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Chapter 2 Student Notes Friday, 2/3/12 Dress for Success for Extra Credit 2.1 Inductive Reasoning and Conjecture Conjecture Make a conjecture from the given statement. Given: The toast is burnt. Conjecture: ___________________________ Given: It is winter. Conjecture: ___________________________ Given: Angle A is a right angle. Conjecture: ___________________________ Counterexample Write a counterexample for each conjecture. Conjecture: The sky is blue. Counterexample: ________________________________ Conjecture: Angle 1 and Angle 2 are congruent. Counterexample: ________________________________ Conjecture: l and m are parallel Counterexample: ________________________________ Determine if each conjecture is true or false. Give a counterexample for any false conjecture. 1. Given: A, B, C are collinear. Conjecture: A, B, C are on the same line. 2. Given: 1 is a right angle. Conjecture: m1 = 90. 3. Given: AB = BC. Conjecture: B is the midpoint of AC. T/F T/F T/F Determine if each conjecture is true or false. Give a counterexample for any false conjecture. 1. Given: The dog is brown. Conjecture: It is a chocolate lab. T/F 2. Given: 3 and 4 form a linear pair. Conjecture: 3 4 T/F 3. Given: 1 and 2 are complementary Conjecture: m1 = 45, m2 = 45. m1 = 48, m2 = 42 T/F 2.2 Logic Statement Truth Value – Negation Compound Statement – Conjunction Symbol for And: Disjunction Symbol for Or: Circle the statement that is true. p: Angle A is a right angle. r: Angle A is an obtuse angle. r: Angle A is an acute angle. p q r 2. p q r > > 1. > > Truth Table Examples of Truth Tables. p ~p T F F T Disjunction p q T T T F F T F F p q p q T T T F F T F F p q > Negation Conjunction 2.3 Conditional Statements Conditional Statement Statement: A right angle has a measure of 90 degrees. If-then: Statement: A car has four wheels. If-then: Statement: A triangle has 3 sides. If-then: Parts of a Conditional Statement Hypothesis Conclusion If it is a car, then it has four wheels. Converse Conditional: If it is a car then it has 4 wheels. Converse: Conditional: If it is a pig, then it can fly. Converse: Conditional: If it is a right angle, then it measure 90. Converse: Inverse Conditional: If it is a car then it has 4 wheels. Inverse: Conditional: If it is a pig, then it can fly. Inverse: Conditional: If it is a right angle, then it measure 90. Inverse: Contrapositive Conditional: If it is a car, then it has 4 wheels. Contrapositive: Conditional: If it is a pig, then it can fly. Contrapositive: Conditional: If it is a right angle, then it measure 90. Contrapositive: Identify the converse, inverse and contrapositive of each conditional statement. Determine if each statement is true or false. T / F If you go to WMHS, then you are a hornet. T / F Converse: __________________________ __________________________ T / F Inverse: __________________________ __________________________ T / F Contrapositive: ______________________ ______________________ Identify the converse, inverse and contrapositive of each conditional statement. Determine if each statement is true or false. T / F If it is a right angle, then it measures 90. T / F Converse: __________________________ __________________________ T / F Inverse: __________________________ __________________________ T / F Contrapositive: ______________________ ______________________ 2.4 Deductive Reasoning Deductive Reasoning - Law of Detachment 1. 2. 3. Law of Syllogism 1. 2. 3. Examples of the Laws of Detachment and Syllogism. Detachment 1. If it is a triangle, then it has 3 sides. Syllogism 1. If it is a Jeep, then it has 4 wheel drive. Determine whether the 3rd statement is valid based on the given information. If not, write invalid. 1. If it is a dog, then it has 4 legs. 2. Rover is a dog. 3. Rover has 4 legs. Is it valid? Does it follow one of our Laws? Determine whether the 3rd statement is valid based on the given information. If not, write invalid. 1. If you are 18 or older, then you are an adult. 2. If you are an adult, then you can vote. 3. If you are 18 or older, then you can vote. Is it valid? Does it follow one of our Laws? Use the Law of Detachment or the Law of Syllogism to determine if a valid conclusion can be reached. If it can, state it and the law used. If not, write no conclusion. 1. If it is a car, then it has 4 wheels. 2. A Ferrari is a car. 3. __________________________ 1. If you go to the store, then you will go to the post office. 2. If you go to the post office, then you will buy stamps. 3. _________________________________________ Use the Law of Detachment or the Law of Syllogism to determine if a valid conclusion can be reached. If it can, state it and the law used. If not, write no conclusion. 1. If you are in college, then you are at least 18. 2. Pete is in college. 