1. point, line, and plane 2a. always 2b. always 2c. sometimes 2d. always a. True 7b. True 7c. True 7d. True 7e. True 8.
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1 1. point, line, and plane 2a. always 2b. always 2c. sometimes 2d. always a. True 7b. True 7c. True 7d. True 7e. True 8. 3 and a 4, c a. EG FH 17b o o 21a. 5 21b. 30 o MPS
2 22. x 20, y F x, y x, y 24. F x, y x, y 25. F x, y y, x 26. F x, y x, y 27. F x, y x 5, y F x, y 6 x, y a. 30b. y x 30c. 30d. 3, 2, 3, 2,, 4 P x y x y The reflection across the y-axis makes the x-coordinate the opposite, while the translation downward subtracts 4 from the y- coordinate. MPS
3 31a. warm blooded dog 31b. If an animal is warm-blooded, then the animal is a dog. 31c. If an animal is not a dog, then the animal is not warm-blooded. 31d. If an animal is not warm-blooded, then the animal is not a dog. 31e. The contrapositive 32a. If I am a teenager, then I am between 13 and 19 years old, inclusive. 32b. If I am not between the ages of 13 and 19 years old, inclusive, then I am not a teenager. 32c. If I am not a teenager, then I am not between the ages of 13 and 19, inclusive. 32d. ll three statements, converse, inverse, and contrapositive are all true. 33a. You can vote You are 19 yrs old 33b. If X represents a voter, then the X is inside the box. It may or may not be in the oval, so the statement is not necessarily true. 34a. q,r hris buys gas. hris drives to Rockville 34b. r hris drives to Rockville 34c. no conclusion 34d. p hris does not earn $20 34e. no conclusion 35. Triangle is equiangular. 36. Sally does not study for the test. 37. hris will go to the game, and bring Jane. 38. The far-right columns of the two truth tables have identical truth-values. MPS
4 P Q P P Q P Q P Q T T F T T T T F F F F T F T T T F T F F T T F F P Q Q P Q P Q Q P Q Q P T T F F T T T F T T T T F T F F T F F F T F F T indirect I did not earn $20 this week. 45. Inductive reasoning 46. Inductive reasoning 47. eductive reasoning 48. eductive reasoning 49. Inductive reasoning 50. If it is sunny outside today, I will go to the store If I go to the store, I will buy candy If I buy candy, I will not eat my dinner. 51. If I do my homework, then I will do well on my next test. If I do well on my next test, then I will get a reward. If I get a reward, then I will buy a car. If I buy a new car, then it will have four-wheel drive. 52a. 4 52b. 8 52c. an infinite number MPS
5 53. harlie is correct. The basic rotational symmetry is 360 degrees divided by the number of sides , so any multiple of 60 degrees will work Property Parallelogram Rectangle Square Rhombus Trapezoid Opposite sides congruent x x x x Only one pair of opposite sides parallel x Opposite angles congruent x x x x Each diagonal forms 2 congruent triangles x x x x iagonals bisect each other x x x x iagonals congruent x x iagonals perpendicular x x diagonal bisects two angles x x ll angles are right angles x x ll sides are congruent x x x 20, y a. lines n and p. orresponding angles are congruent. 58b. lines l and m. lternate interior angles are congruent. 58c. lines l and m. Same side adjacent interior angles are supplementary o o o sides sides 66. 4,5,6,7,8,9,10,11,12,13,14 MPS
6 67. y o 69. x x y x The quadrilateral is a parallelogram. The slopes of & equal 2, so. The slopes of & equal 3, so. The quadrilateral is not a rectangle or a square, since the slopes of & are not opposite reciprocals (do not have a product of 1). The quadrilateral is not a rhombus since the slopes of the diagonals are not 1 opposite reciprocals. m, m a. 3 parallelograms 72b. 1, 1, 3, 3, and ( 5, 1) 73. The triangle is a right triangle ,13 75a. SS 75b. cannot be proven congruent 75c. S 75d. SSS 75e. cannot be proven congruent 75f. S 2 7 m, m. So. 7 2 MPS
7 76. two-column proof is given. paragraph or flowchart proof is also acceptable. Statements Reasons 1. is the perp. bisector of 1. Given 2. E and E are right angles 2. efinition of perpendicular 3. E E 3. ll right angles are congruent 4. E E 4. efinition of bisector 5. E E 5. Reflexive Property of ongruence 6. E E 6. SS PT column proof is given. paragraph or flowchart proof is also acceptable. Statement Reasons 1. E E 1. Given 2. E E 2. In a triangle, if two angles are congruent, the corresponding sides opposite those angles are also congruent. 3. E 3. Given 4. E is a right angle 4. efinition of perpendicular 5. E 5. Given 6. E is a right angle 6. efinition of perpendicular 7. E E 7. ll right angles are congruent 8. E E 8. Given 9. E E 9. S 10. E E 10. PT column proof is given. paragraph or flowchart proof is also acceptable. Statements Reasons 1. EG 1. Given 2. F FG 2. If 2 lines are cut by a transversal, alternate interior angles are congruent. 3. FG 3. Given 4. F 4. Reflexive property of congruence 5. F GF 5. SS 6. F GF 6. PT lternative proof: Given that EGand FG, then one pair of opposite sides of quadrilateral FG is parallel and congruent. Therefore FG is a parallelogram. Since opposite angles of a parallelogram are congruent, then F GF. MPS
8 79a. parallelogram 79b. rhombus 79c. rectangle 79d. none of the figures 80a. E Justification: ongruent circles were constructed with centers at points and. Since radii of congruent circles are congruent, ; therefore is a rhombus. In the rhombus, the diagonals are perpendicular, therefore. Since is a parallelogram the diagonals bisect each other. Therefore E E, so is the perpendicular bisector of. 80b. Justification: since they are the radii of the same circle. since they are constructed using the same compass setting. by the reflexive property of congruence. Therefore by SSS. by PT, and by the definition of angle bisector bisects. MPS
9 80c. E F Justification: E, F since they were drawn by the same compass setting. by SSS. Therefore EF EF since they were drawn with the same compass setting. Therefore EF by PT. Finally by the converse of the corresponding angles postulate, F //. 80d. Justification. I drew segments between and and and. I constructed the perpendicular bisector of. Every point on that line is equidistant from points and. I constructed the perpendicular bisector of. Every point on that line is equidistant from points and. Therefore, point, the intersection of those two perpendicular bisectors, is equidistant from points,, and MPS
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