Unit2ReviewGlencoe.pdf

Vocabulary and Concept Check biconditional (p. 81) counterexample (p. 63) informal proof (p. 90) postulate (p. 89) compound statement (p. 67) deductive argument (p. 94) inverse (p. 77) proof (p. 90) conclusion (p. 75) deductive reasonin...

Vocabulary and Concept Check biconditional (p. 81) counterexample (p. 63) informal proof (p. 90) postulate (p. 89) compound statement (p. 67) deductive argument (p. 94) inverse (p. 77) proof (p. 90) conclusion (p. 75) deductive reasoning (p. 82) Law of Detachment (p. 82) related conditionals (p. 77) conditional statement (p. 75) disjunction (p. 68) Law of Syllogism (p. 83) statement (p. 67) conjecture (p. 62) formal proof (p. 95) logically equivalent (p. 77) theorem (p. 90) conjunction (p. 68) hypothesis (p. 75) matrix logic (p. 88) truth table (p. 70) contrapositive (p. 77) if-then statement (p. 75) negation (p. 67) truth value (p. 67) converse (p. 77) inductive reasoning (p. 62) paragraph proof (p. 90) two-column proof (p. 95) A complete list of postulates and theorems can be found on pages R1–R8. Exercises Choose the correct term to complete each sentence. 1. A (counterexample, conjecture ) is an educated guess based on known information. 2. The truth or falsity of a statement is called its (conclusion, truth value ). 3. Two or more statements can be joined to form a (conditional, compound ) statement. 4. A conjunction is a compound statement formed by joining two or more statements using (or, and ). 5. The phrase immediately following the word if in a conditional statement is called the ( hypothesis , conclusion). 6. The (inverse, converse ) is formed by exchanging the hypothesis and the conclusion. 7. (Theorems, Postulates ) are accepted as true without proof. 8. A paragraph proof is a (an) ( informal proof , formal proof ). 2-1 Inductive Reasoning and Conjecture See pages 62–66. Concept Summary Conjectures are based on observations and patterns. Counterexamples can be used to show that a conjecture is false. Example Given that points P, Q, and R are collinear, determine whether the conjecture that Q is between P and R is true or false. If the conjecture is false, give a counterexample. The figure at the right can be used to disprove the P R Q conjecture. In this case, R is between P and Q. Since we can find a counterexample, the conjecture is false. Exercises Make a conjecture based on the given information. Draw a figure to illustrate your conjecture. See Example 2 on page 63. 9.
A and
B are supplementary. 10. X, Y, and Z are collinear and XY
YZ. 11. In quadrilateral LMNO, LM
LO
MN
NO, and m
L
90. www.geometryonline.com/vocabulary_review Chapter 2 Study Guide and Review 115 7/23/2003 10:45 PM 115-121 GEO C2SGT-829637 Extra Practice, see pages xxx-xxx. Chapter 2 X Study Guide and Review Mixed Problem Solving, see page xxx. 2-2 Logic See pages Concept Summary 67–74. The negation of a statement has the opposite truth value of the original statement. Venn diagrams and truth tables can be used to determine the truth values of statements. Example Use the following statements to write a compound statement for each conjunction. Then find its truth value. p:
155 q: The measure of a right angle equals 90. a. p and q
15 
5, and the measure of a right angle equals 90. p and q is false because p is false and q is true. b. p
q
15 
5, or the measure of a right angle equals 90. p  q is true because q is true. It does not matter that p is false. Exercises Use the following statements to write a compound statement for each conjunction. Then find its truth value. See Examples 1 and 2 on pages 68 and 69. p: 1  0 q: In a right triangle with right angle C, a2
