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Deductivereasoning and bicond and algebraic proof updated 2014s

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Deductivereasoning and bicond and algebraic proof updated 2014s

  1. 1. PUT HW ON THE CORNER OF YOUR DESK! GT Geometry Drill 10/21/14 1. Identify the hypothesis of the conditional statement “Two angles are complementary if the sum of their measures is 90 degrees.” 2. Provide a counterexample to show that each statement is false. If the sum of two integers is even, then the integers are even 3. Write the converse of each and then decided if it is true or false. Angle is a right angle if its measure is 90 degrees
  2. 2. Is the conclusion a result of inductive or deductive reasoning? There is a myth that you can balance an egg on its end only on the spring equinox. A person was able to balance an egg on July 8, September 21, and December 19. Therefore this myth is false. Since the conclusion is based on a pattern of observations, it is a result of inductive reasoning.
  3. 3. Is the conclusion a result of inductive or deductive reasoning? There is a myth that the Great Wall of China is the only man-made object visible from the Moon. The Great Wall is barely visible in photographs taken from 180 miles above Earth. The Moon is about 237,000 miles from Earth. Therefore, the myth cannot be true. The conclusion is based on logical reasoning from scientific research. It is a result of deductive reasoning.
  4. 4. There is a myth that an eelskin wallet will demagnetize credit cards because the skin of the electric eels used to make the wallet holds an electric charge. However, eelskin products are not made from electric eels. Therefore, the myth cannot be true. Is this conclusion a result of inductive or deductive reasoning? The conclusion is based on logical reasoning from scientific research. It is a result of deductive reasoning.
  5. 5. 2-4 Biconditional Statements and Definitions Review properties of equality and use them to write algebraic proofs. Identify properties of equality and congruence. Holt McDougal Geometry Objectives Write and analyze biconditional statements.
  6. 6. 2-4 Biconditional Statements and Definitions deductive reasoning Holt McDougal Geometry Vocabulary biconditional statement definition polygon triangle quadrilateral
  7. 7. 2-4 Biconditional Statements and Definitions Deductive reasoning is the process of using logic to draw conclusions from given facts, definitions, and properties. Holt McDougal Geometry
  8. 8. 2-4 Biconditional Statements and Definitions When you combine a conditional statement and its converse, you create a biconditional statement. A biconditional statement is a statement that can be written in the form “p if and only if q.” This means “if p, then q” and “if q, then p.” Holt McDougal Geometry
  9. 9. 2-4 Biconditional Statements and Definitions p q means p q and q p Writing Math The biconditional “p if and only if q” can also be written as “p iff q” or p  q. Holt McDougal Geometry
  10. 10. 2-4 Biconditional Statements and Definitions Example 1A: Identifying the Conditionals within a Biconditional Statement Write the conditional statement and converse within the biconditional. An angle is obtuse if and only if its measure is greater than 90° and less than 180°. Let p and q represent the following. p: An angle is obtuse. q: An angle’s measure is greater than 90° and less than 180°. Holt McDougal Geometry
  11. 11. 2-4 Biconditional Statements and Definitions Let p and q represent the following. p: An angle is obtuse. q: An angle’s measure is greater than 90° and less than 180°. Holt McDougal Geometry Example 1A Continued The two parts of the biconditional p  q are p  q and q  p. Conditional: If an  is obtuse, then its measure is greater than 90° and less than 180°. Converse: If an angle's measure is greater than 90° and less than 180°, then it is obtuse.
  12. 12. 2-4 Biconditional Statements and Definitions For a biconditional statement to be true, both the conditional statement and its converse must be true. If either the conditional or the converse is false, then the biconditional statement is false. Holt McDougal Geometry
  13. 13. 2-4 Biconditional Statements and Definitions In geometry, biconditional statements are used to write definitions. A definition is a statement that describes a mathematical object and can be written as a true biconditional. Holt McDougal Geometry
  14. 14. 2-4 Biconditional Statements and Definitions In the glossary, a polygon is defined as a closed plane figure formed by three or more line segments. Holt McDougal Geometry
  15. 15. 2-4 Biconditional Statements and Definitions A triangle is defined as a three-sided polygon, and a quadrilateral is a four-sided polygon. Holt McDougal Geometry
  16. 16. 2-4 Biconditional Statements and Definitions Example 4: Writing Definitions as Biconditional Write each definition as a biconditional. A figure is a pentagon if and only if it is a 5-sided polygon. Holt McDougal Geometry Statements A. A pentagon is a five-sided polygon. B. A right angle measures 90°. An angle is a right angle if and only if it measures 90°.
