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Name ___________ Geometry 1 Unit 2: Reasoning and Proof August 25 2.1 Conditional Statements 26 27 2.2 Definitions and Biconditional Statements 2.1 Conditional Statements Page 75 #10 - 16 even, Page 77 #50, Page 78 #62, 64, 72, Pages 76 - 77 #18 74 32 even, 40-44 all, 28 2.3 Deductive Reasoning 29 2.3 Deductive Reasoning Pages 92 - 93 #22 - 30 Pages 82 - 83 14 - 28 even, #34 - 40 even, even, 40, 41, 42 Pages 91 #8 - 14 all, 50 September Monday Tuesday 1 2 NO SCHOOL Review 2.1-2.3 Labor Day ASVAB testing in Sept. Wednesday Page 94 # 56 - 63, Page 95 #1 - 5 8 Thursday 4 Progress Rpts Due 3 2.4 Reasoning and Properties from Algebra Friday 2.5 Proving Statements about Segments 2.5 Proving Statements about Segments Page 99 - 100 #10 16all, page 103 # 40, Page 105-107 #6, 7, 42, 45 16, 19, 31-34 Page 105 #8-11all 9 10 11 2.6 Proving Statements about Angles 2.6 Proving Statements about Angles Page 113 #10- 18 even, Page 114 #20, 22, 27, 28 Page 121 #1 - 19 Review Unit 2 Review Unit 2 Study 5 12 Geometry 1 Chapter 2 Reasoning and Proof Exam 1 Write the directions for making a Peanut Butter and Jelly Sandwich. 2 Geometry 1 2.1: Conditional statements Conditional Statement Unit 2: Reasoning and Proof If-then form Hypothesis Conclusion Examples of Conditional Statements Example 1 Circle the hypothesis and underline the conclusion in each conditional statement Example 2 A line contains at least two points. When two planes intersect their intersection is a line. Two angles that add to 90° are complementary. 3 Label the following statement as true or false. If the statement is false, then give a counterexample. If it is Monday, then it is a school day. Write a true statement. Write a false statement and provide a counterexample to show that it is false. 4 Counterexample Example 3 Determine if the following statements are true or false. If false, give a counterexample. If x + 1 = 0, then x = -1 If a polygon has six sides, then it is a decagon. If the angles are a linear pair, then the sum of the measure of the angles is 90º. Negation Examples of Negations Example 4 Write the negation of each statement. Determine whether your new statement is true or false. Yuma is the largest city in Arizona. All triangles have three sides. Dairy cows are not purple. Some CGUHS students have brown hair. 5 Write the following as an if- then statement, then write the converse, inverse, and contrapositive of the conditional statement. “A line contains at least two points.” Statement: Converse: Inverse: Contrapositive: Is the statement true? Is the converse true? Is the inverse true? Is the contrapositive true? 6 Converse Inverse Contrapositive Examples Write in if…then form. Write the converse, inverse and contrapositive of each statement. I will wash the dishes, if you dry them. If…then form Converse Inverse Contrapositive A square with side length 2 cm has an area of 4 cm2. If…then form Converse Inverse Contrapositive 7 Sketch an example to represent each of the 5 postulates learned in this lesson: 1. 2. 3. 4. 5. 8 Point-line Postulate Point-line converse Intersecting lines postulate Point-plane postulate Point-plane converse Line-plane postulate Intersecting planes postulate 9 Determine whether each statement and its converse are true or false. If the statement and the converse are true, write as a biconditional statement, if false, write a counterexample. 1. If an angle measures 90 degrees, then it is a right angle 2. If it is Saturday, then there is no school 3. If you do your homework, then you will pass the class. 4. If a figures contains at least two points, then the figure is a line 10 Geometry 1 2.2- Biconditional Statements Biconditional Statement Example 1 Unit 2: Reasoning and Proof Determine whether the biconditional is true Perpendicular Lines A line perpendicular to a plane Example 2 Example 3 Example 4 Example 5 11 Use symbols to represent (a) a conditional statement, (b) the converse, (c) the inverse, and (d) the contrapositive. a) b) c) d) Write a statement Similar to example 2 on pg. 13, then make up 3 problems for that statement Statement: Example: Write p ^ q for the given statement 1. 2. 3. 