 6 X = 110◦
21. m6 A = m6 C = 28◦, m6 ABE = m6 DBC = 90◦, m 6 D = m6 E = 62◦
22. m6 B = m6 D = 153◦, m 6 BAC = m6 ACD = 15◦, m 6 BCA = m6 CAD = 12◦
TABLE 4.3:
Statement Reason1. 6 A∼= 6 D, 6 B∼= 6 E Given2. m6 A = m 6 D, m6 B = m 6 E ∼= angles have = measures3. m6 A+m 6 B+m6 C = 180◦
m6 D+m 6 E +m6 F = 180◦ Triangle Sum Theorem4. m6 A+m 6 B+m6 C = m6 D+m 6 E +m6 F Substitution PoE5. m6 A+m 6 B+m6 C = m6 A+m 6 B+m6 F Substitution PoE6. m6 C = m6 F Substitution PoE7. 6 C ∼= 6 F ∼= angles have = measures
24. Transitive PoC25. Reflexive PoC26. Symmetric PoC27. Reflexive PoC28. 4ABC is either isosceles or equiangular because the congruence statement tells us that 6 A∼= 6 B.
38
www.ck12.org Chapter 4. Triangles and Congruence, Answer Key
4.3 Basic Geometry, Triangle Congruence us-ing SSS and SAS, Review Answers
1. Yes,4DEF ∼=4IGH, SSS2. No, HJ and ED are not congruent because they have different tic marks3. No, the angles marked are not in the same place in the triangles.4. Yes,4ABC ∼=4RSQ, SAS5. No, this is SSA, which is not a congruence postulate6. No, one triangle is SSS and the other is SAS.7. Yes,4ABC ∼=4FED, SSS8. Yes,4ABC ∼=4Y XZ, SAS9. No, these are both SSA, which is not a congruent postulate.
10. Yes,4AT D∼=4ET D, SSS11. AB∼= EF12. AB∼= HI13. 6 C ∼= 6 G14. 6 C ∼= 6 K15. AB∼= JL16. AB∼= ON
TABLE 4.4:
Statement Reason1. AB∼= DC, BE ∼=CE Given2. 6 AEB∼= 6 DEC Vertical Angles Theorem3. 4ABE ∼=4ACE SAS
TABLE 4.5:
Statement Reason1. AB∼= DC, AC ∼= DB Given2. BC ∼= BC Reflexive PoC3. 4ABC ∼=4DCB SSS
TABLE 4.6:
Statement Reason1. B is a midpoint of DC, AB⊥ DC Given2. DB∼= BC Definition of a midpoint3. 6 ABD and 6 ABC are right angles ⊥ lines create 4 right angles4. 6 ABD∼= 6 ABC All right angles are ∼=5. AB∼= AB Reflexive PoC6. 4ABD∼=4ABC SAS
39
4.3. Basic Geometry, Triangle Congruence using SSS and SAS, Review Answers www.ck12.org
TABLE 4.7:
Statement Reason1. AB is an angle bisector of 6 DAC, AD∼= AC Given2. 6 DAB∼= 6 BAC Definition of an Angle Bisector3. AB∼= AB Reflexive PoC4. 4ABD∼=4ABC SAS
TABLE 4.8:
Statement Reason1. B is the midpoint of DC, AD∼= AC Given2. DB∼= BC Definition of a Midpoint3. AB∼= AB Reflexive PoC4. 4ABD∼=4ABC SSS
TABLE 4.9:
Statement Reason1. B is the midpoint of DE and AC, 6 ABE is a rightangle
Given
2. DB∼= BE, AB∼= BC Definition of a Midpoint3. m6 ABE = 90◦ Definition of a Right Angle4. m6 ABE = m6 DBC Vertical Angle Theorem5. 4ABE ∼=4CBD SAS
TABLE 4.10:
Statement Reason1. DB is the angle bisector of 6 ADC, AD∼= DC Given2. 6 ADB∼= 6 BDC Definition of an Angle Bisector3. DB∼= DB Reflexive PoC4. 4ABD∼=4CBD SAS
24. 4ABC :√
90,√
73,√
61, 4DEF :√
90,√
73,√
61. Yes the triangles are congruent.25. 4ABC :
√10,√
80,√
130, 4DEF :√
10,√
130,√
80. Yes26. 4ABC :
√18,√
52,√
58, 4DEF :√
58,√
137,√
17. No27. 4ABC :
√37,√
34,√
45, 4DEF :√
37,√
34,√
45. Yes
40
www.ck12.org Chapter 4. Triangles and Congruence, Answer Key
4.4 Basic Geometry, Triangle Congruence us-ing ASA, AAS, and HL, Review Answers
1. Yes, AAS,4ABC ∼=4FDE2. Yes, ASA,4ABC ∼=4IHG3. Yes, SAS,4ABC ∼=4KLJ4. No5. Yes, SAS,4ABC ∼=4RQP6. Yes, HL,4ABC ∼=4QPR7. Yes, SAS,4ABE ∼=4DBC8. No9. Yes, SSS,4WZY ∼=4Y XW
10. Yes, AAS,4WXY ∼=4QPO11. 6 DBC ∼= 6 DBA because the are both right angles.12. 6 CDB∼= 6 ADB13. DB∼= DB
TABLE 4.11:
Statement Reason1. DB⊥ AC, DB is the angle bisector of 6 CDA Given2. 6 DBC and 6 ADB are right angles Definition of perpendicular3. 6 DBC ∼= 6 ADB All right angles are ∼=4. 6 CDB∼= 6 ADB Definition of an angle bisector5. DB∼= DB Reflexive PoC6. 4CDB∼=4ADB ASA
15. CPCTC16. 6 L∼= 6 O and 6 P∼= 6 N by the Alternate Interior Angles Theorem17. 6 LMP∼= 6 NMO by the Vertical Angles Theorem
TABLE 4.12:
Statement Reason1. LP||NO, LP∼= NO Given2. 6 L∼= 6 O, 6 P∼= 6 N Alternate Interior Angles Theorem3. 4LMP∼= 6 OMN ASA
19. CPCTC
TABLE 4.13:
Statement Reason1. LP||NO, LP∼= NO Given2. 6 L∼= 6 O, 6 P∼= 6 N Alternate Interior Angles3. 4LMP∼= 6 OMN ASA4. LM ∼= MO CPCTC
41
4.4. Basic Geometry, Triangle Congruence using ASA, AAS, and HL, Review Answers www.ck12.org
TABLE 4.13: (continued)
Statement Reason5. M is the midpoint of PN. Definition of a midpoint
21. 6 A∼= 6 N22. 6 C ∼= 6 M23. PM ∼= MN24. LM ∼= MO or LP∼= NO25. UT ∼= FG26. SU ∼= FH
TABLE 4.14:
Statement Reason1. SV ⊥WU , T is the midpoint of SV and WU Given2. 6 STW and 6 UTV are right angles Definition of perpendicular3. 6 STW ∼= 6 UTV All right angles are ∼=4. ST ∼= TV , WT ∼= TU Definition of a midpoint5. 4STW ∼=4UTV SAS6. WS∼=UV CPCTC
TABLE 4.15:
Statement Reason1. 6 K ∼= 6 T , EI is the angle bisector of 6 KET Given2. 6 KEI ∼= 6 T EI Definition of an angle bisector3. EI ∼= EI Reflexive PoC4. 4KEI ∼=4T EI AAS5. 6 KIE ∼= 6 T IE CPCTC6. EI is the angle bisector of 6 KIT Definition of an angle bisector
29. 1. 50◦
2.
1. 50◦
2.
30. No, the triangles are not congruent.
42
www.ck12.org Chapter 4. Triangles and Congruence, Answer Key
4.5 Basic Geometry, Isosceles and EquilateralTriangles, Review Answers
All of the constructions are drawn to scale with the appropriate arc marks.
1.
2.
3.
4.
5.6. x = 10,y = 77. x = 148. x = 13◦
9. x = 16◦
10. x = 7◦
11. x = 112. y = 313. y = 11◦,x = 4◦
14. x = 25◦,y = 19◦
1. 90◦
2. 30◦
3. 60◦
4. 2
15. True
43
4.5. Basic Geometry, Isosceles and Equilateral Triangles, Review Answers www.ck12.org
16. False17. True18. False19. True20. a = 46◦, b = 88◦, c = 46◦, d = 134◦, e = 46◦, f = 67◦, g = 67◦
TABLE 4.16:
Statement Reason1. Isosceles 4CIS, with base angles 6 C and 6 SIO isthe angle bisector of 6 CIS
Given
2. 6 C ∼= 6 S Base Angles Theorem3. 6 CIO∼= 6 SIO Definition of an Angle Bisector4. IO∼= IO Reflexive PoC5. 4CIO∼=4SIO ASA6. CO∼= OS CPCTC7. 6 IOC ∼= 6 IOS CPCTC8. 6 IOC and 6 IOS are supplementary Linear Pair Postulate9. m6 IOC = m6 IOS = 90◦ Congruent Supplements Theorem10. IO is the perpendicular bisector of CS Definition of a ⊥ bisector (Steps 6 and 9)
TABLE 4.17:
Statement Reason1. Equilateral4RST with RT ∼= ST ∼= RS Given2. 6 R∼= 6 S Base Angles Theorem3. 6 S∼= 6 T Base Angles Theorem4. 6 R∼= 6 T Transitive PoC5. 4RST is equiangular Defintion of an Equiangular4
TABLE 4.18:
Statement Reason1. Isosceles 4ICS with 6 C and 6 S, IO is the perpen-dicular bisector of CS
Given
2. 6 C ∼= 6 S Base Angle Theorem3. CO∼= OS Definition of a ⊥ bisector4. m6 IOC = m6 IOS = 90◦ Definition of a ⊥ bisector5. 4CIO∼=4SIO ASA6. 6 CIO∼= 6 SIO CPCTC7. IO is the angle bisector of 6 CIS Definition of an Angle Bisector
25. Prove: 4ABC ∼=4XY Z
44
www.ck12.org Chapter 4. Triangles and Congruence, Answer Key
TABLE 4.19:
Statement Reason1. Isosceles 4ABC with base angles 6 B and 6 C,Isosceles 4XY Z with base angles 6 Y and 6 Z, 6 C ∼=6 Z, BC ∼= Y Z
Given
2. 6 B∼= 6 C, 6 Y ∼= 6 Z Base Angles Theorem3. 6 B∼= 6 Y Transitive PoC4. 4ABC ∼=4XY Z ASA
26. Isosceles27. Scalene28. Scalene29. Isosceles30. Isosceles
45
4.6. Chapter Review Answers www.ck12.org
4.6 Chapter Review Answers
For 1-5, answers will vary.
