Reasoning and Proof Review Questions

Transcription

1 1 Reasoning and Proof Review Questions Inductive Reasoning from Patterns 1. What is the next term in the pattern: 1, 4, 9, 16, 25, 36, 49...? (a) 81 (b) 64 (c) 121 (d) What is the next term in the pattern 2, 6, 18, 54, 162? (a) 486 (b) 324 (c) Complete the sequence: 1, -2, 4, (a) 32 (b) -16 (c) 16 (d) What is the next number in the sequence: 243, 81, 27, 9... (a) 0 (b) 3 (c) 1 (d) 1/3 5. Examples of inductive reasoning include. (a) Visual patterns (b) Number patterns (c) Both A, B 6. Deductive reasoning is the process of drawing conclusions based on examples and patterns. ( True/False ) 7. Inductive reasoning differs from deductive reasoning in that a general conclusion is drawn by observing specific examples. ( True/False ) 8. Inductive reasoning is the opposite of induction or inductive logic. ( True/False )

2 2 Conjectures and Counterexamples 1. are conjectures disproven through counterexample. (a) Pseudo conjectures (b) False conjectures (c) Proxy conjectures 2. An example that disproves a conjecture is a (a) A counterexample (b) Pseudo conjectures (c) Proxy conjectures 3. Find the best counterexample from the given options to show that the following conjecture is false: the difference of two integers is less than either integer. (a) Difference between 7 and 8 (b) Difference between 9 and 2 (c) Difference between 3 and 4 4. Which of the following proves that a conjecture is true? (a) Deductive reasoning (b) Counterexample (c) Both A, B 5. All shapes with four sides of the same length are squares. ( True/False ) 6. The counterexample to the statement, "All students are lazy," is, "Any hard working student." ( True/False ) 7. If the product of two numbers is even, then the two numbers must be even. The counterexample to this statement are 7 and 4. ( True/False ) 8. 2 is a counterexample to the statement, "All prime numbers are odd numbers." ( True/False ) Deductive Reasoning 1. Determine the right conclusion from the following statements: I. A shape that has more than 2 sides is a polygon. II. A regular polygon s sides and angles are all congruent. III. An equilateral triangle has 3 congruent sides and 3 congruent angles.

3 3 (a) All triangles are polygons. (b) An equilateral triangle is a regular polygon. (c) A rectangle with sides 2, 2, 4, and 4 is not a regular polygon. (d) All of the above can be concluded. 2. Using the two equations, determine the relationship between p and q. 2p = p + 3 II. Q 5 2 = 1. Therefore: (a) (b) (c) (d) q > p p > q p = q None 3. Which of the following laws would be used to draw a logical conclusion from the statements: If you are not in London, then you can t be on the bridge. Ben is in London. (a) Law of Contrapositive (b) Law of Detachment (c) Both A, B 4. Under which of the following laws is a conclusion reached by combining the hypothesis of one statement with the conclusion of another? (a) Law of detachment (b) Law of syllogism (c) Law of induction 5. Determine the conclusion from the following statements: If it is snowing, then I wear a cap. I am not wearing a cap. (a) It is snowing (b) It is not snowing (c) It is winter 6. The difference between inductive and deductive arguments lies in the strength of the evidence that the premises do not provide for the conclusion, and have nothing to do with the content or subject matter of the argument. ( True/False ) 7. P, Q, and R, are collinear points. Q is equidistant from P and R. Therefore, Q must be the midpoint between P and R. ( True/False ) 8. In deductive reasoning, a specific conclusion can be drawn from a general truth. ( True/False )