  17. 17. 2-4 Biconditional Statements and Definitions Check It Out! Example 4 Write each definition as a biconditional. 4a. A quadrilateral is a four-sided polygon. A figure is a quadrilateral if and only if it is a 4-sided polygon. 4b. The measure of a straight angle is 180°. An  is a straight  if and only if its measure is 180°. Holt McDougal Geometry
  18. 18. 2-4 Biconditional Statements and Definitions A proof is an argument that uses logic, definitions, properties, and previously proven statements to show that a conclusion is true. An important part of writing a proof is giving justifications to show that every step is valid. Holt McDougal Geometry
  19. 19. 2-4 Biconditional Statements and Definitions Holt McDougal Geometry
  20. 20. 2-4 Biconditional Statements and Definitions Remember! The Distributive Property states that a(b + c) = ab + ac. Holt McDougal Geometry
  21. 21. 2-4 Biconditional Statements and Definitions Example 1: Solving an Equation in Algebra Solve the equation 4m – 8 = –12. Write a justification for each step. 4m – 8 = –12 Given equation +8 +8 Addition Property of Equality 4m = –4 Simplify. m = –1 Simplify. Holt McDougal Geometry Division Property of Equality
  22. 22. 2-4 Biconditional Statements and Definitions Check It Out! Example 1 Solve the equation . Write a justification for each step. t = –14 Simplify. Holt McDougal Geometry Given equation Multiplication Property of Equality.
  23. 23. 2-4 Biconditional Statements and Definitions Like algebra, geometry also uses numbers, variables, and operations. For example, segment lengths and angle measures are numbers. So you can use these same properties of equality to write algebraic proofs in geometry. Helpful Hint Holt McDougal Geometry A B AB represents the length AB, so you can think of AB as a variable representing a number.
  24. 24. 2-4 Biconditional Statements and Definitions Example 3: Solving an Equation in Geometry Write a justification for each step. NO = NM + MO 4x – 4 = 2x + (3x – 9) Holt McDougal Geometry Segment Addition Post. Substitution Property of Equality 4x – 4 = 5x – 9 Simplify. –4 = x – 9 5 = x Subtraction Property of Equality Addition Property of Equality
  25. 25. 2-4 Biconditional Statements and Definitions Check It Out! Example 3 Write a justification for each step. mABC = mABD + mDBC  Add. Post. 8x° = (3x + 5)° + (6x – 16)° Subst. Prop. of Equality 8x = 9x – 11 Simplify. –x = –11 Subtr. Prop. of Equality. x = 11 Holt McDougal Geometry Mult. Prop. of Equality.
  26. 26. 2-4 Biconditional Statements and Definitions You learned in Chapter 1 that segments with equal lengths are congruent and that angles with equal measures are congruent. So the Reflexive, Symmetric, and Transitive Properties of Equality have corresponding properties of congruence. Holt McDougal Geometry
  27. 27. 2-4 Biconditional Statements and Definitions Holt McDougal Geometry
  28. 28. 2-4 Biconditional Statements and Definitions Remember! Numbers are equal (=) and figures are congruent (). Holt McDougal Geometry
  29. 29. 2-4 Biconditional Statements and Definitions Example 4: Identifying Property of Equality and Identify the property that justifies each statement. A. QRS  QRS B. m1 = m2 so m2 = m1 C. AB  CD and CD  EF, so AB  EF. D. 32° = 32° Holt McDougal Geometry Congruence Symm. Prop. of = Trans. Prop of  Reflex. Prop. of . Reflex. Prop. of =
  30. 30. 2-4 Biconditional Statements and Definitions Check It Out! Example 4 Identify the property that justifies each statement. 4a. DE = GH, so GH = DE. 4b. 94° = 94° 4c. 0 = a, and a = x. So 0 = x. 4d. A  Y, so Y  A Holt McDougal Geometry Sym. Prop. of = Reflex. Prop. of = Trans. Prop. of = Sym. Prop. of 
  31. 31. 2-4 Biconditional Statements and Definitions Holt McDougal Geometry Lesson Quiz: Part I Solve each equation. Write a justification for each step. 1. Given z – 5 = –12 Mult. Prop. of = z = –7 Add. Prop. of =
  32. 32. 2-4 Biconditional Statements and Definitions Holt McDougal Geometry Lesson Quiz: Part II Solve each equation. Write a justification for each step. 2. 6r – 3 = –2(r + 1) Given 6r – 3 = –2r – 2 8r – 3 = –2 Distrib. Prop. Add. Prop. of = 6r – 3 = –2(r + 1) 8r = 1 Add. Prop. of = Div. Prop. of =
  33. 33. 2-4 Biconditional Statements and Definitions Holt McDougal Geometry Lesson Quiz: Part III Identify the property that justifies each statement. 3. x = y and y = z, so x = z. 4. DEF  DEF 5. AB  CD, so CD  AB. Trans. Prop. of = Reflex. Prop. of  Sym. Prop. of 
  34. 34. 2-4 Biconditional Statements and Definitions Converse: If an  measures 90°, then the  is right. Biconditional: An  is right iff its measure is 90°. Holt McDougal Geometry Lesson Quiz 1. For the conditional “If an angle is right, then its measure is 90°,” write the converse and a biconditional statement. 2. Determine if the biconditional “Two angles are complementary if and only if they are both acute” is true. If false, give a counterexample. False; possible answer: 30° and 40° 3. Write the definition “An acute triangle is a triangle with three acute angles” as a biconditional. A triangle is acute iff it has 3 acute s.

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