1. One leg and the hypotenuse from each are congruent,4ABC ∼=4Y XZ2. Two angles and the side between them,4ABC ∼= EDC3. Two angles and a side that is NOT between them,4ABC ∼=4SRT4. All three sides are congruent,4ABC ∼=4CDA5. Two sides and the angle between them,4ABF ∼=4ECD6. Linear Pair Postulate7. Base Angles Theorem8. Exterior Angles Theorem9. Property of Equilateral Triangles
10. Triangle Sum Theorem11. Equilateral Triangle Theorem12. Property of an Isosceles Right Triangle
46
www.ck12.org Chapter 5. Relationships with Triangles, Answer Key
CHAPTER 5 Relationships withTriangles, Answer Key
Chapter Outline5.1 BASIC GEOMETRY, MIDSEGMENTS, REVIEW ANSWERS
5.2 BASIC GEOMETRY, PERPENDICULAR BISECTORS AND ANGLE BISECTORS IN
TRIANGLES, REVIEW ANSWERS
5.3 BASIC GEOMETRY, MEDIANS AND ALTITUDES IN TRIANGLES, REVIEW AN-SWERS
5.4 BASIC GEOMETRY, INEQUALITIES IN TRIANGLES, REVIEW ANSWERS
5.5 BASIC GEOMETRY, EXTENSION: INDIRECT PROOF, REVIEW ANSWERS
5.6 CHAPTER REVIEW ANSWERS
47
5.1. Basic Geometry, Midsegments, Review Answers www.ck12.org
5.1 Basic Geometry, Midsegments, ReviewAnswers
1. True2. True3. False4. False5. True6. RS = TU = 67. TU = 88. x = 5, TU = 109. x = 2
10. y = 1811. x = 1212. x = 5.513. x = 614. x = 14, y = 2415. x = 6, z = 2616. x = 5, y = 317. x = 1, z = 11
1. 13, 19, 212. 533. 1064. The perimeter of the larger triangle is double the perimeter of the midsegment triangle.
18. (7, 1), (3, 6), (1, 3)19. (3, 6), (2, 2), (-5, -3)20. (2, 5), (7, 1), (4, -1)21. (-1, -8), (-9, 5), (-12, -2)22. AB =−7
3 , BC =−13 , AC = 2
323. BC
25 and 26: Here is a graph of what to do with the slope triangles. The vertices are (-8, 8), (10, 2) and (-2, -6).
48
www.ck12.org Chapter 5. Relationships with Triangles, Answer Key
27. GH = 13 , HI = 2, GI =−1
2
28.29. (3, 4), (15, -2), (-3, -8)30. GH =
√90≈ 9.49,
√360≈ 18.97 Yes, GH is half of this side
49
5.2. Basic Geometry, Perpendicular Bisectors and Angle Bisectors in Triangles, Review Answers www.ck12.org
5.2 Basic Geometry, Perpendicular Bisectorsand Angle Bisectors in Triangles, ReviewAnswers
1.
2.
3.
The 6 triangles are congruent.
4-5. Construct the point of intersection using investigation 5-4.
6. The constructions would look the same. We can conclude that the perpendicular bisector and the angle bisectorare the same line for an equilateral triangle.
7. x = 168. x = 89. x = 5
10. x = 12
11. x = 31◦
12. x = 34
1. AE = EB, AD = DB2. No, AC 6=CB3. Yes, AD = DB
13. No, we don’t have enough information.</math>14. No, we don’t know if T is the midpoint of XY .15. x = 616. x = 317. x = 818. x = 719. x = 9
50
www.ck12.org Chapter 5. Relationships with Triangles, Answer Key
20. x = 921. No, the line segment must be perpendicular to the sides of the angle also.22. No, it doesn’t matter if the bisector is perpendicular to the interior ray.23. Yes, the angles are marked congruent.24. m = 1
225. (4, 2)26. -227. y =−2x+10
TABLE 5.1:
Statement Reason1.←→CD is the perpendicular bisector of AB Given
2. D is the midpoint of AB Definition of a perpendicular bisector3. AD∼= DB Definition of a midpoint4. 6 CDA and 6 CDB are right angles Definition of a perpendicular bisector5. 6 CDA∼= 6 CDB Definition of right angles6. CD∼=CD Reflexive PoC7. 4CDA∼=4CDB SAS8. AC ∼=CB CPCTC
TABLE 5.2:
Statement Reason1. AD∼= DC Given2.−→BA⊥ AD and
−→BC ⊥ DC The shortest distance from a point to a line is perpen-
dicular.3. 6 DAB and 6 DCB are right angles Definition of perpendicular lines4. 6 DAB∼= 6 DCB All right angles are congruent5. BD∼= BD Reflexive PoC6. 4ABD∼=4CBD HL7. 6 ABD∼= 6 DBC CPCTC8.−→BD bisects 6 ABC Definition of an angle bisector
51
5.3. Basic Geometry, Medians and Altitudes in Triangles, Review Answers www.ck12.org
5.3 Basic Geometry, Medians and Altitudes inTriangles, Review Answers
1-3. Use Investigation 5-3 to find the centroid.
4. The centroid will always be inside of a triangle because medians are always on the interior of a triangle.
5-6. Use Investigation 5-4 and 3-2 to find the altitude.
7. GE = 10BE = 15
8. GF = 8CF = 24
9. AG = 20GD = 10
10. GC = 2xCF = 3x
11. x = 2, AD = 2712. centroid13. XY = 1014. XM = 915. CN = 1516. XC = 817. ZY = 2y18. x = 7
19.20. (0, -4)21. -422. y =−4x−4
52
www.ck12.org Chapter 5. Relationships with Triangles, Answer Key
23.24. (3, 1)25. -126. y =−x+427. True28. False29. False30. True31. True32. False33. True34. True
TABLE 5.3:
Statement Reason1. Isosceles4ABC with legs AB and ACBD⊥ DC and CE ⊥ BE
Given
2. 6 DBC ∼= 6 ECB Base Angles Theorem3. 6 BEC and 6 CEB are right angles Definition of perpendicular lines4. 6 BEC ∼= 6 CEB All right angles are congruent5. BC ∼= BC Reflexive PoC6. 4BEC ∼=4CDB AAS7. BD∼=CE CPCTC
53
5.4. Basic Geometry, Inequalities in Triangles, Review Answers www.ck12.org
5.4 Basic Geometry, Inequalities in Triangles,Review Answers
1. AB, BC, AC2. BC, AB, AC3. AC, BC, AB4. 6 B, 6 A, 6 C5. 6 B, 6 C, 6 A6. 6 C, 6 B, 6 A
7.
8.
8 is the longest side.
9.10. No, 6+6 < 1311. No, 1+2 = 312. Yes13. Yes14. No, 23+56 < 8515. Yes16. Yes17. No, 7+8 = 1518. Yes19. 1 < 3rd side < 1720. 11 < 3rd side < 1921. 12 < 3rd side < 5222. 3 < 3rd side < 723. 2 < 3rd side < 1824. x < 3rd side < 3x25. Both legs must be longer than 1226. 0 < base < 2427. 0 < x < 10.328. m6 1 > m 6 2 because 7 > 629. IJ, IG, GJ, GH, JH
54
www.ck12.org Chapter 5. Relationships with Triangles, Answer Key
30. m6 1 < m 6 2, m6 3 > m6 4
55
5.5. Basic Geometry, Extension: Indirect Proof, Review Answers www.ck12.org
5.5 Basic Geometry, Extension: Indirect Proof,Review Answers
Answers will vary. Here are some hints.
1. Assume n is odd, therefore n = 2a+1.2. Assume4ABC is equilateral. What can you then conclude about 6 A and 6 B?3. Remember the square root of a number can be negative or positive.4. Use the definition of an isosceles triangle to lead you towards a contradiction.5. If x+ y is even, then x+ y = 2n, where n is any integer.6. Use the Triangle Sum Theorem to lead you towards a contradiction.7. With the assumption of the opposite of AB+BC = AC, these three lengths could make a triangle, thus making
A, B, and C non-collinear.8. Assume the third side of the first triangle is less than or equal to the length of the second triangle. Then,
if the third sides are equal, the two triangles will be congruent and all the angles are congruent by CPCTC.This contradicts the hypothesis of the theorem that “the included angle of the first triangle is greater than theincluded angle of the second triangle.”
56
www.ck12.org Chapter 5. Relationships with Triangles, Answer Key
5.6 Chapter Review Answers
1. BE2. AE3. AH4. CE5. AG6. x−7 < third side < 3x+57. midpoints8. altitude9. centroid
10. largest11. perpendicular, equidistant12. angle, equidistant13. inscribed14. indirect15. AC > DF
57
www.ck12.org
CHAPTER 6 Polygons andQuadrilaterals, Answer Key
Chapter Outline6.1 BASIC GEOMETRY, ANGLES IN POLYGONS, REVIEW ANSWERS
6.2 BASIC GEOMETRY, PROPERTIES OF PARALLELOGRAMS, REVIEW ANSWERS
6.3 BASIC GEOMETRY, PROVING QUADRILATERALS ARE PARALLELOGRAMS, RE-VIEW ANSWERS
6.4 BASIC GEOMETRY, RECTANGLES, RHOMBUSES AND SQUARES, REVIEW AN-SWERS
6.5 BASIC GEOMETRY, TRAPEZOIDS AND KITES, REVIEW ANSWERS
6.6 CHAPTER REVIEW ANSWERS
58
www.ck12.org Chapter 6. Polygons and Quadrilaterals, Answer Key
6.1 Basic Geometry, Angles in Polygons, Re-view Answers
TABLE 6.1:
# of sides # of 4s from onevertex
4s×180◦ (sum) Each angle in aregular n−gon
Sum of the exteriorangles
3 1 180◦ 60◦ 360◦
4 2 360◦ 90◦ 360◦
5 3 540◦ 108◦ 360◦
6 4 720◦ 120◦ 360◦
7 5 900◦ 128.57◦ 360◦
8 6 1080◦ 135◦ 360◦
9 7 1260◦ 140◦ 360◦
10 8 1440◦ 144◦ 360◦
11 9 1620◦ 147.27◦ 360◦
12 10 1800◦ 150◦ 360◦
2. An interior angle can never be 180◦ because that would be a straight line. Theoretically, an interior anglecould be 179◦, but the polygon would have 360 sides! Very close to a circle.