b2  c 2. r: The sum of the measures of two supplementary angles is 180. 12. p and q 13. q or r 14. r  p  15. p  (q r)  16. q (p r) 17. (q  r)  p 2-3 Conditional Statements See pages Concept Summary 75–80. Conditional statements are written in if-then form. Form the converse, inverse, and contrapositive of an if-then statement by using negations and by exchanging the hypothesis and conclusion. Example Identify the hypothesis and conclusion of the statement The intersection of two planes is a line. Then write the statement in if-then form. Hypothesis: two planes intersect Conclusion: their intersection is a line If two planes intersect, then their intersection is a line. Exercises Write the converse, inverse, and contrapositive of each conditional statement. Determine whether each related conditional is true or false. If a statement is false, find a counterexample. See Example 4 on page 77. 18. If an angle measure equals 120, then the angle is obtuse. 19. If the month is March, then it has 31 days. 20. If an ordered pair for a point has 0 for its x-coordinate, then the point lies on the y-axis. 116 Chapter 2 Reasoning and Proof  Chapter 2 Study Guide and Review Determine the truth value of the following statement for each set of conditions. If the temperature is at most 0°C, then water freezes. See Example 3 on page 76. 21. The temperature is 10°C, and water freezes. 22. The temperature is 15°C, and water freezes. 23. The temperature is 2°C, and water does not freeze. 24. The temperature is 30°C, and water does not freeze. 2-4 Deductive Reasoning See pages Concept Summary 82–87. The Law of Detachment and the Law of Syllogism can be used to determine the truth value of a compound statement. Example Use the Law of Syllogism to determine whether a valid conclusion can be reached from the following statements. (1) If a body in our solar system is the Sun, then it is a star. (2) Stars are in constant motion. p: a body in our solar system is the sun q: it is a star r: stars are in constant motion Statement (1): p → q Statement (2): q → r Since the given statements are true, use the Law of Syllogism to conclude p → r. That is, If a body in our solar system is the Sun, then it is in constant motion. Exercises Determine whether the stated conclusion is valid based on the given information. If not, write invalid. Explain your reasoning. See Example 1 on page 82. If two angles are adjacent, then they have a common vertex. 25. Given:
1 and
2 are adjacent angles. Conclusion:
1 and
2 have a common vertex. 26. Given:
3 and
4 have a common vertex. Conclusion:
3 and
4 are adjacent angles. Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not follow, write invalid. See Example 3 on page 83. 27. (1) If a student attends North High School, then the student has an ID number. (2) Josh Michael attends North High School. (3) Josh Michael has an ID number. 28. (1) If a rectangle has four congruent sides, then it is a square. (2) A square has diagonals that are perpendicular. (3) A rectangle has diagonals that are perpendicular. 29. (1) If you like pizza with everything, then you’ll like Cardo’s Pizza. (2) If you like Cardo’s Pizza, then you are a pizza connoisseur. (3) If you like pizza with everything, then you are a pizza connoisseur. Chapter 2 Study Guide and Review 117  Chapter 2 Study Guide and Review 2-5 Postulates and Paragraph Proofs See pages Concept Summary 89–93. Use undefined terms, definitions, postulates, and theorems to prove that statements and conjectures are true. Example Determine whether the following statement is always, sometimes, or never true. Explain. Two points determine a line. According to a postulate relating to points and lines, two points determine a line. Thus, the statement is always true. Exercises Determine whether the following statements are always, sometimes, or never true. Explain. See Example 2 on page 90. 30. The intersection of two lines can be a line. Y 31. If P is the midpoint of X , then XP
PY. 32. If MX
MY, then M is the midpoint of XY. 33. Three points determine a line. 34. Points Q and R lie in at least one plane. 35. If two angles are right angles, they are adjacent. 36. An angle is congruent to itself. 37. Vertical angles are adjacent. 38. PROOF Write a paragraph proof to prove that A Q M B B if M is the midpoint of A and Q is the midpoint 1 , then AQ
AB. M of A 4 2-6 Algebraic Proof See pages Concept Summary 94–100. The properties of equality used in algebra can be applied to the measures of segments and angles to verify and prove statements. 5 Example Given: 2x  6
3   x 3 Prove: x
9 Proof: Statements Reasons 5 1. 2x  6
3  x 1. Given 3 2. 3(2x  6)
33  x 5 2. Multiplication Property 3 3. 6x  18
9  5x 3. Distributive Property 4. 6x  18  5x
9  5x  5x 4. Subtraction Property 5. x  18
9 5. Substitution 6. x  18  18
9  18 6. Subtraction Property 7. x
