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Chapter 2

Chapter 2. Reasoning and Proof. 2.1 Conditional Statements. If – then statement. Example: If it rains tonight, then I am going to the mall. Hypothesis. Conclusion. State the hypothesis and conclusion: If all angles of a triangle are congruent, then it is equiangular.

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Chapter 2

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  1. Chapter 2 Reasoning and Proof

  2. 2.1 Conditional Statements If – then statement Example: If it rains tonight, then I am going to the mall. Hypothesis Conclusion State the hypothesis and conclusion: If all angles of a triangle are congruent, then it is equiangular.

  3. How to write a conditional statement Identify the hypothesis Identify the conclusion If hypothesis then conclusion Example: A square has four congruent sides. Hypothesis: It is a square Conclusion: There are four congruent sides. Conditional: If it is a square, then it has four congruent sides.

  4. Write a conditional statement for each. On Halloween, I Trick-or-Treat. There is no practice when it rains. Rectangles have four right angles. Parallel planes do not intersect.

  5. Truth value True: every time the hypothesis is true, the conclusion is true False: find one counterexample Examples: If it is a right triangle, then it has one right angle. If it is a rectangle, then it is a square.

  6. Converse – switch the hypothesis and conclusion Example: If it rains tonight, then I am going to the mall. Hypothesis Conclusion Converse: If I am going to the mall, then it rains tonight.

  7. Write a converse to each of the following conditionals: If it is a pentagon then it is a five-sided polygon. If two angles are complementary, then they sum to 90º. On Thursday there are no activity buses.

  8. DOGS poodle ... a dog ...A horse is not a dog .poodle horse ... NOT a dog . horse Venn diagrams: • show relationships between different sets of data. • can represent conditional statements. • is usually drawn as a circle. • Every point IN the circle belongs to that set. • Every point OUT of the circle does not. Example:

  9. For all..., every..., if...then... All right angles are congruent. Example1: Congruent Angles Example 2: Every rose is a flower. Right Angles Flower • lines that do • not intersect Rose parallel lines Example 3: If two lines are parallel, then they do not intersect.

  10. Using a Venn Diagram to Write a Conditional Statement Write the conditional statement implied by each Venn Diagram. Then determine the truth value.

  11. Section 2.2 Biconditionals and Definitions

  12. Warm Up:

  13. Biconditional Statements: • If both a conditional statement and its converse are TRUE, they can be combined as a biconditional. …if and only if… Conditional: If two angles have the same measure, then they are congruent. Converse: If two angles are congruent, then they have the same measure. Biconditional: Two angles have the same measure if and only if they are congruent.

  14. Write the converse and biconditional: Conditional: If the intersection of two lines creates a right angle, then they are perpendicular. Converse: If two lines are perpendicular, then their intersection is a right angle. Biconditional: The intersection of two lines creates a right angle if and only if they are perpendicular.

  15. Determine if the converse is true. If so, write a biconditional statement. • If three points are collinear, then they lie on the same line. • If 2 angles are vertical, then they are congruent. • If two angles are supplementary and adjacent, then they form a linear pair. • If x = 3, then x = 9. 2

  16. If a biconditional can be formed, then the original conditional statement is considered an acceptable definition.

  17. Splitting up a Biconditional Statement into two conditional statements. • A triangle is equilateral if and only if all of its sides are congruent. • Conditional #1: • If a triangle is equilateral, then all of its sides are congruent. • Conditional #2: • If all sides of a triangle are congruent, then it is equilateral.

  18. Splitup theBiconditional Statements into two conditional statements. • 3x – 4 = 11 if and only if x = 5. • A number is divisible by 3 if and only if the sum of its digits is divisible by 3.

  19. Activity: Come up with a true conditional statement that has a true converse. Combine your statements into one biconditional statement.

  20. Section 5.4 Negations, Inverses, and Contrapositives

  21. Warm Up:

  22. Negation: Example: It is cool outside. Negation: It is not cool outside. If it is something, negating makes it not something. If it is not something, negating makes it is something.