4 4 If-Then Statements 1. If a = b and b = c, then (a) (b) (c) a > c a < c a = c (d) None o f the above 2. Two lines are perpendicular lines if they intersect to form a. (a) Obtuse angle (b) Right Angle (c) Supplementary angle 3. A conclusion is the result of a(n). (a) Hypothesis (b) Premise (c) Argument 4. If x = 10, then 10x is 100. (a) TRUE (b) FALSE (c) You should meditate 5. If a rectangle has four equal side lengths, then it is a. (a) Square (b) Rectangle (c) Polygon 6. Give the hypothesis and conclusion of the following statement: If x = 29, then x 9 = 20. (a) No conclusion (b) The statement is ambiguous. (c) Hypothesis: x = 29; Conclusion: x - 9 = If x = 4, then 3x = 12. ( True/False ) 8. A triangle has three sides. ( True/False )

5 5 Converse, Inverse, and Contrapositive 1. Consider the statement, "If two angles are congruent, then they have the same measure". What is the converse of the statement? (a) If two angles have the same measure, then they are congruent. (b) If two angles are not congruent, then they do not have the same measure. (c) If two angles do not have the same measure, then they are not congruent. 2. In the statement, "If it rains, then they will cancel school",[u+009d] what is the hypothesis? (a) They will cancel School (b) It rains (c) They will 3. Consider the statement, "If it rains, then I do not go outside." Which of the following could be a logical conclusion that we can draw from this statement? (a) If it does not rain, then I can go fishing (b) If I do not go fishing, then it rains (c) If I go fishing, then it does not rain 4. For any statement where A implies B, not-b implies not-a. (a) Always (b) Not always (c) Sometimes 5. Consider the statement, "All blue objects have color." Its inverse is. (a) If an object has color, then it is blue (b) There exists a blue object that does not have the properties of color (c) If an object is not blue, then it does not have color 6. "If it is cold, then I will wear my woolen clothes." The converse of the preceding statement is, "If I wear my woolen clothes, then it is not cold." ( True/False ) 7. "If something is a bat, then it is a mammal." The contrapositive to this statement is, "If something is a mammal, then it is not a bat." ( True/False ) 8. The converse and inverse of a statement are always true. ( True/False )

6 6 Properties of Equality and Congruence 1. If DE = ED then it is a. (a) Reflexive property of equality (b) Symmetric property of equality (c) Addition Property of equality 2. According to the Right Angle Congruence Theorem, all angles are congruent. (a) Right (b) Acute (c) Obtuse 3. If 3b = 18, then 3b 3 = 18 3 or b = 6. What property makes the statement true? (a) Division Property of Equality (b) Substitution property of equality (c) Symmetric property of equality (d) Addition Property of equality 4. If two angles form a linear pair, then they are: (a) Right angles (b) Obtuse angles (c) Supplementery angles 5. In a triangle, the largest angle is across from the. (a) Shortest side (b) Longest side (c) Shortest angle 6. The sum of interior angles of triangle is. (a) 90 degrees (b) 45 degrees (c) 180 degrees 7. If a = b, then b = a is a symmetric property. ( True/False ) 8. If a quadrilateral is a parallelogram, then the opposite sides are perpendicular. ( True/False )

7 7 Two-Column Proofs 1. We need to explicitly state that BD = BD. This idea that something is congruent to itself is called the: (a) Reflexive property of equality (b) Symmetric property of equality (c) Addition Property of equality (d) Transitive property 2. In higher-level mathematics, are usually written in paragraph form. (a) Proofs (b) Hypothesis (c) Axioms 3. If 3x + 4 = 16 then x =. (a) 12 (b) 4 (c) 13 (d) 3 4. In parallelogram ABCD, angle ABD is congruent to angle BDC because if we have parallel lines and a transversal, we know that: (a) Alternate interior angles are congruent (b) same-side interior angles are congruent (c) corresponding angles are congruent (d) exterior angles are congruent 5. The Vertical Angles Theorem states that vertical angles are congruent. ( True/False ) 6. A statement that is proved is often called a hypothesis. ( True/False ) 7. If a = b, and a + b = 10, and a = 5, then b = 10. ( True/False ) 8. To clarify and emphasize the effectiveness of an argument in geometry, a two-column approach has a list of statements and the corresponding reasons supporting the truth of these statements is often adapted. ( True/False )