9 7. Substitution 118 Chapter 2 Reasoning and Proof  Chapter 2 Study Guide and Review Exercises State the property that justifies each statement. See Example 1 on page 94. 39. If 3(x  2)
6, then 3x  6
6. 40. If 10x
20, then x
2. 41. If AB  20
45, then AB
25. 42. If 3
CD and CD
XY, then 3
XY. PROOF Write a two-column proof. See Examples 2 and 4 on pages 95 and 96. 1 43. If 5
2  x, then x
6. 2 x  10 44. If x  1
 , then x
4. 2 45. If AC = AB, AC = 4x  1, and AB
6x  13, then x
7. 46. If MN
PQ and PQ
RS, then MN
RS. 2-7 Proving Segment Relationships See pages Concept Summary 101–106. Use properties of equality and congruence to write proofs involving segments. Example Write a two-column proof. P Q Given: QT
RT, TS
TP Prove: QS
RP T S R Proof: Statements Reasons 1. QT
RT, TS
TP 1. Given 2. QT  TS
RT  TS 2. Addition Property 3. QT  TS
RT  TP 3. Substitution 4. QT  TS
QS, RT  TP
RP 4. Segment Addition Postulate 5. QS
RP 5. Substitution Exercises Justify each statement with a property of equality or a property of congruence. See Example 1 on page 102. 47. PS
PS 48. If XY
OP, then OP
XY. 49. If AB  8
CD  8, then AB
CD. 50. If EF
GH and GH
LM, then EF
LM. 1 51. If 2(XY)
AB, then XY
(AB). 2 52. If AB
CD, then AB  BC
CD  BC. Chapter 2 Study Guide and Review 119  Extra Practice, see pages 756–758. Mixed Problem Solving, see page 783. PROOF Write a two-column proof. See Examples 1 and 2 on pages 102 and 103. 53. Given: BC
EC, CA
CD 54. Given: AB
CD Prove: BA
DE Prove: AC
BD B E A B C D C D A 2-8 Proving Angle Relationships See pages Concept Summary 107–114. The properties of equality and congruence can be applied to angle relationships. Example Find the measure of each numbered angle. 2 1 m
1
55, since
1 is a vertical angle to the 55° angle. 55°
2 and the 55° angle form a linear pair. 55 + m
2
180 Def. of supplementary  m
2
125 Subtract 55 from each side. Exercises Find the measure of each numbered angle. See Example 2 on page 108. 55. m
6 56. m
7 7 157° 6 35° 57. m
8 8 58. PROOF Copy and complete the proof. See Example 3 on page 109. Given:
1 and
2 form a linear pair. m
2
2(m
1) Prove: m
1
60 Proof: Statements Reasons a.
1 and
2 form a linear pair. a. ? b.
1 and
2 are supplementary. b. ? c. ? c. Definition of supplementary angles d. m
2
2(m
1) d. ? e. ? e. Substitution f. ? f. Substitution 3(m
1) 180 g. 
 g. ? 3 3 h. ? h. Substitution 120 Chapter 2 Reasoning and Proof  Vocabulary and Concepts 1. Explain the difference between formal and informal proofs. 2. Explain how you can prove that a conjecture is false. 3. Describe the parts of a two-column proof. Skills and Applications Determine whether each conjecture is true or false. Explain your answer and give a counterexample for any false conjecture. 4. Given:
A 
B 5. Given: y is a real number 6. Given: 3a2
48 Conjecture:
B 
A Conjecture: y  0 Conjecture: a
4 Use the following statements to write a compound statement for each conjunction or disjunction. Then find its truth value. p: 3  2 q: 3x  12 when x  4. r: An equilateral triangle is also equiangular. 7. p and q 8. p or q 9. p  (q  r) Identify the hypothesis and conclusion of each statement and write each statement in if-then form. Then write the converse, inverse, and contrapositive of each conditional. 10. An apple a day keeps the doctor away. 11. A rolling stone gathers no moss. 12. Determine whether statement (3) follows from statements (1) and (2) by the Law of Detachment or the Law of Syllogism. If it does, state which law was used. If it does not, write invalid. (1) Perpendicular lines intersect. (2) Lines m and n are perpendicular. (3) Lines m and n intersect. Find the measure of each numbered angle. 95˚ 13.
1 2 3 14.
2 1 73˚ 15.
3 16.Write a two-column proof. 17. Write a paragraph proof. If y
4x  9 and x
2, then y
17. Given: AM
CN, MB
ND Prove: AB
CD A M B D N C 18. ADVERTISING Identify the hypothesis and conclusion of the following statement, then write it in if-then form. Hard working people deserve a great vacation. 19. STANDARDIZED TEST PRACTICE If two planes intersect, their intersection can be FCAT Practice I a line. II three noncollinear points. III two intersecting lines. A I only B II only C III only D I and II only www.geometryonline.com/chapter_test/fcat Chapter 2 Practice Test 121