3. 2340◦
4. 3780◦
5. 266. 207. 157.5◦
8. 165◦
9. 1510. 411. 30◦
12. 10◦
13. 360◦
14. 163◦
15. 168◦
16. 120◦
17. x = 60◦
18. x = 90◦,y = 20◦
19. x = 35◦
20. y = 115◦
21. x = 105◦
22. x = 51◦,y = 108◦
23. x = 70◦,y = 70◦,z = 90◦
24. x = 72.5◦,y = 107.5◦
25. x = 90◦,y = 64◦
26. x = 52◦,y = 128◦,z = 123◦
27. x = 60◦,y = 120◦,z = 60◦
28. x = 67.5◦
29. x = 36◦
30. x = 117.5◦
59
6.2. Basic Geometry, Properties of Parallelograms, Review Answers www.ck12.org
6.2 Basic Geometry, Properties of Parallelo-grams, Review Answers
1. 62. 83. 100◦
4. 45◦
5. 62◦
6. 87. m6 A = 108◦,m6 C = 108◦,m6 D = 72◦
8. m6 P = 37◦,m6 Q = 143◦,m6 D = 37◦
9. all angles are 90◦
10. m6 E = m 6 G = (180− x)◦,m6 H = x◦
11. a = b = 53◦
12. c = 613. d = 10,e = 1414. f = 5,g = 315. h = 25◦, j = 11◦,k = 8◦
16. m = 25◦,n = 19◦
17. p = 8,q = 318. r = 1,s = 219. t = 3,u = 420. 96◦
21. 85◦
22. 43◦
23. 42◦
24. yes)25. no26. no27. yes
TABLE 6.2:
Statement Reason1. ABCD is a parallelogram with diagonals BD Given2. AB‖DC,AD‖BC Definition of a parallelogram3. 6 ABD∼= 6 BDC, 6 ADB∼= 6 DBC Alternate Interior Angles Theorem4. DB∼= DB Reflexive PoC5. 4ABD∼=4CDB ASA6. 6 A∼= 6 C CPCTC
TABLE 6.3:
Statement Reason1. ABCD is a parallelogram with diagonals BD and AC Given2. AB‖DC,AD‖BC Definition of a parallelogram3. 6 ABD∼= 6 BDC, 6 CAB∼= 6 ACD Alternate Interior Angles Theorem
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www.ck12.org Chapter 6. Polygons and Quadrilaterals, Answer Key
TABLE 6.3: (continued)
Statement Reason4. AB∼= DC Opposite Sides Theorem5. 4DEC ∼=4BEA ASA6. AE ∼= EC,DE ∼= EB CPCTC
30. x = 16,y = 105◦,z = 60◦
61
6.3. Basic Geometry, Proving Quadrilaterals are Parallelograms, Review Answers www.ck12.org
6.3 Basic Geometry, Proving Quadrilateralsare Parallelograms, Review Answers
1. No2. Yes3. Yes4. Yes5. No6. No7. Yes8. No9. Yes
10. Yes11. No12. No13. x = 1914. x = 65◦,y = 115◦
15. x = 23,y = 1516. x = 517. x = 8◦,y = 10◦
18. x = 4,y = 319. Yes20. Yes21. No22. Yes
TABLE 6.4:
Statement Reason1. 6 A∼= 6 C, 6 D∼= 6 B Given2. m6 A = m 6 C,m 6 D = m6 B ∼= angles have = measures3. m6 A+m 6 B+m6 C+m6 D = 360◦ Definition of a quadrilateral4. m6 A+m 6 A+m6 B+m6 B = 360◦ Substitution PoE5. 2m6 A+2m 6 B = 360◦
2m 6 A+2m6 D = 360◦ Combine Like Terms6. m6 A+m 6 B = 180◦
m6 A+m6 D = 180◦ Division PoE7. 6 A and 6 B are supplementary6 A and 6 D are supplementary
Definition of Supplementary Angles
8. AD‖BC,AB‖DC Consecutive Interior Angles Converse9. ABCD is a parallelogram Definition of a Parallelogram
TABLE 6.5:
Statement Reason1. AE ∼= EC,DE ∼= EB Given2. 6 AED∼= 6 BEC6 DEC ∼= 6 AEB Vertical Angles Theorem
62
www.ck12.org Chapter 6. Polygons and Quadrilaterals, Answer Key
TABLE 6.5: (continued)
Statement Reason3. 4AED∼=4CEB4AEB∼=4CED SAS4. AB∼= DC,AD∼= BC CPCTC5. ABCD is a parallelogram Opposite Sides Converse
TABLE 6.6:
Statement Reason1. 6 ADB∼= 6 CBD,AD∼= BC Given2. AD‖BC Alternate Interior Angles Converse3. ABCD is a parallelogram Theorem 5-10
63
6.4. Basic Geometry, Rectangles, Rhombuses and Squares, Review Answers www.ck12.org
6.4 Basic Geometry, Rectangles, Rhombusesand Squares, Review Answers
1. 132. 263. 244. 105. 90◦
1. 122. 21.43. 114. 54◦
5. 90◦
1. 90◦
2. 90◦
3. 45◦
4. 45◦
4. Rhombus5. Parallelogram6. Rectangle7. Rectangle8. Rhombus9. None
10. Parallelogram11. Square12. Rectangle13. None14. Square15. Parallelogram16. Sometimes, when the figure is a square.17. Always18. Sometimes, when it is a square.19. Always20. Sometimes, when it is a square.21. Never
22.
64
www.ck12.org Chapter 6. Polygons and Quadrilaterals, Answer Key
23.24. Draw one diagonal vertically and the other horizontally, so they intersect at each other’s midpoints. Connect
the four endpoints.25. Draw one side and then a 90◦ angle at each endpoint. Mark two inches on each side. Connect these two
endpoints.26. Square27. Rhombus28. Rectangle29. Parallelogram
1. Rectangle2. 83. SC = 14.42,RC = 16.97
65
6.5. Basic Geometry, Trapezoids and Kites, Review Answers www.ck12.org
6.5 Basic Geometry, Trapezoids and Kites, Re-view Answers
1. 1. 55◦
2. 125◦
3. 90◦
4. 110◦
1. 50◦
2. 50◦
3. 90◦
4. 25◦
5. 115◦
2. No, if the parallel sides were congruent, then it would be a parallelogram. By the definition of a trapezoid, itcan never be a parallelogram (exactly one pair of parallel sides).
3. Yes, the diagonals do not have to bisect each other.
5. 336. 287. 88. 119. 37
10. 511. x = 114◦,y = 44◦
12. x = y = 102.5◦
13. x = 10,y = 614. x = 5,y = 1215. x = 8,y = 1716. x =
√130,y = 41
17. x = 418. x = 5,y =
√73
19. x = 11,y = 1720. y = 5◦
21. y = 45◦
22. x = 12◦,y = 8◦
23. no24. yes25. parallelogram26. square27. kite
66
www.ck12.org Chapter 6. Polygons and Quadrilaterals, Answer Key
28. trapezoid
67
6.5. Basic Geometry, Trapezoids and Kites, Review Answers www.ck12.org
TABLE 6.7:
Statement Reason1. KE ∼= T E and KI ∼= T I Given2. EI ∼= EI Reflexive PoC3. 4EKI ∼=4ET I SSS4. 6 KES∼= 6 T ES and 6 KIS∼= 6 T IS CPCTC5. EI is the angle bisector of 6 KET and 6 KIT Definition of an angle bisector
TABLE 6.8:
Statement Reason1. KE ∼= T E and KI ∼= T I Given2. 4KET and4KIT are isosceles triangles Definition of isosceles triangles3. EI is the angle bisector of 6 KET and 6 KIT Theorem 6-224. EI is the perpendicular bisector of KT Isosceles Triangle Theorem5. KT⊥EI Definition of perpendicular lines.
