Set Operations and Venn Diagram Interpretation

CHAPTERS 1-3 {1, 9} {0, 1, 2, 4, 6, 8, 9} point: 0 line: 1 FIRST QUARTER EXAM 1. Given D 5 {1, 3, 5, 7, 9}, S 5 {1, 4, 9}, and U 5 {x u x is a digit}, find the result of each set operation. Then shade the Venn diagram to illustrate the result. [1.1] a. D > S D S D S U b. S < D9 U 2. Name the three undefined terms of geometry and state the number of dimensions exhibited by each term. [1.2] plane: 2 skew line 3. Fill in the blanks to complete each definition, postulate, or theorem. [1.2, 1.3] a. Lines that are not coplanar are ___ lines. b. Any two points in space lie in exactly one ___. point c. If two lines intersect, then their intersection is exactly one ___. plane d. A line and a point not on that line are contained in one and only one ___. 4. Match each statement to the described characteristic of a postulate system. [1.3] II I. consistent II. independent III. complete a. No postulate can be deduced or proved from the other postulates. I b. Postulates do not contradict one another. III c. Every statement expressed within the system can be proved or disproved from the postulates. I d. Every useful postulate system in mathematics is ___. 5. State the result of each set operation. [1.4] A [ a. h1 > h2 5 ___ p b. h1 < h2 < AB 5 ___ 4 AC S S S S B C h2 c. AB > CA 5 ___ AB d. AB < BC 5 ___ S or AC 6 S p h1 6. State the number of sides of each polygon and the number of faces of each polyhedron. [1.6, 1.7] a. hexagon 5 b. pentagon 4 c. tetrahedron 12 d. dodecahedron (48 2 12p) cm < 10.3 cm 7. Find the difference between the perimeter of the square and the circumference of its inscribed circle. [1.6] p 5 4(12) 5 48; c 5 2p(6) 5 12p [ (48 2 12p) cm < 10.3 cm 6 cm 8. Sketch, draw, and construct a right angle. Explain how the three figures differ from each other. [1.8] Sketch Drawing Construction The sketch is a freehand picture made only with a writing instrument; the drawing is an accurate figure made with any available tools; and the construction is a figure made with the aid of only an unmarked straightedge and a compass. Inductive deductive inductive deductive 9. Fill in the blanks. [2.1] a. ___ reasoning reaches a general conclusion by finding a pattern in several specific examples, but ___ reasoning applies given premises to reach a specific conclusion. b. While ___ reasoning is frequently used to discover a relationship and make a conjecture, ___ reasoning is used to prove the relationship. 10. Use the following to identify each statement. [2.2, 2.3] p: Bob is tall. q: Mary is short. III a. p ~ q I b. conjunction VI c. p S , q III. Bob is tall or Mary is short. II d. biconditional IV. If Bob is tall, then Mary is short. I. Bob is tall and Mary is short. II. Bob is tall if and only if Mary is short. V. If Mary is short, then Bob is not tall. VI. none of the above 11. Complete the truth table. [2.2, 2.3] p q T T F F T F T F q p p q pSq p4q T T T T T T F F F F F T T F F T 12. State whether the following conditional is true or false; then write its contrapositive and the equivalent disjunction of the original conditional and state whether each is true or false. [2.3] true If a figure is a circle, then it is a curve. true contrapositive: If a figure is not a curve, then it is not a circle. true equivalent disjunction: A figure is not a circle or it is a curve. I III II IV 13. Identify each argument or fallacy given the premise “If I study, then I learn.” [2.4] a. I am studying. Therefore, I am learning. I. modus ponens b. I am learning. Therefore, I am studying. II. modus tollens c. I am not learning. Therefore, I am not studying. III. assuming the converse d. I am not studying. Therefore, I am not learning. IV. assuming the inverse V. none of the