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Geometry Chapter 2: Reasoning and Proof 2.1 Inductive objectives: -make conjectures based on inductive reasoning Reasoning -find counterexamples and __________________________________________________________________ Conjecture Answer question on transparency. Show work here and circle your final answer: Bell Ringer: __________________________________________________________________ conjecture inductive reasoning counterexample Make a conjecture about the next item in the sequence. Examples Text Page 64 Determine whether each conjecture is true or false. Give a counterexample for any false conjecture. Assignment 2.1 page 64 #12- 36 even (omit#18),48,53, ______________________________________________________________________________ 59 Summary: How do you prove that a conjecture is false? Provide one example. Honors: Additionally, #38-40, 43 1 2.3 Conditional Statements Bell Ringer objective: -analyze statements in if-then form -write the converse, inverse, and contrapositive of if-then statements _____________________________________________________________________________ Refer to the transparency (show your work and circle your answer: _____________________________________________________________________________ conditional statement hypothesis conclusion Examples Write each statement in the if-then form. Identify the hypothesis and conclusion. a. A 32-ounce pitcher holds a quart of liquid b. The sum of the measures of supplementary angles is 180. c. An angle formed by perpendicular lines is a right angle Other Types of statements Summary: Statement Formed by Given hypothesis and conclusion Exchanging the Converse hyp. and concl.of the conditional Negating both the Inverse hypo. and concl. of the conditional Contrapositive Negating both the hyp. and concl. of the converse Conditional Examples Truth Value If an animal is a dog, then it is a mammal If an animal is a mammal, then it is a dog. If an animal is not a dog, then it is not a mammal If an animal is not a mammal, then it is not a dog Assignment 2.3 page 78 #18-44 even, 61-62 Honors: Additionally, 46,47,50 2 2.4 Deductive objective: Reasoning -use deductive reasoning to draw conclusions _____________________________________________________________________________ Bell Ringer Refer to the transparency. Show work here and circle your answer: _____________________________________________________________________________ Deductive reasoning the process of using facts, rules, definitions, or properties to reach conclusions. Law of If p q is true and p is true, then q is true. Detachment Determine whether each conclusion is valid based on the true conditional given. If not, Examples write invalid. Explain your reasoning. If two angles are complementary to the same angle, then the angles are congruent. 1. Given: A and C are complementary to B. Conclusion: A is congruent to C. 2. Given: A C Conclusion: A and C are complements of B. 3. Given: E and F are complementary to G. Conclusion: E and F are vertical angles. Law of If p q is true and q r is true, then p r is also true. Syllogism Determine whether you can reach a valid conclusion from each set of statements. Examples 1. If a dog eats Superdog Dog Food, he will be happy. Rover is happy. 2. If an angle is supplementary to an obtuse angle, then it is acute. If an angle is acute, then its measure is less than 90. 3. If the measure of A is less than 90, then A is acute. If A is acute, then A B. _____________________________________________________________________________ Summary: What did I learn today? (2 or more sentences) Assignment 2.4 Pg 85 12-26 even 3 2.5 Postulates & Paragraph Proofs Objective: -identify and use basic postulates about points, lines and planes _____________________________________________________________________________ Refer to the transparency. Show work here and circle your answer: Bell Ringer: _____________________________________________________________________________ Postulate a statement that is accepted as true. (axiom) Postulate 2.1: Through any two points, there is exactly one line. Postulate 2.2: Through any three points not on the same line, there is exactly one plane. Postulate 2.3: A line contains at least two points. Postulate 2.4: A plane contains at least three points not on the same line. Postulate 2.5: If two points lie in a plane, then the line containing those points lies in the plane. Postulate 2.6: If two lines intersect, then their intersection is exactly one point. Postulate 2.7: If two planes intersect, then their intersection is a line. Examples Theorem A statement that can be proved true Midpoint Theorem If M is the midpoint of AB , then AM MB. Summary: Describe the difference between a postulate and a theorem. Assignment 2.5 page 92 #16-27 Honors: #28 too 4 2.6. Algebraic Proofs Objective: -Use algebra to write two-column proofs Bell Ringer (To be collected): Please show your work on a sheet of looseleaf. Reflexive For every number a, a = a. property Symmetric For all numbers a and b, if a = b then b = a. Property Transitive For all numbers a, b, and c, if a= b and b = c then a = c. property Addition & For all numbers a, b, and c, if a = b then a + c = b + c and a - c = b - c. Subtraction Property Multiplication / For all numbers a, b, and c, if a = b then a • c = b• c, and if c ≠ 0 then Division a/c = b/c. Property Substitution For all numbers a and b, if a = b then a may be replaced by b in any equation Property or expression. Distributive Property Reflexive Property For all numbers a, b, and c, a(b + c) = ab + ac. Segments AB = AB Angles m1 = m1 Symmetric Property If AB = CD, then CD = AB If m1 = m2, then m2 = m1 Transitive If AB = CD and CD = EF, Property then AB = EF If m1 = m2 and m2 = m3, then m1 = m3 Two-Column Contains statements and reasons organized in two columns. Each step is called a statement and Proof the properties, definitions, postulates, or theorems that justify each step are called reasons. 5 Examples Complete each proof. Assignment 2.6 Part 1, page 97 #14-23 all Assignment 2.6 Part 2, page 98 #24-27, 37, 38 Summary (below): Describe a two column proof in your own words. 6 Objectives: 2.7 Proving Segment Relationships _____________________________________________________________________________ Refer to the transparency. Show work here and circle your answer: Bell Ringer _____________________________________________________________________________ Segment addition B is between A and C if and only if AB+ BC = AC. Postulate Congruence of segments is reflexive, symmetric, and transitive. AB AB Symmetric Property: If AB CD, then CD AB. Transitive Property: If AB CD and CD EF, then AB EF. Theorem 2.2 Reflexive Property: Examples (page 103 #4-7) Assignment 2.7 page 104 #12-21 all, 29, 30 Honors: Additionally, 22,23, & 26 Summary (on back of notes): Describe the theorems that relate to congruence of segments. 7 Objective: 2.8 Proving -Write proofs involving supplementary and complementary angles Angle Relationships -Write proofs involving congruent and right angles _____________________________________________________________________________ Bell Ringer: Refer to the transparency. Show work here and circle your answer: _____________________________________________________________________________ Angle Addition R is in the interior of PQS if and only if Postulate mPQR + mRQS = mPQS. Supplement If two angles form a linear pair, then they are supplementary angles. If 1 and 2 form a linear pair, then m1 + m2 = 180. Theorem Complement If the non-common sides of two adjacent angles form a right angle, Theorem then the angles are complementary angles. If GF GH, then m3 + m4 = 90. Vertical Angle If two angles are vertical angles, then they are congruent. Theorem m6 m7 Examples (See Wkst 2.8 Skills Practice) Assignment page 112 #1624, 27-32, 38, 39 Summary: What did I learn today? What do I need help with? 8