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www.ck12.org Chapter 6. Polygons and Quadrilaterals, Answer Key
6.6 Chapter Review Answers
TABLE 6.9:
Oppositesides ‖
Diagonalsbisect eachother
Diagonals ⊥ Oppositesides ∼=
Opposite an-gles ∼=
Diagonals ∼=
Trapezoid One set No No No No NoIsoscelesTrapezoid
One set No No Non-parallelsides
No,bases6 s∼=
Yes
Kite No No Yes No Non-vertex6 s
No
Parallelogram Both sets Yes No Yes Yes NoRectangle Both sets Yes No Yes All 6 s∼= YesRhombus Both sets Yes Yes All sides ∼= Yes NoSquare Both sets Yes Yes All sides ∼= All 6 s∼= Yes
1. a. 180◦
b. 360◦
c. 540◦
d. 720◦
2. a = 64◦,b = 118◦,c = 82◦,d = 99◦,e = 106◦, f = 88◦,g = 150◦,h = 56◦, j = 74◦,k = 136◦
69
www.ck12.org
CHAPTER 7 Similarity, Answer KeyChapter Outline
7.1 BASIC GEOMETRY, RATIOS AND PROPORTIONS, REVIEW ANSWERS
7.2 BASIC GEOMETRY, SIMILAR POLYGONS, REVIEW ANSWERS
7.3 BASIC GEOMETRY, SIMILARITY BY AA, REVIEW ANSWERS
7.4 BASIC GEOMETRY, SIMILARITY BY SSS AND SAS, REVIEW ANSWERS
7.5 BASIC GEOMETRY, PROPORTIONALITY RELATIONSHIPS, REVIEW ANSWERS
7.6 BASIC GEOMETRY, SIMILARITY TRANSFORMATIONS, REVIEW ANSWERS
7.7 BASIC GEOMETRY, EXTENSION: SELF-SIMILARITY, REVIEW ANSWERS
7.8 CHAPTER REVIEW ANSWERS
70
www.ck12.org Chapter 7. Similarity, Answer Key
7.1 Basic Geometry, Ratios and Proportions,Review Answers
1. 1. 4:32. 5:83. 6:194. 6:8:5
2. 2:13. 1:34. 2:15. 1:16. 5:4:37. 4 gal8. 24 ft9. 600 cm
10. 512
11. 12
12. 11
13. 1930
14. angles are 54◦,54◦,72◦
15. length is 12, width is 2016. length is 64, width is 11217. 20 girls18. 240 boys19. 30 seniors20. x = 1221. x =−522. y = 1623. x = 12,−1224. y =−2125. z = 3.7526. x = 13.9 gallons27. The president makes $800,000, vice president makes $600,000 and the financial officer makes $400,000.28. False29. True30. True31. False32. 2833. 1834. 735. 24
71
7.2. Basic Geometry, Similar Polygons, Review Answers www.ck12.org
7.2 Basic Geometry, Similar Polygons, ReviewAnswers
1. True2. False3. False4. False5. True6. True7. False8. True9. 6 B∼= 6 H, 6 I ∼= 6 A, 6 G∼= 6 T, BI
HA = IGAT = BG
HT10. 3
5 or 53
11. HT = 3512. IG = 2713. 57,95, 3
5 or 53
14. Not similar15. No16. m6 E = 113◦,m6 Q = 112◦
17. 23 or 3
218. 1219. 2120. 621. No, 32
26 6=1812
22. Yes,4ABC ∼4NML23. Yes, ABCD∼ STUV24. Yes,4EFG∼4LMN25. Yes, QRST ∼ BCDA26. No, angles are not the same.27. No, 5
10 6=8
1828. Yes,4EFG∼4MLN29. Yes, ABDC ∼ EFGH30. No, we do not know any angle measures.
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www.ck12.org Chapter 7. Similarity, Answer Key
7.3 Basic Geometry, Similarity by AA, ReviewAnswers
1. 4T RI2. T R,T I,AM3. 124. 65. 6, 126. 4ABE ∼4CDE because 6 BAE ∼= 6 DCE and 6 ABE ∼= 6 CDE by the Alternate Interior Angles Theorem.7. Answers will vary. One possibility: AE
CE = BEDE
8. One possibility: 4AED and4BEC9. AC = 22.4
10. Yes, right angles are congruent and solving for the missing angle in each triangle, we find that the other twoangles are congruent as well.
11. FE = 34 k
12. k = 1613. right, right, similar14. Congruent triangles have the same shape AND size. Similar triangles only have the same shape. Congruent
triangles are always similar. Similar triangles are not always congruent.15. Yes,4DEG∼4FDG∼4FED16. Yes,4HLI ∼4HKJ17. No only vertical angles are congruent18. Yes,4LNK ∼4JNM19. Yes,4FIH ∼4GIF20. Yes,4AEB∼4ADC21. No, no congruent angles.22. Yes,4TUW ∼4XUV23. No, EG ∦ DC.24. Yes, they are ⊥ to the same line.25. Yes, the two right angles are congruent and 6 OEC and 6 NEA are vertical angles.26. x = 48 f t.27. Yes, we can use the Pythagorean Theorem to find EA. EA = 93.3 f t.28. 13,000 ft.29. 32.5 ft. tall30. 1666.67 ft. long
73
7.4. Basic Geometry, Similarity by SSS and SAS, Review Answers www.ck12.org
7.4 Basic Geometry, Similarity by SSS andSAS, Review Answers
1. congruent2. proportional3. congruent4. proportional5. proportional, corresponding, congruent, similar6. Yes, SSS. The side lengths are proportional.7. No. One is much larger than the other.8. There are 2.2 cm in an inch, so that is the scale factor.9. 4DFE
10. DF,EF,DF11. DH = 7.512. Perimeter4ABC = 36, Perimeter4DEF = 27, Ratio = 4:313. 4DBE14. SAS15. 2716. AB,BE,AC17. Yes, 7
21 = 824 . This proportion will be valid as long as AC‖DE.
18. No, 721 6=
3627 .
19. Yes,4ABC ∼4DFE, SAS20. No, the angle is not between the given sides21. Yes,4ABC ∼4DFE, SSS22. Yes,4ABE ∼4DBC, SAS23. No, 10
20 6=1525
24. No, 2432 6=
1620
25. x = 326. x = 6,y = 3.527. x = 228. x = 529. The building is 10 ft tall.30. The child’s shadow is 105 inches long.
74
www.ck12.org Chapter 7. Similarity, Answer Key
7.5 Basic Geometry, Proportionality Relation-ships, Review Answers
1. 4ECF ∼4BCD2. DF3. CD4. FE5. DF,DB6. 14.47. 21.68. 16.89. 45
10. 2:311. 3:512. 2:3 is the ratio of the segments created by the parallel lines, 3:5 is the ratio of the similar triangles.13. yes14. no15. yes16. no17. yes18. no19. x = 920. y = 1021. y = 1622. z = 423. x = 824. x = 2.525. a = 626. b = 12.827. y = 328. x = 429. a = 4.8,b = 9.630. a = 4.5,b = 4,c = 10
75
7.6. Basic Geometry, Similarity Transformations, Review Answers www.ck12.org
7.6 Basic Geometry, Similarity Transforma-tions, Review Answers
1. (2, 6)2. (-8, 12)3. (4.5, -6.5)4. k = 3
25. k = 96. k = 1
2
7.
8.
9.
10.11. 20, 26, 3412. 2 2
3 , 3, 513. 7.5, 10, 12.514. 2, 3, 415. k = 2
516. k = 14
1117. k = 5
918. k = 4
719. A′(6,12),B′(−9,21),C′(−3,−6)20. A′(9,6),B′(−3,−12),C′(0,−7.5)
76
www.ck12.org Chapter 7. Similarity, Answer Key
21.22. k = 223. A′′(4,8),B′′(48,16),C′′(40,40)24. k = 225. 2.3426. 2.3427. 6.7128. 4.6829. 8.9430. 11.1831. 22.3632. 44.7233. OA : OA′ = 1 : 2
AB : A′B′ = 1 : 234. OA : OA′′ = 1 : 4
AB : A′′B′′ = 1 : 4
77
7.7. Basic Geometry, Extension: Self-Similarity, Review Answers www.ck12.org
7.7 Basic Geometry, Extension: Self-Similarity, Review Answers
1.
TABLE 7.1:
Number of Segments Length of each Segment Total Length of the Seg-ments
Stage 0 1 1 1Stage 1 2 1
323
Stage 2 4 19
49
Stage 3 8 127
827
Stage 4 16 181
1681
Stage 5 32 1243
32243
3. There will be 2n segments.
4.
5.
TABLE 7.2:
Stage 0 Stage 1 Stage 2 Stage 3Color 0 1 9 73No Color 1 8 64 512
7. Answers will vary. Many different flowers (roses) and vegetables (broccoli and cauliflower) are examples offractals in nature.
78
www.ck12.org Chapter 7. Similarity, Answer Key
7.8 Chapter Review Answers
1. 1. x = 122. x = 14.5
2. x = 10◦;50◦,60◦,70◦
3. 3.75 gallons4. yes5. no6. yes, AA7. yes, SSS8. no9. yes
10. A′(10.5,3),B′(6,13.5),C′(−1.5,6)11. x = 6.312. x = 113. z = 614. a = 5,b = 7.5
79
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CHAPTER 8 Right TriangleTrigonometry, Answer Key
Chapter Outline8.1 BASIC GEOMETRY, THE PYTHAGOREAN THEOREM, REVIEW ANSWERS
8.2 BASIC GEOMETRY, CONVERSE OF THE PYTHAGOREAN THEOREM, REVIEW
ANSWERS
8.3 BASIC GEOMETRY, USING SIMILAR RIGHT TRIANGLES, REVIEW ANSWERS
8.4 BASIC GEOMETRY, SPECIAL RIGHT TRIANGLES, REVIEW ANSWERS
8.5 BASIC GEOMETRY, TANGENT, SINE AND COSINE, REVIEW ANSWERS
8.6 BASIC GEOMETRY, INVERSE TRIGONOMETRIC RATIOS, REVIEW ANSWERS
8.7 CHAPTER REVIEW ANSWERS
80
www.ck12.org Chapter 8. Right Triangle Trigonometry, Answer Key
8.1 Basic Geometry, The Pythagorean Theo-rem, Review Answers
1. 4√
52. 2√
63. 1084. 32
√5
5. 1206. 8√
5
7. 2√
303
8. 6√
105
9. 7√
39
10.√
50511. 9
√5
12.√
79913. 1214. 1015. 10
√14
16. 2617. 3
√41
18. 16√
219. 9
√2
20. yes21. no22. no23. yes24. yes25. no26. 2
√39
27.√
42928.√
25329. 4
√5
30.√
49331. 5
√10
32. 36.6×20.633. 33.6×25.234. 4
√3
35. s√
32
81
8.2. Basic Geometry, Converse of the Pythagorean Theorem, Review Answers www.ck12.org
8.2 Basic Geometry, Converse of thePythagorean Theorem, Review Answers
1. right2. right3. no4. right5. no6. right7. acute8. right9. obtuse
10. right11. acute12. acute13. right14. obtuse15. obtuse16. acute17. obtuse18. right19. right20. The slopes of AB and AC are -1 and 1, respectively, making them perpendicular.21. c = 1322. d =
√194
23. 4√
224. 4
√3
25. The sides of4ABC are a multiple of 3, 4, 5 which is a right triangle. 6 A is opposite the largest side, which isthe hypotenuse, making it 90◦.