12 Geometry 1 2.3: Deductive Reasoning Symbolic logic Unit 2: Reasoning and Proof ~ Symbols → Example 1 Let p be “the measure of two angles is 180º Let q be “two angles are supplementary p→q q→p Example 2 P: Jen Cleaned her room q: Jen is going to the mall What does p→q mean? What does q→p mean? What does ~q mean? What does p ^ q mean? Example 3 t: Jeff has a math test today s: Jeff Studied t s s→t ~s→t What does ~q mean? 13 If Ana completes all of her homework, then she will go to the movies Ana completed all of her homework What will Ana do now? If Joe wins the football game, he will get a new movie John did not win the football game Will John get a new movie? Law of Syllogism If Derrick cleans his room, he will go to the mall If Derrick goes to the mall, he will get new shoes Derrick cleaned his room, does he get new shoes? Write your own system of expressions using the law of syllogism 14 Deductive Reasoning Law of Detachment Example 4 Determine if the argument is valid Example 5 Determine if the argument is valid Law of Syllogism Example 6 Example 7 15 Write an example to illustrate each of the algebraic properties of equality. Addition Property Subtraction Property Multiplication Property Division Property Reflexive Property Symmetric Property Transitive Property Substitution Property Distributive Property 16 Geometry 1 2.4: Reasoning with properties from Algebra Objectives: Unit 2: Reasoning and Proof 1. 2. 3. Algebraic properties of equality Addition property Subtraction property Multiplication property Division property Reflexive property Symmetric Property Transitive property Substitution Property Distributive Property Example 1 Statement Reason 17 Create an algebraic equation. Then solve the equation step by step and write the reason for each step. Equation: Step 1. Reason 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 7. 7. 18 Example 2: Solve 2(x – 3) = 6x + 6 Statement Valid or invalid algebraic equations Reason Determine if the equations are valid or invalid. (x + 2) (x + 2) = x2 + 4 X3x3 = x6 – (x + y) = x – y Geometric Properties of Equality Reflexive property Symmetric Property Transitive Property Example 3 In the diagram, AB = CD. Show that AC = BD A Statement B Reason C D AB = CD AB + BC = BC + CD AC = AB + BC BD = BC + CD AC = BD 19 Write an example of each of the three properties of segment congruence, and draw an example that illustrates each of these geometric properties of equality 1. 2. 3. 20 Geometry 1 2.5: Proving Statements about line segments Unit 2: Reasoning and Proof 2-column proof Theorem Properties of Segment congruence Reflexive Symmetric Transitive Example 1 Triangle segment proof In Triangle JKL, Given: LK = 5, JK = 5, JK = JL, Prove: LK = JL Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. 5. 5. 6. 6. 21 Draw 2 different size line segments, then label the segments x and y Duplicate each segment Construct a segment that is 2x + y Construct a segment that’s length is the difference of the 2 original segments 22 Duplicate segment AB using construction steps A B C D Create a segment 2AB Create a Segment 3CD - AB 23 In your own words, explain each of the angle theorems or postulates. Draw an example to represent each theorem or postulate: Right Angle Congruence Theorem: Congruent Supplements Theorem: Congruent Complements Theorem: Linear Pair Postulate: Vertical Angles Theorem: 24 Geometry 1 2.6: Proving Statements about angles Properties of Angle Congruence Reflexive Unit 2: Reasoning and Proof Symmetric Transitive Right Angle Congruence Congruent Supplements Congruent Complements Linear Pair Postulate Vertical Angles Theorem 25 Investigating Complementary Angles Activity McDougall Littell Geometry Page 108 26 Example 1 Given: Prove: Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. Example 2 Given: Prove: Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. 27 To close a pair of scissors, you close the handles. Will the angle formed by the blades be the same as the angle formed by the handles? Explain. 28 Example 3 Given: Prove: Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. Example 4 Given: Prove: Statement Reason 1. 1. 2. 2. 3. 3. 4. 4. 29 Make a sketch using the given information. Then state all pairs of congruent angles. Two lines intersect to form angles 1, 2, 3, 4. 1 and 2 are vertical angles. 3 and 4 are vertical and supplementary angles. 2. Solve for each variable. (4b + 43)º (7a + 8)º (8a – 3)º (6b + 17)º 30 Example 5 Example 6 Given: Prove Statement Reason 1. 1. 2. 2. 3. 3. 31