  23. Write the negation of each statement. • 1. A triangle has four sides. • A triangle does not have four sides. • 2. It is raining outside. • It is not raining outside. • 3. Mrs. Smith is not going to be a Tyrannosaurus Rex for Halloween. • Mrs. Smith is going to be a Tyrannosaurus Rex for Halloween.

  24. Inverse • The negation of the hypothesis and conclusion of the conditional. Example: Conditional: If the football game is home then Mrs. Smith will go. Inverse: If the football game is not home, then Mrs. Smith will not go.

  25. Write the inverse of the following conditionals: 1. If the figure is a square, then it has four congruent sides and angles. If the figure is not a square, then it does not have four congruent sides and angles. 2. If the water is 100˚C, then it will boil. If the water is not 100˚C, then it will not boil. 3. If it is Halloween, then I will go Trick or Treating. If it is not Halloween, then I will not go Trick or Treating.

  26. Contrapositive: • Negation of the hypothesis and conclusion of the converse. Example: Conditional: If the football game is home, then Mrs. Smith will go. Converse: If Mrs. Smith goes to the football game, then it is home. Contrapositive: If Mrs. Smith does not go to the football game, then it is not home.

  27. Write the contrapositive to each conditional: 1. If it is Tuesday, then pasta is for lunch. If pasta is not for lunch, then it is not Tuesday. 2. If it is Columbus Day, then we do not have school. If we have school, then it is not Columbus Day. 3. If you are in Mrs. Smith’s 5th or 6th period class, then you take Geometry. If you do not take Geometry, then you are not in Mrs. Smith’s 5th or 6th period class.

  28. How to rearrange a conditional statement: I can afford a movie ticket if I have $15. p = hypothesis q = conclusion Conditional: If p, then q or p q If I have $15, then I can afford a movie ticket. Converse: If q, then p or q p If I can afford a movie ticket, then I have $15. Negation (~) Inverse: If p, then q or p q “not” “not” ~ ~ If I DO NOT have $15, then I CANNOT afford a movie ticket. Contrapositive: If “not” q, then “not” p or ~ q ~ p If I CANNOT afford a movie ticket, then I DO NOT have $15.

  29. The conditional and contrapositive will always have the same truth value. If the conditional is true, then the contrapositive will be true. The converse and inverse will always have the same truth value. If the converse is true, then the inverse will be true. If the converse is false, then the inverse will be false.

  30. Conditional: If a triangle is acute, then each interior angle has a measure between 0 and 90 degrees. Converse: Inverse: Contrapositive: If each interior angle of a triangle has a measure between 0 and 90 degrees, then the triangle is acute. If a triangle is not acute, then each interior angle does not have a measure between 0 and 90 degrees. If each interior angle of a triangle does not have a measure between 0 and 90 degrees, then the triangle is not acute.

  31. Summary: Conditional: If p, then q. Converse: If q, then p. Inverse: If –p, then –q . Contrapositive: If –q, then –p.

  32. Section 2.3 Deductive Reasoning

  33. Warm Up:

  34. Deductive Reasoning • Reasoning logically from given statements to a conclusion • If the given statements are true, deductive reasoning produces a true conclusion. Example: Statement: To hang a picture on the wall, a hammer and nail are needed. Deductive Reasoning: I want to hang a picture on the wall, so I will need a hammer and nail.

  35. Law of Detachment • If a conditional is true and its hypothesis is true, then its conclusion is true. Example: Conditional: If R is the midpoint of a segment, then it divides the segment into two congruent segments. Given: R is the midpoint of AB. Conclusion:

  36. Make a conclusion: 1) If water is 32ºF, then it is frozen. The water is 32ºF. Conclusion: The water is frozen. 2) If the regular polygon has eight sides, then it is an octagon. A stop sign is regular and has eight sides. Conclusion: A stop sign is an octagon. If p q is a true statement and p is true, then q is true.