82
www.ck12.org Chapter 8. Right Triangle Trigonometry, Answer Key
8.3 Basic Geometry, Using Similar Right Trian-gles, Review Answers
1. 4ACD,4BCA2. AC,AC3. BD4. CD5. 4ADC ∼4ABD∼4DBC6. 4EHG∼4HFG∼4EFH7. 4KLM ∼4KJL∼4LJM8. 4KML∼4JML∼4JKL9. KM = 6
√3
10. JK = 6√
711. KL = 3
√21
12. 16√
213. 15
√7
14. 2√
3515. 14
√6
16. 20√
1017. 2
√102
18. x = 12√
519. y = 5
√5
20. z = 9√
221. x = 422. y =
√465
23. z = 14√
524. x = 9.625. y = 2.1226. z = 8
√2
27. x = 325 ,y =
8√
415 ,z = 2
√41
28. x = 9,y = 3√
34
29. x = 9√
48120 ,y = 81
40 ,z = 40
TABLE 8.1:
Statement Reason1. 4ABD with AC ⊥ DB and 6 DAB is a right angle. Given2. 6 DCA and 6 ACB are right angles Definition of perpendicular lines.3. 6 DAB∼= 6 DCA∼= 6 ACB All right angles are congruent.4. 6 D∼= 6 D Reflexive PoC5. 4CAD∼=4ABD AA Similarity Postulate6. 6 B∼= 6 B Reflexive PoC7. 4CBA∼=4ABD AA Similarity Postulate8. 4CAD∼=4CBA Transitive PoC
83
8.4. Basic Geometry, Special Right Triangles, Review Answers www.ck12.org
8.4 Basic Geometry, Special Right Triangles,Review Answers
1. 4√
22. 5√
3, 103. x√
24. x√
3,2x5. 15
√2
6. 11√
27. 128. 10
√3
9. a = 2√
2,b = 210. c = 6
√2,d = 12
11. e = f = 13√
212. g = 10
√3,h = 20
13. k = 12, j = 12√
314. x = 11
√3,y = 22
√3
15. m = 9,n = 1816. q = 14, p = 14
√2
17. s = 9, t = 3√
318. x = w = 9
√2
19. a = 9√
3,b = 18√
320. p = 18,q = 6
√3
21. s = 2√
2, t = 422. v = 15,w = 10
√3
23. f = g = 7.5√
2 or 15√
22
24. y≈ 21.8625. y≈ 13.86
84
www.ck12.org Chapter 8. Right Triangle Trigonometry, Answer Key
8.5 Basic Geometry, Tangent, Sine and Cosine,Review Answers
1. df
2. fe
3. fd
4. de
5. de
6. fe
7. D,D8. reciprocals9. 0.4067
10. 0.707111. 28.636312. 0.682013. 0.212614. 0.190815. 0.990316. 1.000017. sinA = 4
5 ,cosA = 35 , tanA = 4
3
18. sinA =
√2
2 ,cosA =
√2
2 , tanA = 1
19. sinA = 13 ,cosA = 2
√2
3 , tanA =
√2
420. sinA = 4
5 ,cosA = 35 , tanA = 4
3
21. sinA = 12 ,cosA =
√3
2 , tanA =
√3
322. sinA = 8
17 ,cosA = 1517 , tanA = 8
1523. x = 9.4,y = 17.724. x = 14.1,y = 19.425. x = 20.8,y = 22.326. x = 19.3,y = 5.227. x = 5.9,y = 12.528. x = 20.9,y = 13.429. x = 435.9 f t.30. x = 56 m
85
8.6. Basic Geometry, Inverse Trigonometric Ratios, Review Answers www.ck12.org
8.6 Basic Geometry, Inverse TrigonometricRatios, Review Answers
1. 33.7◦
2. 31.0◦
3. 44.7◦
4. 39.4◦
5. 46.6◦
6. 36.9◦
7. 34.6◦
8. 82.9◦
9. 70.2◦
10. 10.1◦
11. 86.4◦
12. 51.7◦
13. m6 A = 38◦,BC = 9.38,AC = 15.2314. AB = 4
√10,m6 A = 18.4◦,m6 B = 71.6◦
15. BC =√
51,m6 A = 45.6,m 6 C = 44.4◦
16. m6 A = 60◦,BC = 12,AC = 12√
317. CB = 7
√5,m6 A = 48.2◦,m6 B = 41.8◦
18. m6 B = 50◦,AC = 38.14,AB = 49.7819. AB = 18.6,BC = 19.3,m 6 C = 75◦
20. AC = 29.7,BC = 47.5,m 6 A = 58◦
21. BC = 15,m6 A = 36.7◦,m6 B = 53.1◦
22. 47.6◦
23. 1.6◦
24. 44.0◦
25. cos50◦
26. sin20◦
27. As the angle measures increase, the sine value increases.28. As the angle measures increase, the cosine value decreases.29. The sine and cosine values are between 0 and 1.30. tan85◦ = 11.43, tan89◦ = 57.29, and tan89.5◦ = 114.59. As the tangent values get closer to 90◦, they get
larger and larger. There is no maximum, the values approach infinity.
86
www.ck12.org Chapter 8. Right Triangle Trigonometry, Answer Key
8.7 Chapter Review Answers
1. b2. B3. A4. A5. B6. A7. f8. d9. a,a
10. e11. BC = 4.4,AC = 10.0,m6 A = 26◦
12. AB = 5√
10,m6 A = 18.4◦,m6 B = 71.6◦
13. BC = 6√
7,m6 A = 41.4◦,m6 C = 48.6◦
14. m6 A = 30◦,AC = 25√
3,BC = 2515. BC = 7
√13,m 6 A = 31◦,m6 B = 59◦
16. m6 B = 45◦,AC = 32,AB = 32√
217. m6 B = 63◦,BC = 19.1,AB = 8.718. m6 C = 19◦,AC = 22.7,AB = 7.819. BC = 4
√13,m 6 B = 33.7◦,m6 C = 56.3◦
20. acute21. right, Pythagorean triple22. obtuse23. right24. acute25. obtuse26. x = 227. x = 2
√110
28. x = 6√
729. 2576.5 ft.30. x = 29.2◦
87
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CHAPTER 9 Circles, Answer KeyChapter Outline
9.1 BASIC GEOMETRY, PARTS OF CIRCLES AND TANGENT LINES, REVIEW AN-SWERS
9.2 BASIC GEOMETRY, PROPERTIES OF ARCS, REVIEW ANSWERS
9.3 BASIC GEOMETRY, PROPERTIES OF CHORDS, REVIEW ANSWERS
9.4 BASIC GEOMETRY, INSCRIBED ANGLES, REVIEW ANSWERS
9.5 BASIC GEOMETRY, ANGLES OF CHORDS, SECANTS, AND TANGENTS, REVIEW
ANSWERS
9.6 BASIC GEOMETRY, SEGMENTS OF CHORDS, SECANTS, AND TANGENTS, RE-VIEW ANSWERS
9.7 BASIC GEOMETRY, EXTENSION: WRITING AND GRAPHING THE EQUATIONS OF
CIRCLES, REVIEW ANSWERS
9.8 CHAPTER REVIEW ANSWERS
88
www.ck12.org Chapter 9. Circles, Answer Key
9.1 Basic Geometry, Parts of Circles and Tan-gent Lines, Review Answers
1. diameter2. secant3. chord4. point of tangency5. common external tangent6. common internal tangent7. center8. radius9. the diameter
10. 4 lines
11. 3 lines
12. none
13. radius of⊙
B = 4, radius of⊙
C = 5, radius of⊙
D = 2, radius of⊙
E = 214.
⊙D∼=
⊙E because they have the same radius length.
15. 2 common tangents16. CE = 717. y = x−218. yes19. no
89
9.1. Basic Geometry, Parts of Circles and Tangent Lines, Review Answers www.ck12.org
20. yes21. 4
√10
22. 4√
1123. x = 924. x = 325. x = 526. x = 8
√2
27. Yes, by AA. m 6 CAE = m6 DBE = 90◦ and 6 AEC ∼= 6 BED by vertical angles.28. 2529. 1230. 1531. 37 and 3532. 3233. BDFH is a kite because adjacent sides are congruent.34. The perimeter would be 40.
35. No, because the opposite sides are congruent and at most a circle can touch three sides.
36.
37.
90
www.ck12.org Chapter 9. Circles, Answer Key
TABLE 9.1:
Statement Reason1. AB and CB with points of tangency at A and C. ADand DC are radii.
Given
2. AD∼= DC All radii are congruent.3. DA⊥ AB and DC ⊥CB Tangent to a Circle Theorem4. m6 BAD = 90◦ and m6 BCD = 90◦ Definition of perpendicular lines5. Draw BD. Connecting two existing points6. 4ADB and4DCB are right triangles Definition of right triangles (Step 4)7. DB∼= DB Reflexive PoC8. 4ABD∼=4CBD HL9. AB∼=CB CPCTC
39. 1. kite2. center, bisects
40. AT ∼= BT ∼=CT by theorem 9-2 and the transitive property.
91
9.2. Basic Geometry, Properties of Arcs, Review Answers www.ck12.org
9.2 Basic Geometry, Properties of Arcs, Re-view Answers
1. minor2. major3. semicircle4. major5. minor6. semicircle7. yes, CD∼= DE8. 66◦
9. 228◦
10. mBC = 100◦, mBDC = 260◦
11. mBC = 175◦, mBDC = 185◦
12. mBC = 38◦, mBDC = 322◦
13. mBC = 109◦, mBDC = 251◦
14. mBC = 124◦, mBDC = 236◦
15. mBC = 34◦, mBDC = 326◦
16. yes, they are in the same circle with equal central angles17. yes, the central angles are vertical angles, so they are equal, making the arcs equal18. no, we don’t know the measure of the corresponding central angles.19. 90◦
20. 49◦
21. 82◦
22. 16◦
23. 188◦
24. 172◦
25. 196◦
26. 270◦
27. x = 54◦
28. x = 47◦
29. x = 25◦
30.⊙
A∼=⊙
B
92
www.ck12.org Chapter 9. Circles, Answer Key
9.3 Basic Geometry, Properties of Chords, Re-view Answers
1. No, see picture. The two chords can be congruent and perpendicular, but will not bisect each other.2. If a diameter bisects a chord and its corresponding arc, then it is perpendicular chord. This statement is true
as long as it says “bisects a chord AND its corresponding arc.” If it only said “bisects the chord,” then thediameter does not have to be perpendicular to the chord.