above V; S 14. State whether each argument is valid (V) or invalid (I); then classify it as sound (S) or unsound (U). [2.4] a. All dogs are mammals. Fido is a dog. Therefore, Fido is a mammal. I; U b. All dogs are mammals. Fido is a mammal. Therefore, Fido is a dog. V; U c. All dogs are horses. Fido is a dog. Therefore, Fido is a horse. I; U d. All dogs are horses. Fido is a horse. Therefore, Fido is a dog. 15. Supply each missing statement or reason. [1.5, 2.6, 2.7] a. 1. given 1. AB 5 XY b. c. d. 2. AB > XY 2. definition of congruent segments 1. /X > /Y 1. given 2. m/X 5 m/Y 2. definition of congruent angles 1. /J and /K are right angles. 1. given 2. /J > /K 2. All right angles are congruent. 1. Point M is in the interior of /KLN. 2. m/KLM 1 m/MLN 5 m/KLN 1. given 2. Angle Addition Postulate 16. Complete the proof. If X-Y-Z and XZ 5 24, then a 5 8. [2.5, 2.6] 3a 2 7 X Z Y Statements 68 15 2 a Reasons 1. X-Y-Z; XZ 5 24; XY 5 3a 2 7; YZ 5 15 2 a 1. given 2. XY 1 YZ 5 XZ 2. definition of between 3. 3a 2 7 1 15 2 a 5 24 3. substitution 4. 2a 1 8 5 24 4. simplify 5. 2a 5 16 5. Addition Property of Equality 6. a 5 8 6. Multiplication Property of Equality 17. Find the coordinate of V if W is the midpoint of UV. [1.5, 2.8] U 220 214 u1v 2 5w 214 1 v 5 27 2 214 1 v 5 54 v 5 68 W 0 20 27 40 18. Complete the proof of Theorem 2.7.7: Supplements of congruent angles are congruent. [2.7] Given: /1 > /2; /3 is a supplement of /1; /4 is a supplement of /2. Prove: /3 > /4 3 Statements 1 4 2 Reasons 1. /1 > /2; /3 is a supplement of /1; /4 is a supplement of /2. 1. given 2. m/1 1 m/3 5 1808; m/2 1 m/4 5 1808 2. definition of supplementary 3. m/1 1 m/3 5 m/2 1 m/4 3. Transitive Property of Equality (or substitution) 4. m/1 5 m/2 4. definition of congruent angles 5. m/3 5 m/4 5. Addition Property of Equality 6. /3 > /4 6. definition of congruent angles 19. Complete the following postulates and theorems. [3.1, 3.2, 3.4] a. If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the second line also. b. If two lines are parallel to the same line, they are parallel to each other. c. Nonvertical lines are parallel if and only if they have equal slopes. d. Nonvertical and nonhorizontal lines are perpendicular if and only if the product of their slopes is 21. t Use the figure for questions 20–21. alternate exterior 20. Classify each pair of angles. [3.1] a. /1 and /8 alternate interior b. /4 and /5 consecutive interior c. /3 and /5 corresponding d. /2 and /6 798 m n 21. Find m/5 if m/6 5 (4x 1 1)8 and m/4 5 (3x 1 4)8. [3.1] m/6 1 m/4 5 180 (4x 1 1) 1 (3x 1 4) 5 180 7x 1 5 5 180 7x 5 175 x 5 25 m/5 5 m/4 5 [3(25) 1 4] 5 798 7 5 3 1 4 2 8 6 22. Given: a i b, /1 > /15 Prove: c i d [3.2] b 1 2 4 3 a 5 6 8 7 14 13 15 16 Statements c 10 9 11 12 d Reasons 1. a i b; /1 > /15 1. given 2. /1 > /7 2. Parallel lines imply congruent alternate interior angles. 3. /15 > /7 3. Transitive Property of Congruent Angles 4. c i d 4. Congruent corresponding angles imply parallel lines. 13 (14, 0.5) 23. Given RS with R (20, 22) and S (8, 3), find each of the following. [3.4] a. RS " 1 8 2 20 2 2 1 1 3 1 2 2 2 5 " 1 2 12 2 2 1 52 5 13 b. the coordinates of the midpoint of RS Q y 5 232x 2 1 20 1 8 22 1 3 1 R 5 Q14, 2 R 5 (14, 0.5) 2 , 2 24. Given X (24, 5) and Y (2, 24), find each of the following. [3.5] 4 a. the slope-intercept form of the equation of XY m 5 24 12 5 2 5 232 2 2 24 y 5 mx 1 b: 5 5 232(24) 1 b 5561b 21 5 b y 5 232x 21 y 5 23x 2 14 b. the slope-intercept4form of the equation of the line through Z (3, 212) and perpendicular to XY m' 5 2 1 3 Q22 R 2 53 y 5 mx 1 b 212 5 23(3) 1 b 212 5 2 1 b 214 5 b y 5 23x 214 25. Construct each of the following. a. the bisector of /FGH [2.8] F G H b. the line parallel to l through P [3.3] P l