3. ______________________________ 1. Right angles are congruent. 2. Angle 1 and Angle 2 are congruent. 3. _______________________________ 2.5 Postulates Postulate – Statement that is accepted without proof. Postulate 2.1 - A B Postulate 2.2 - A P B C Plane P Plane ABC Postulate 2.3 _________________________ _____________________________________ Postulate 2.4 _________________________ _____________________________________ _____________________________________ Postulate 2.5 ________________________ ____________________________________ ____________________________________ ____________________________________ Postulate 2.6 _________________________ _____________________________________ Postulate 2.7 _________________________ _____________________________________ P R Midpoint Theorem - A M B Determine if each statement is always, sometimes or never true. 1. A, B, and C are collinear. 2. A, B, and C, are coplanar. 3. RST is a right angle. 4. Two planes intersect to form a line. 5. If AB = BC, the B is the midpoint of AC. 6. Vertical angles are adjacent. 7. If B is the midpoint of AC, then AB = BC. Determine the number of segments that can be drawn connecting each pair of points. 1. 2. 2.6 Algebraic Proof Properties Reflexive: Symmetric Transitive Substitution Properties Distribution Addition / Subtraction Multiplication / Division Identify each property that justifies each statement. 1. If 7 = x, then x = 7. 2. If x + 5 = 7, then x = 2 3. If x = 7 and 7 = y, then x = y. 4. If m1 + m2 = 180 and m2 = m3, then m1 + m3 = 180. Identify each property that justifies each statement. 1. 2x + 1 = 2x + 1 2. If x – 6 = 7, then x = 13 3. If 2(x + 3) = 7, then 2x + 6 = 7. 4. If 2x = 16, then x = 8. Given: 2x – 5 = 13 Prove: x = 9 Statements Reasons 1. _____________ 1. ___________ 2. _____________ 2. ___________ 3. _____________ 3. ___________ Given: 8 – n = 4 – 2 n 3 Prove: n = 12 Statements Reasons 1. ___________________ 1. ______________ 2. ___________________ 2. ______________ 3. ___________________ 3. ______________ 4. ___________________ 4. ______________ 5. ___________________ 5. ______________ Given: 2x + 1 = 7 3 Statements Prove: x = 10 Reasons 1. 2x + 1 = 7 3 1. Given 2. _________________ 2. _________________ 3. _________________ 3. _________________ 4. _________________ 4. _________________ 2.7 Proving Segment Relationships Ruler Postulate The points on any line or line segment can be _______________________________________________________ ______________________________________ Betweenness of Points A point can only be between two _________ _________________________________ C A B Segment Addition Postulate If B is between A and C, C B A Segment Addition Postulate Statements Make a statement using the previous postulate about each figure. Y M X J R K L XY = JK = T S RS = Theorem 2.2 Reflexive Symmetric Transitive Given: AB XY, AC XZ Prove: BC YZ A B Z Y Reasons Statements 1. AB XY, AC XZ 2. ______________ 3. ______________ 1. ______________ 2. ______________ 3. ______________ 4. 5. 6. 7. 4. 5. 6. 7. ______________ ______________ ______________ ______________ C ______________ ______________ ______________ ______________ X O Given: MO PO, MN PR Prove: NO RO R N Statements M Reasons 1. MO PO, MN PR 2. ________________ 3. ________________ 1. Given 2. ______________ 3. ______________ 4. 5. 6. 7. 4. 5. 6. 7. ________________ ________________ ________________ ________________ ______________ _____________ ______________ ______________ P 2.8 Proving Angle Relationships Angle Addition Postulate If R is in the interior of PQS, then P OR Q 1 R 2 mPQS = S Make an angle addition postulate statement about each figure. P mMNO = O M N K J M L mJKL = Supplement Theorem m1 + m2 = ____o Complement Theorem 3 4 1 2 If the noncommon sides of two adjacent angles form a right angle, then the angles are complementary. m3 + m4 = ____o Theorem 2.5 Reflexive Symmetric Transitive Theorem 2.6 2 3 1 If m1 + m2 = 180o, Theorem 2.7 3 2 1 If m1 + m2 = 90o, Vertical Angles Theorem 1 4 2 3 Abbreviation: Theorem 2.9 Theorem 2.10 Theorem 2.13 Given: MNO RST, MNQ RSP Prove: QNO PST Q O M Statements N S R T P Reasons 1. MNQ RSP, MNQ RSP 1. Given 2. _________________________________ 2. _______________ 3. _________________________________ 3. _______________ 4. _________________________________ 4. _______________ 5. _________________________________ 5. _______________ 6. _________________________________ 6. _______________ 7. _________________________________ 7. _______________ Find the measure of each numbered angle. 2. 3 and 4 are complementary, m4 = 48. 1. m1 = 72 2 1 3 5 4 m2 = m3 = m5 = Find the measure of each numbered angle. 3. m6 = x – 5, m7 = 2x – 4 6 7 m6 = m7 = Find the measure of each numbered angle. 4. m8 = 52. 9 8 m9 = Find the measure of each numbered angle. 5. 10 and 11 are complementary. 13 11, m12= 38. m13 = 12 13 m11 = 10 11 m10 =