3. Theorem 9-3: If minor arcs are congruent, then corresponding chords are congruent. If chords are congruent,then the corresponding minor arcs are congruent. Theorem 9-6: If two chords are congruent, then they areequidistant from the center. If two chords are equidistant from the center, then they are congruent.
4. AC5. DF6. JF7. DE8. 6 HGC9. 6 AGC
10. AG, HG, CG, FG, JG, DG11. 107◦
12. 8◦
13. 118◦
14. 133◦
15. 140◦
16. 120◦
17. x = 64◦, y = 418. x = 8, y = 1019. x = 3
√26, y≈ 12.3
20. x = 9√
521. x = 9◦, y = 622. x = 4.523. x = 324. x = 725. x = 2
√11
26. mAB = 121.3◦
27. mAB = 112.9◦
28. BF ∼= FD and BF ∼= FD by Theorem 9-5.29. CA∼= AF by Theorem 9-6.30. QS is a diameter by Theorem 9-4.
93
9.4. Basic Geometry, Inscribed Angles, Review Answers www.ck12.org
9.4 Basic Geometry, Inscribed Angles, ReviewAnswers
1. inscribed2. half3. equal to4. semicircle5. congruent6. opposite, supplementary7. chords8. m6 JKL = 1
2 m 6 JML9. 48◦
10. 120◦
11. 54◦
12. 45◦
13. 87◦
14. 27◦
15. 100.5◦
16. 95.5◦
17. 79.5◦
18. 84.5◦
19. 51◦
20. 46◦
21. x = 180◦, y = 21◦
22. x = 60◦, y = 49◦
23. x = 30◦, y = 60◦
24. x = 72◦, y = 92◦
25. x = 200◦, y = 100◦
26. x = 68◦, y = 99◦
27. x = 93◦, y = 97◦
28. x = 37◦
29. x = 42◦
30. x = 24◦
31. x = 6◦
32. x = 35◦
33. x = 10◦
TABLE 9.2:
Statement Reason1. Inscribed 6 ABC and diameter BDm6 ABE = x◦ and m6 CBE = y◦
Given
2. x◦+ y◦ = m 6 ABC Angle Addition Postulate3. AE ∼= EB and EB∼= EC All radii are congruent4. 4AEB and4EBC are isosceles Definition of an isosceles triangle5. m6 EAB = x◦ and m 6 ECB = y◦ Isosceles Triangle Theorem6. m6 AED = 2x◦ and m6 CED = 2y◦ Exterior Angle Theorem7. mAD = 2x◦ and mDC = 2y◦ The measure of an arc is the same as its central angle.
94
www.ck12.org Chapter 9. Circles, Answer Key
TABLE 9.2: (continued)
Statement Reason8. mAD+mDC = mAC Arc Addition Postulate9. mAC = 2x◦+2y◦ Substitution10. mAC = 2(x◦+ y◦) Distributive PoE11. mAC = 2m 6 ABC Subsitution12. m6 ABC = 1
2 mAC Division PoE
95
9.5. Basic Geometry, Angles of Chords, Secants, and Tangents, Review Answers www.ck12.org
9.5 Basic Geometry, Angles of Chords, Se-cants, and Tangents, Review Answers
1. 1.
2.
3.2. No, by definition a tangent line cannot pass through a circle, so it can never intersect with any line inside of
one.
1.
2.
3. center, equal4. inside, intercepted5. on, half6. outside, half7. x = 103◦
8. x = 25◦
9. x = 100◦
10. x = 44◦
11. x = 38◦
12. x = 54.5◦
13. x = 63◦, y = 243◦
96
www.ck12.org Chapter 9. Circles, Answer Key
14. x = 216◦
15. x = 42◦
16. x = 150◦
17. x = 66◦
18. x = 113◦
19. x = 70◦
20. x = 152◦
21. x = 180◦, y = z = 45◦
22. x = 60, y = 40◦
23. x = 180◦, y = 60◦
24. x = 35◦, y = 90◦, z = 55◦
25. x = 27◦
26. x = 39.5◦
TABLE 9.3:
Statement Reason1. Intersecting chords AC and BD. Given2. Draw BC Construction
3. m6 DBC = 12 mDC Inscribed Angle Theorem
4. m6 ACB = 12 mAB Inscribed Angle Theorem
5. m 6 a = m6 DBC+m6 ACB Exterior Angle Theorem6. m6 a = 1
2 mDC+ 12 mAB Substitution
TABLE 9.4:
Statement Reason1. Intersecting secants
−→AB and
−→AC. Given
2. Draw BE. Construction
3. m6 BEC = 12 mBC Inscribed Angle Theorem
4. m6 DBE = 12 mDE Inscribed Angle Theorem
5. m6 a+m 6 DBE = m6 BEC Exterior Angle Theorem6. m6 a = m6 BEC−m 6 DBE Subtraction PoE7. m6 a = 1
2 mBC− 12 mDE Substitution
8. m6 a = 12
(mBC−mDE
)Distributive Property
97
9.6. Basic Geometry, Segments of Chords, Secants, and Tangents, Review Answers www.ck12.org
9.6 Basic Geometry, Segments of Chords, Se-cants, and Tangents, Review Answers
1. 5, 10, x = 22. 3+ x, x = 33. 102, x = 54. 20, 8+7, x = 65. x2, 4, x = 66. x+15, x = 57. x = 128. x = 1.59. x = 12
10. x = 7.511. x = 6
√2
12. x = 1013. x = 1014. x = 815. x = 816. x = 22.417. x = 1118. x = 4
√41
19. x = 1207 ≈ 17.14
20. x = 4√
6621. x = 622. x =
√231
23. x = 4√
4224. x = 1025. The error is in the set up. It should be 10 ·10 = y · (15+ y). The correct answer is y = 5.26. x = 1027. x = 928. x = 429. 10 inches
TABLE 9.5:
Statement Reason1. Intersecting chords AC and BE with segmentsa, b, c, and d.
Given
2. 6 AED∼= 6 BCD6 EAD∼= 6 CBD Theorem 9-83. 4ADE ∼4BDC AA Similarity Postulate4. a
c = db Corresponding parts of similar triangles are propor-
tional5. ab = cd Cross multiplication
98
www.ck12.org Chapter 9. Circles, Answer Key
TABLE 9.6:
Statement Reason1. Secants PR and RT with segments a, b, c, and d. given2. 6 R∼= 6 R Reflexive PoC3. 6 QPS∼= 6 ST Q Theorem 9-84. 4RPS∼4RT Q AA Similarity Postulate5. a
c+d = ca+b Corresponding parts of similar triangles are propor-
tional6. a(a+b) = c(c+d) Cross multiplication
99
9.7. Basic Geometry, Extension: Writing and Graphing the Equations of Circles, Review Answers www.ck12.org
9.7 Basic Geometry, Extension: Writing andGraphing the Equations of Circles, ReviewAnswers
1. center: (-5, 3), radius = 42. center: (0, -8), radius = 23. center: (7, 10), radius = 2
√5
4. center: (-2, 0), radius = 2√
25. (x−4)2 +(y+2)2 = 166. (x+1)2 +(y−2)2 = 497. (x−2)2 +(y−2)2 = 48. (x+4)2 +(y+3)2 = 259. yes
10. no11. yes12. no13. (x−2)2 +(y−3)2 = 5214. (x−10)2 + y2 = 2915. (x+3)2 +(y−8)2 = 20016. (x−6)2 +(y+6)2 = 325
100
www.ck12.org Chapter 9. Circles, Answer Key
9.8 Chapter Review Answers
1. I2. A3. D4. G5. C6. B7. H8. E9. J
10. F
101
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CHAPTER 10Perimeter and Area, AnswerKey
Chapter Outline10.1 BASIC GEOMETRY, TRIANGLES AND PARALLELOGRAMS, REVIEW ANSWERS
10.2 BASIC GEOMETRY, TRAPEZOIDS, RHOMBI, AND KITES, REVIEW ANSWERS
10.3 BASIC GEOMETRY, AREAS OF SIMILAR POLYGONS, REVIEW ANSWERS
10.4 BASIC GEOMETRY, CIRCUMFERENCE AND ARC LENGTH, REVIEW ANSWERS
10.5 BASIC GEOMETRY, AREAS OF CIRCLES AND SECTORS, REVIEW ANSWERS
10.6 CHAPTER REVIEW ANSWERS
102
www.ck12.org Chapter 10. Perimeter and Area, Answer Key
10.1 Basic Geometry, Triangles and Parallelo-grams, Review Answers
1. A = 144 in2,P = 48 in2. A = 144 cm2,P = 50 cm3. A = 360 m2
4. A = 112 u2,P = 44 u5. A = 324 f t2,P = 72 f t6. P = 36 f t7. A = 36 in2
8. A = 210 cm2
9. 12 m10. Possible answers: 10×6,12×411. Possible answers: 9×10,3×3012. If the areas are congruent, then the figures are congruent. We know this statement is false, #11 would be a
counterexample.13. 8
√2 cm
14. P≈ 57.45 cm15. A = 96
√2≈ 135.8 cm2
16. 15 in17. P≈ 73.98 in18. A = 180 in2
19. 315 units2
20. 90 units2
21. 14 units2
22. 187.5 units2
23. 672 units2
24. 30 units2
1.2. 20, 12.5, 3753. 407.5 units2
1.2. 504, 315, 67.53. 796.5 units2
25. 144 units2
26. 18 units2
27. 72 units2
28. 72 units2
103
10.1. Basic Geometry, Triangles and Parallelograms, Review Answers www.ck12.org
29. 30-60-90 triangle, AD = 4,BD = 4√
330. 16
√3
31. AD = x2 ,BD = x
2
√3
32. 12 · x ·
x√
32 = x2
√3
4
33. 144√
34 = 36
√3
34. 25√
34 = 6.25
√3
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www.ck12.org Chapter 10. Perimeter and Area, Answer Key
10.2 Basic Geometry, Trapezoids, Rhombi, andKites, Review Answers
1. If a kite and a rhombus have the same diagonal lengths the areas will be the same. This is because bothformulas are dependent upon the diagonals. If they are the same, the areas will be the same too. This does notmean the two shapes are congruent, however.
