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Warm-Up 1/21/09 “Early to bed and early to rise, makes a man healthy, wealthy and wise.” -Benjamin Franklin Rewrite the statement above: 1. Conditional 2. Inverse 3. Converse 4. Contrapositive The plan •Warm-up •Homework Check •Notes 2.2, 2.3 & possibly 2.4 •Practice •Homework Chapter 2 will have a quiz at the end, not a test 2.1 Conditional Statements Objective: Recognize and analyze a conditional statement and write postulates about points, lines, planes using conditional statements. Conditional Statements • • • • Called if-then statements Have 2 parts Hypothesis- The part after if. Conclusion- The part after then. * Do not include if and then in the hypothesis and conclusion. A compound statement joined with an if…then. p: A figure is a rectangle. q: The diagonals are congruent. If p then q: If a figure is a rectangle, then the diagonals are congruent. Symbol: p q Try These. Identify the Hypothesis and the conclusion. • If it is Saturday, then Elise plays soccer. • Hypothesis- it is Saturday • Conclusion- Elise plays soccer • If points are collinear, then they lie on the same line. • Hypothesis• Conclusion- points are collinear they lie on the same line Negation A statement can be altered by negation by writing the negative of the statement Symbol: ~ Inverse When you negate the hypothesis and conclusion of a conditional statement, you form the inverse. Negating a conditional. p: A figure is a rectangle. q: The diagonals are congruent. If ~p then ~q: If the figure is not a rectangle, then the diagonals are not congruent. Symbol: ~ p ~ q • The converse of a conditional statement swaps the hypothesis and the conclusion. • Conditional- If a figure is a triangle, then it has three angles. • Converse- If a figure has three angles, then it is a triangle. Converse Exchange the p and q of a conditional statement. p: A figure is a rectangle. q: The diagonals are congruent. If q then p: If the diagonals are congruent, then the figure is a rectangle. Symbol: q p * Converses are not always true. • Conditional- If a figure is a square, then it has four sides. • Converse- If a figure has four sides, then it is a square. Contrapositive When you negate the hypothesis and conclusion of the converse of a conditional statement, you form the contrapositive. Negating the converse. p: A figure is a rectangle. q: The diagonals are congruent. If ~q then ~p: If the diagonals are not congruent, then the figure is not a rectangle. For the conditional statement (a) rewrite it in ifthen form, (b) write the inverse, (c) write the converse, and (d) write the contrapositive. I will dry the dishes if you will wash them. (a) If you will wash the dishes, then I will dry them. (b) If you will not wash the dishes, then I will not dry them. (c) If I will dry the dishes, then you will wash them. (d) If I will not dry the dishes, then you will not wash them. Decide whether the statement is true or false. If false, give a counterexample. 1.Through any 3 points there exists exactly one line. FALSE The points could be noncollinear and make no line at all. Biconditional Statement • A biconditional statement is a statement that contains the phrase “if and only if.” Also abbreviated as iff. Writing a biconditional statement is equivalent to writing a conditional statement and its converse at the same time. Rewrite the biconditional statement as a conditional statement and its converse. A number is a perfect square if and only if it is the product of some number times itself. Conditional: If a number is a perfect square, then it is the product of some number times itself. Converse: If a number is the product of some number and itself, then it is a perfect square. Consider the following statement: x2 < 49 iff x<7 A) Is this a biconditional statement? Yes, because it contains “if and only if” B) Is it true? (conditional & converse must be true.) Conditional: If x2 < 49 then x < 7 is true. Converse: If x < 7 then x2 < 49 is not true. (x=-9) So, No. It is not true. Can this true statement be written as a true biconditional statement? If x2 = 4, then x = 2 or –2. 1st: Is the converse true? Yes. If x = 2 or –2, then x2 = 4 2nd: Write as a biconditional statement 2 x = 4, if and only if x = 2 or –2. 