  37. Exception: • If the conditional is true but you are given a true conclusion, you cannot conclude that the hypothesis is true. • Example: • If it is Columbus Day, then we do not have school. • We do not have school today. • No Conclusion because it could be Thanksgiving, Election Day, etc.

  38. Give a conclusion. If not possible, write “Not Possible” • If it rains tonight, then I am going to the mall. I am going to the mall. • If I study for the math test, I will get an A. I am studying for the math test. • If you share a drink with someone who is sick, then you will get sick. You got sick. • If you are 16, then you can get your permit. Jack is having his 16th birthday today.

  39. If the car’s battery is dead, then the car won’t start. The car’s battery is dead. The car won’t start. If you do not get enough sleep, you will be tired. Snoopy is tired. NO CONCLUSION! When it snows, the temperature is less than 32 degrees. It is 20 degrees outside. NO CONCLUSION! If M is the midpoint of a segment, then it divides the segment into two congruent segments. M is the midpoint of . Law of Detachment

  40. Law of Syllogism • If p q and q r are true statements, then p r is also a true statement. Example: If 2x = 6, then x is 3. If x is 3, then 5x = 15. Therefore: If 2x = 6 then 5x = 15.

  41. Make a conclusion: If it is a square, then it is a rectangle. If it is a rectangle, then it has four right angles. If it is a square, then it has four right angles. A triangle with three equal sides is equilateral. Equilateral triangles are also equiangular. A triangle with three equal sides is equiangular.

  42. Make a conclusion: If Jake gets an A in math, he will get the new COD. If Jake plays COD all day, then he will fail his classes. If Jake fails his classes, then he will be grounded for a month. If Jake gets the new COD, then he will play all day. If Jake gets an A in math, then he will be grounded for a month. To be able to make a conclusion, the conclusion of one statement must be the hypothesis of another!

  43. Make a conclusion: If Ashley goes skiing, she will ride the lift. If Ashley skis down the mountain, then she will ride the lift. If Ashley gets hot cocoa, she will be happy. If Ashley skis down the mountain, she will get hot cocoa. No conclusion. Skiing down the mountain is two hypothesis statements and riding the lift is two conclusions.

  44. If a person eats chicken, then they eat meat. If a person eats meat, then they are not a vegetarian. Joe eats chicken. Joe is not a vegetarian. On Sundays, GHS is not in session. When GHS is not in session, no homework is assigned. Yesterday was Sunday. There was no homework assigned yesterday. If a number is prime, then it does not have repeated factors. If a number does not have repeated factors, then it is not a perfect square. 2161 is a prime number. 2161 is not a perfect square. Law of Syllogism

  45. Detachment or Syllogism? If the circus is in town, then Alison works as a ticket collector. The circus is in town today. Law of Detachment Alison is working as a ticket collector. Geometry class is so much fun. If I am having fun, I smile. If I am in Geometry class, then I am smiling. Law of Syllogism Someone that is 5’ 11” is taller than Ethan. Jason is 5’ 11”. Jason is taller than Ethan. Law of Detachment If you are 17, then you are old enough to get your permit. You can drive if you have a permit. Law of Syllogism If you are 17, then you are old enough to drive.

  46. Section 2.4 Reasoning in Algebra

  47. Properties of Equality Addition: If a = b, then a + c = b + c Subtraction: If a = b, then a – c = b – c Multiplication: If a = b, then ac = bc Division: If a = b and c ≠ 0, then

  48. Properties Continued Reflexive Property: a = a Symmetric Property: If a = b, then b = a. Transitive Property: If a = b and b = c, then a = c. Substitution Property: If a = b, b can replace a in any expression Distributive Property: a (b + c) = ab + ac

  49. Justify each step to solving the equation using properties of equality.Given: 2x + 18 = 12, Prove x = -3 Subtraction How do we get this from the first equation? Always start with the given. Division

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