2. 160 units2
3. 558 units2
4. 96 units2
5. 77 units2
6. 86.60 units2
7. 84 units2
8. 1000 units2
9. 63 units2
10. 62.5 units2
11. A = 480 units2
P = 104 units12. A = 98.35 units2
P = 40.97 units13. A = 108 units2
P = 52.97 units14. A = 119.29 units2
P = 52 units15. A = 685.89 units2
P = 116 units16. A = 572.43 units2
P = 96 units17. mAB = mDC = 1, trapezoid18. mAD =−1, yes they are perpendicular.19. AB = 2
√2,DC = 4
√2,AD = 2
√2
20. A = 12 units2
21. mHE = mGF = 34 ,mHG = mEF =−3
4 ,mHF = 0,mEG = unde f ined, rhombus22. HF = 8,EG = 623. 24 units2
24. 6425. 8×8,4×1626. 10827. 12×9,6×1828. 10 cm and 6 cm29. 30 cm2
30. 90 cm2
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10.3. Basic Geometry, Areas of Similar Polygons, Review Answers www.ck12.org
10.3 Basic Geometry, Areas of Similar Poly-gons, Review Answers
1. 925
2. 116
3. 494
4. 36121
5. 16
6. 29
7. 73
8. 512
9. 14
10. 12
11. 5 units2
12. 24 units13. 12, 814. 2
315. 2
316. 4, 917. 4
918. 3
√2,2√
2, 23 , the same as the scale factor.
19. 96 units2
20. 198 f t2
21. 54 in22. 32 units23. 4
924. 2
325. Diagonals are 12 and 18. The length of the sides are 6
√2 and 9
√2.
26. Because the diagonals of these rhombi are congruent, the rhombi are actually squares.
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www.ck12.org Chapter 10. Perimeter and Area, Answer Key
10.4 Basic Geometry, Circumference and ArcLength, Review Answers
TABLE 10.1:
diameter radius circumference1. 15 7.5 15π
2. 8 4 8π
3. 6 3 6π
4. 84 42 84π
5. 18 9 18π
6. 25 12.5 25π
7. 2 1 2π
8. 36 18 36π
9. r = 44π
in10. C = 20 cm11. 1612. The diameter is the same length as the diagonals of the square.13. 32
√2
14. 16π
15. 9π
16. 80π
17. 15π
18. 25π
19. 3π
20. 10π
21. r = 10822. r = 3023. r = 7224. 120◦
25. 162◦
26. 15◦
27. 81.7 in, the circumference28. 775 complete rotations29. 5.25 miles30. 23270901 rotations
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10.5. Basic Geometry, Areas of Circles and Sectors, Review Answers www.ck12.org
10.5 Basic Geometry, Areas of Circles and Sec-tors, Review Answers
TABLE 10.2:
radius Area circumference1. 2 4π 4π
2. 4 16π 8π
3. 5 25π 10π
4. 12 144π 24π
5. 9 81π 18π
6. 3√
10 90π 6√
10π
7. 17.5 306.25π 35π
8. 7π
49π
149. 30
π
900π
6010. 6√
π36 12
√π
11. 54π
12. 1.042π
13. 189π
14. 7.5π
15. 1188π
16. 16π
17. r = 8√
318. r = 3019. r = 1520. 120◦
21. 10◦
22. 198◦
23. 123.6124. 1033.5825. 54.9426. 6π
27. 9√
328. 3.2629. 64π
30. 12831. 73.06
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www.ck12.org Chapter 10. Perimeter and Area, Answer Key
10.6 Chapter Review Answers
1. A = 225P = 60
2. A = 198P = 58
3. A = 124.71P = 48
4. A = 249.42P = 72
5. A = 3000P = 232
6. A = 27.59P = 22
7. 728. 2249. 162
√3
10. C = 34π
A = 289π
11. C = 30π
A = 225π
12. 54 units2
13. 1070.1214. 1220.3915. 70.06
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CHAPTER 11 Surface Area and Volume,Answer Key
Chapter Outline11.1 BASIC GEOMETRY, EXPLORING SOLIDS, REVIEW ANSWERS
11.2 BASIC GEOMETRY, SURFACE AREA OF PRISMS AND CYLINDERS, REVIEW
ANSWERS
11.3 BASIC GEOMETRY, SURFACE AREA OF PYRAMIDS AND CONES, REVIEW AN-SWERS
11.4 BASIC GEOMETRY, VOLUME OF PRISMS AND CYLINDERS, REVIEW ANSWERS
11.5 BASIC GEOMETRY, VOLUME OF PYRAMIDS AND CONES, REVIEW ANSWERS
11.6 BASIC GEOMETRY, SURFACE AREA AND VOLUME OF SPHERES, REVIEW AN-SWERS
11.7 BASIC GEOMETRY, EXTENSION: EXPLORING SIMILAR SOLIDS, REVIEW AN-SWERS
11.8 CHAPTER REVIEW ANSWERS
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www.ck12.org Chapter 11. Surface Area and Volume, Answer Key
11.1 Basic Geometry, Exploring Solids, ReviewAnswers
1. V = 82. F = 93. E = 304. F = 65. E = 66. V = 67. F = 98. V = 69. Yes, hexagonal pyramid. F = 7,V = 7,E = 12
10. No, a cone has a curved face.11. Yes, hexagonal prism. F = 8,V = 12,E = 1812. No a hemisphere has a face.13. Yes, trapezoidal prism. F = 6,V = 8,E = 1214. Yes, concave decagonal prism. F = 10,V = 16,E = 2415. Rectangle16. Circle17. Trapezoid
18.
19.
20.
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11.1. Basic Geometry, Exploring Solids, Review Answers www.ck12.org
21.
22.
23.24. Cube25. Square Pyramid26. Cube27. Regular Icosahedron28. Decagonal Pyramid29. Trapezoidal Prism30. The truncated icosahedron has 60 vertices, by Euler’s Theorem.
F +V = E +2
32+V = 90+2
V = 60
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www.ck12.org Chapter 11. Surface Area and Volume, Answer Key
11.2 Basic Geometry, Surface Area of Prismsand Cylinders, Review Answers
1. rectangular prism2. 6 rectangles: 2 are 6×7, 2 are 2×6, and 2 are 2×7.3. 42 in2 each4. 52 in2
5. 136 in2
6. A = 67. The rectangles are 3×6,4×6, and 6×5. Their areas are 18, 24, and 30.8. 72units2
9. 84units2
10. Lateral surface area is the area of all the sides, total surface area includes the bases.
1. 96 in2
2. 192 in2
11. 350π cm2
12. 1607.3713. 830.0514. 486π
15. 18216. 34π
17. 280818. x = 819. x = 3220. x = 2521. 60π in2
22. 6080π cm2
23. 3 cm24. 29 ft.25. 4060 f t2
26. 2940 f t2
27. 2800 f t2
28. 2520 f t2
29. 9380 f t2
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11.3. Basic Geometry, Surface Area of Pyramids and Cones, Review Answers www.ck12.org
11.3 Basic Geometry, Surface Area of Pyramidsand Cones, Review Answers
1. vertex2. y3. lateral edge4. w5. z6. t7. vertex8. y9. circle, base
10. slant height11. 5
√10 cm
12. 12.82 in13. The height of the base is 3
√3.
14. 3015. 1716. 4517. 67118. 6819. 6420. 1413.7221. 36022. 422.3523. 1847.2624. 89625. 1507.96
1. 3√
32. 9√
33. 36
√3
26. 6 cm27. 7 cm28. 3 ft.29. 25 in.30. 21 in.31. 576, 321.5332. 1159.2533. 1146.2934. 1467.82
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www.ck12.org Chapter 11. Surface Area and Volume, Answer Key
11.4 Basic Geometry, Volume of Prisms andCylinders, Review Answers
1. No, the volumes do not have to be the same. One cylinder could have a height of 8 and a radius of 4, whileanother could have a height of 22 and a radius of 2. Both have a surface area of 96π, but the volumes are notthe same.
2. 960 cubes, yes this is the same as the volume.3. 280 in3
4. 4π in3
5. 6 in6. 512 in3
7. 64π cm3
8. d = 189. 5
10. 36 units3
1. 64 in3
2. 128 in3
11. 882π cm3
12. 396013. 902.5414. 4580.4415. 14716. 50.2717. 777618. x = 719. x = 2420. x = 3221. 294π in3
22. 24000π cm3
23. 75π m3
24. 41472 cm3
25. 364.5π cm3
26. 40326.89 cm3
27. 13230 cm3
28. 425.25π cm3
29. 14565.96 cm3
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11.5. Basic Geometry, Volume of Pyramids and Cones, Review Answers www.ck12.org
11.5 Basic Geometry, Volume of Pyramids andCones, Review Answers
Unless otherwise specified, all units are units3.