2.3 Deductive Reasoning When you walk out the door, you should be able to apply laws of logic to true statements and make valid conclusions. What do we remember about the symbols? Let p be “Today is Wednesday” and q be “There is school. What is p q ? What is ~p ~q ? What is q p ? What is ~q ~p ? Deductive Reasoning Deductive Reasoning uses fact definitions and accepted properties to write logical arguments (proofs). How is this different from Inductive Reasoning? Law of Detachment If a conditional statement is true and the hypothesis is true, then the conclusion is automatically true. pq p is true q is true EX 1: Use the Law of Detachment to determine a conclusion. If a triangle is equilateral, then the measure of each angle is 60. Triangle ABC is an equilateral triangle. The measure of each angle is 60. EX 2: Use the Law of Detachment to determine a conclusion. If two lines are parallel, then the lines do not intersect. k is parallel to m. k and m do not intersect. EX 3: Use the Law of Detachment to determine a conclusion. If Robby is taller than Ben, then Robby is at least 6 feet tall. Ben is older than Robby. NO VALID CONCLUSION! Law of Syllogism Similar to the transitive property in algebra Has 3 statements: 1st and 2nd are true, 2nd and 3rd are true, so 1st and 3rd must also be true. p q is true q r is true p r is true EX 4: Use the Law of Syllogism to determine a conclusion. If Donnie asks Pam, then she will say yes. If she says yes, then they will get married. If Donnie asks Pam, then they will get married. EX:5 Use the Law of Syllogism to determine a conclusion. If Tim gets stung by a bee, then he will get very ill. If he gets very ill, then he will go to the hospital. If Tim gets stung by a bee, then he will go to the hospital. EX 6: Use the Law of Syllogism to determine a conclusion. If Jay doesn’t work hard, then he won’t start the game. If he doesn’t start the game, then he will quit the team. If Jay doesn’t work hard, then he will quit the team. EX 7: Use the Law of Syllogism to determine a conclusion. If Susan screams, then the dog will run away. If the dog licks Susan, then Susan will scream. NO VALID CONCLUSION!! Ex 8: Use the true statements to determine if the conclusion is true or false. • If it looks like rain, then I will bring my umbrella to school with me. • If there are clouds in the sky and the sky is dark, then it looks like rain. • If I bring my umbrella to school with me, then I will hang it in the classroom closet • This morning, there are clouds in the sky and the sky is dark Conclusion: My umbrella is hanging in the classroom closet. Workbook Page 30 You try it 2.4 Reasoning with Algebraic Properties When you walk out the door, you should be able to use properties of algebra to explain the validity of solving a multi-step equation. Let a, b, and c be real numbers Addition Property If a = b, then a + c = b + c add the same thing to both sides of an eqn. & they will still be equal Subtraction Property If a = b, then a - c = b - c subtract the same thing from both sides of an eqn. & they will still be equal Multiplication Property If a = b, then ac = bc multiply the same thing to both sides of an eqn & they will still be equal Division Property If a = b and c 0, then ac= bc divide the same thing from both sides of an eqn & they will still be equal To solve the equation 10y + 5 = 25 ,put these steps in order 10y + 5 - 5 = 25 - 5 y=2 10y = 20 10 y 20 10 10 10y + 5 = 25 Given 10y + 5 - 5 = 25 - 5 Subtraction Prop. 10y = 20 Simplify 10 y 20 10 10 Division Property 10y + 5 = 25 y=2 Simplify Let a, b, and c be real numbers Reflexive Property For any real number a, a = a A number, length, or measure will always equal itself If a = b, then b = a Symmetric Property If two things are equal then order does not matter If a = b and b = c then a = c Transitive Property This is true by the Law of Syllogism Substitution Property If a = b, then a can be substituted for b in any equation or expression Think about substituting 3 in for x in an equation when x = 3 WY = XZ Show that WX = YZ W WY = XZ WY = WX + XY X Y Given Z Segment Addition Postulate Segment Addition Postulate Substitution Prop. Of Equality Subtraction Prop. Of Equality XZ = XY + YZ WX + XY = XY + YZ WX = YZ Banked turns help the cars travel around the track at high speeds. The angles provide an inward force that helps keep the cars from flying off the track. Given the following information about the four banked turns at the Talladega Superspeedway racetrack in Alabama, find the measure of angle 4 m1 m2 66 m1 m2 m3 99 m3 m1 m1 m4 m1 m2 66 m1 m2 m3 99 66 m3 99 m3 33 = m3 m1, m1 m4 = m3 m4 m4 33 Given Given Substitution prop of equality Subtraction prop of equality Given Transitive prop of equality Substitution prop of equality You try it Homework: Pg 82-84 #15-42 (x3 omit 33&36), 54&55 Pg 92-94 #22-30 even, 46-50 even, Page 99 –101 #10-26 even (this is only 27 problems) You have a quiz Friday over 2.1-2.4

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