1. 96802. 12803. 3392.924. 4005. 15686. 1287. 1884.968. 100.539. 37.70
10. 188.5011. 4212. 20013. 1066.6714. 15.5915. 4.9016. 25.4617. Find the volume of one square pyramid then multiply it by 2.18. 101.8219. h = 13.5 in20. h = 3.6 cm21. r = 3 cm22. h = 15 in23. 80 in3
24. 32 in3
25. 112 in3
26. 216 cm3
27. 25.13 cm3
28. 190.87 cm3
29. 157.08 cm3
30. 314.16 cm3
31. 471.24 cm3
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www.ck12.org Chapter 11. Surface Area and Volume, Answer Key
11.6 Basic Geometry, Surface Area and Volumeof Spheres, Review Answers
1. No, all the cross sections must be circles because there are no edges.2. radius, center, diameter3. great circle, hemisphere4. SA = 256π in2
V = 20483 π in3
5. SA = 324π cm2
V = 972π cm3
6. SA = 1600π f t2
V = 320003 π f t3
7. SA = 16π m2
V = 323 π m2
8. SA = 900π f t2
V = 4500π f t3
9. SA = 1024π in2
V = 163843 π in3
10. SA = 676π cm2
V = 87883 π cm3
11. SA = 2500π yd2
V = 625003 π yd3
12. r = 5.5 in13. r = 33 m14. V = 4
3 π f t3
15. SA = 36π mi2
16. r = 4.31 cm17. r = 7.5 f t.18. 2025π cm2
19. 620π units2
20. 4680π f t2
21. 75.25π units2
22. 47712.94 cm3
23. 7120.94 units3
24. 191134.50 f t3
25. 121.86 units3
26. h = 203 cm
27. 14.14 in3
28. 63.62 in3
29. 21.2 in3
30. The formula for the surface area of a sphere is 4 times the area of a circle with the same radius. This mustmean that you could cover the surface of a sphere with 4 circles (all with the same radius).
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11.7. Basic Geometry, Extension: Exploring Similar Solids, Review Answers www.ck12.org
11.7 Basic Geometry, Extension: ExploringSimilar Solids, Review Answers
1. No, 1410 6=
4235
2. Yes, the scale factor is 4:3.3. Yes, the scale factor is 3:5.4. No, the top base is not in the same proportion as the rest of the given lengths.5. Yes, cubes have the same length for each side. So, comparing two cubes, the scale factor is just the ratio of
the sides.6. 1:167. 8:3438. 125:7299. 8:11
10. 5:1211. 87.48π
12. 4:913. 60 cm14. 512:337515. 2:316. 4:917. y = 8,x = h = 1218. Vs = 170.67,Vl = 57619. Yes, just like the cubes spheres and hemispheres only have a radius to compare. So, all spheres and hemi-
spheres are similar.20. 49:144, 343:1728
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www.ck12.org Chapter 11. Surface Area and Volume, Answer Key
11.8 Chapter Review Answers
1. F2. K3. G4. A5. E6. D7. J8. B9. L
10. C11. H12. I13. H14. G15. A16. B17. D18. J19. I20. E21. F22. C
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www.ck12.org
CHAPTER 12 Rigid Transformations,Answer Key
Chapter Outline12.1 BASIC GEOMETRY, EXPLORING SYMMETRY, REVIEW ANSWERS
12.2 BASIC GEOMETRY, TRANSLATIONS, REVIEW ANSWERS
12.3 BASIC GEOMETRY, REFLECTIONS, REVIEW ANSWERS
12.4 BASIC GEOMETRY, ROTATIONS, REVIEW ANSWERS
12.5 BASIC GEOMETRY, COMPOSITION OF TRANSFORMATIONS, REVIEW ANSWERS
12.6 BASIC GEOMETRY, EXTENSION: TESSELLATING POLYGONS, REVIEW ANSWERS
12.7 CHAPTER REVIEW ANSWERS
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www.ck12.org Chapter 12. Rigid Transformations, Answer Key
12.1 Basic Geometry, Exploring Symmetry, Re-view Answers
1. 32. 43. n4. lines of symmetry5. 90◦
6. false7. true8. true9. true
10. false11. false12. false13. true14. true15. false16. a kite that is not a rhombus17. a circle18. an isosceles trapezoid
19.
20.
21.22. none
23.24. H is the only one with rotational symmetry, 180◦.25. line symmetry
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12.1. Basic Geometry, Exploring Symmetry, Review Answers www.ck12.org
26. rotational symmetry27. line symmetry28. line symmetry29. rotational symmetry30. 2 lines31. 6 lines32. 4 lines33. 180◦
34. 6 times; 60◦
35. 4 times; 90◦
36. none37. 3 times; 120◦
38. 9 times; 40◦
39. 8 lines of symmetry; angle of rotation: 45◦
40. 3 line of symmetry; angle of rotation: 120◦
41. 1 line of symmetry; no rotational symmetry
122
www.ck12.org Chapter 12. Rigid Transformations, Answer Key
12.2 Basic Geometry, Translations, Review An-swers
1. A′(−1,−6)2. B′(9,−1)3. C′(10,−12)4. A′′(4,−15)5. D(7,16)6. A′′′(9,−24)7. All four points are collinear.8. A′(−8,−13),B′(−5,−17),C′(−7,−5)9. A′(5,−3),B′(8,−6),C′(6,6)
10. A′(−6,−10),B′(−3,−13),C′(−5,−1)11. A′(−11,1),B′(−8,−2),C′(−10,10)12. A′(−5,−7),B′(−2,−10),C′(−4,2)13. A′(−3,3),B′(0,0),C′(−2,12)14. (x,y)→ (x−6,y+2)15. (x,y)→ (x+9,y−7)16. (x,y)→ (x−3,y−5)17. (x,y)→ (x+8,y+4)
18 and 19.Using the distance formula, AB = A′B′ =√
26,BC = B′C′ =√
13, and AC = A′C′ =√
13.
20. #18 and #19 tell us that the two triangles are congruent by SSS.21. (x,y)→ (x+6,y−2)22. (x,y)→ (x−9,y+7)23. (x,y)→ (x−6,y−4)24. D′(9,9),E ′(12,7),F ′(10,14),(x,y)→ (x+5,y+11)25. Q′(−9,−6),U ′(−6,0),A′(1,−9),D′(−2,−15),(x,y)→ (x−3,y−7)
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12.3. Basic Geometry, Reflections, Review Answers www.ck12.org
12.3 Basic Geometry, Reflections, Review An-swers
1. (-5, 3)2. (5, -3)3. (3, 5)4. (-3, -5)5. A square
6. d7. p8. (-3, 2), (-8, 4), (-6, 7), (-4, 7)9. (-6, 4), (-2, 6), (-8, 8)
10. (2, 2), (8, 3), (6, -3)11. (2, 6), (-6, 2), (4, -2)12. (2, -2), (8, -6)13. (2, -4), (-4, 2), (-2, -6)14. (2, 3), (4, 8), (7, 6), (7, 4)15. (4, 6), (6, 2), (8, 8)16. (2, 4), (-4, 3), (-2, 9)17. (-4, -14), (4, -10), (-6, -6)18. (-2, -2), (-6, -8)19. (-4, -2), (2, 4), (-6, 2)20. y =−221. y−axis22. y = x
23-25.
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www.ck12.org Chapter 12. Rigid Transformations, Answer Key
26. It is the same as a translation of 8 units down.
27-29.
30. It is the same as a translation of 12 units to the left.
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12.4. Basic Geometry, Rotations, Review Answers www.ck12.org
12.4 Basic Geometry, Rotations, Review An-swers
1. d2. d3. 270◦
4. 90◦
5. 150◦
6. 240◦
7. 20◦
8. Not rotating the figure at all; 0◦
9. They are the same because the direction of the rotation does not matter and it is the same angle measure.10. A protractor only goes to 180◦, so it would make more sense (and easier) to do a rotation of 60◦ clockwise,
even though are supposed to only do counterclockwise rotations. When drawing a figure using a protractor,always perform the rotation (regardless of direction) that is less than 180◦ and then mark it correctly in yourdrawing.
In 11-16, the purple figure is the image.
11.
12.
13.
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www.ck12.org Chapter 12. Rigid Transformations, Answer Key
14.
15.
16.17. (-6, -2)18. (-6, -4)19. (2, -2) and (6, 4)
20.
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www.ck12.org Chapter 12. Rigid Transformations, Answer Key
24.
25.26. x = 327. x = 4.528. x = 2129. 90◦
30. 180◦
31. 270◦
32. 90◦
33. 180◦
34. 180◦
35-37.
129
12.4. Basic Geometry, Rotations, Review Answers www.ck12.org
38. A rotation of 180◦.
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www.ck12.org Chapter 12. Rigid Transformations, Answer Key
12.5 Basic Geometry, Composition of Transfor-mations, Review Answers
1. Every isometry produces a congruent figure to the original. If you compose transformations, each image willstill be congruent to the original.
2. a translation3. a rotation4. (2, -2), (-2, -4), (0, -8), (4, -6)5. (x,y)→ (x+6,−y)6. (x,y)→ (x−6,−y)7. (2, -2), (-2, -4), (0, -8), (4, -6)8. (x,y)→ (x+6,−y)9. No, order does not matter.
10. (-2, -3), (-4, 2), (-9, -3)11. (x,y)→ (−x,y−5)12. (x,y)→ (−x,y+5)13. (2, -10), (10, -6), (8, -4)14. A translation of 12 units down.15. (x,y)→ (x,y+12)16. (2, 14), (10, 18), (8, 20)17. A translation of 12 units up.18. (x,y)→ (x,y+12)19. They are in the opposite direction.20. (4, 3), (8, 2), (10, 6), (2, 8)21. Same as #20.22. We were told that order does not matter. Also, with a rotation of 180◦, it does not matter which direction you
go in, the image will be the same.23. 14 units24. 14 units25. 330◦
26. the point of intersect of the two lines27. 166◦
28. 122◦
29. 49◦
30. 31 units
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12.6. Basic Geometry, Extension: Tessellating Polygons, Review Answers www.ck12.org
12.6 Basic Geometry, Extension: TessellatingPolygons, Review Answers
1.2. A checkerboard3. 3 regular hexagons fit around one point. There are 720◦ in a hexagon, so there are 720◦÷ 6 = 120◦ in each
angle of a regular hexagon. There are 360◦ around a point, so 360◦÷120◦ = 3 hexagons around a point.
4.
5.
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www.ck12.org Chapter 12. Rigid Transformations, Answer Key
12.7 Chapter Review Answers
1. C2. E3. F4. B5. H6. D7. A8. G
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