Deductivereasoning and bicond and algebraic proofs

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Deductivereasoning and bicond and algebraic proofs

  1. 1. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions GT Geometry Drill 10/8/13 Identify the hypothesis and conclusion of each conditional. 1. A mapping that is a reflection is a type of transformation. 2. The quotient of two negative numbers is positive. 3. Determine if the conditional “If x is a number then |x| > 0” is true. If false, give a counterexample. H: A mapping is a reflection. C: The mapping is a transformation. H: Two numbers are negative. C: The quotient is positive. F; x = 0.
  2. 2. GT Geom Drill Solve each equation. 1. 3x + 5 = 17 2. r – 3.5 = 8.7 3. 4t – 7 = 8t + 3 4. 5. 2(y – 5) – 20 = 0 x = 4 r = 12.2 n = –38 y = 15 t = – 5 2
  3. 3. PSAT Practice 1. Which if the following numbers is divisible by 3 and 5, but not by 2? A) 955 B) 975 C) 990 D) 995 E) 999 2. There are n students in biology class and only 6 are seniors. If 7 juniors are added to the class how many students in the class will not be seniors? A) n-3 B) n-2 C) n-1 D) n+1 E) n+2
  4. 4. PSAT Practice 22 ,5&3.3 yxthen yxyxIf
  5. 5. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions Review properties of equality and use them to write algebraic proofs. Identify properties of equality and congruence. Objectives Write and analyze biconditional statements.
  6. 6. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions deductive reasoning Vocabulary biconditional statement definition polygon triangle quadrilateral
  7. 7. Holt McDougal Geometry 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.
  8. 8. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions
  9. 9. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions The Distributive Property states that a(b + c) = ab + ac. Remember!
  10. 10. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions Deductive reasoning is the process of using logic to draw conclusions from given facts, definitions, and properties.
  11. 11. Holt McDougal Geometry 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.”
  12. 12. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions p q means p q and q p The biconditional “p if and only if q” can also be written as “p iff q” or p q. Writing Math
  13. 13. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions Write the conditional statement and converse within the biconditional. Example 1A: Identifying the Conditionals within a Biconditional Statement 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°.
  14. 14. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions 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. 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°.
  15. 15. Holt McDougal Geometry 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.
  16. 16. Holt McDougal Geometry 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.
  17. 17. Holt McDougal Geometry 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.
  18. 18. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions A triangle is defined as a three-sided polygon, and a quadrilateral is a four-sided polygon.
  19. 19. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions Write each definition as a biconditional. Example 4: Writing Definitions as Biconditional Statements A. A pentagon is a five-sided polygon. B. A right angle measures 90°. A figure is a pentagon if and only if it is a 5-sided polygon. An angle is a right angle if and only if it measures 90 .
  20. 20. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions Check It Out! Example 4 4a. A quadrilateral is a four-sided polygon. 4b. The measure of a straight angle is 180°. Write each definition as a biconditional. A figure is a quadrilateral if and only if it is a 4-sided polygon. An is a straight if and only if its measure is 180°.
  21. 21. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions Solve the equation 4m – 8 = –12. Write a justification for each step. Example 1: Solving an Equation in Algebra 4m – 8 = –12 Given equation +8 +8 Addition Property of Equality 4m = –4 Simplify. m = –1 Simplify. Division Property of Equality
  22. 22. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions Check It Out! Example 1 t = –14 Simplify. Solve the equation . Write a justification for each step. Given equation Multiplication Property of Equality.
  23. 23. Holt McDougal Geometry 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. A B AB represents the length AB, so you can think of AB as a variable representing a number. Helpful Hint
  24. 24. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions Write a justification for each step. Example 3: Solving an Equation in Geometry NO = NM + MO 4x – 4 = 2x + (3x – 9) Substitution Property of Equality Segment Addition Post. 4x – 4 = 5x – 9 Simplify. –4 = x – 9 5 = x Addition Property of Equality Subtraction Property of Equality
  25. 25. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions Check It Out! Example 3 Write a justification for each step. x = 11 Subst. Prop. of Equality8x° = (3x + 5)° + (6x – 16)° 8x = 9x – 11 Simplify. –x = –11 Subtr. Prop. of Equality. Mult. Prop. of Equality. Add. Post.m ABC = m ABD + m DBC
  26. 26. Holt McDougal Geometry 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.
  27. 27. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions
  28. 28. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions Numbers are equal (=) and figures are congruent ( ). Remember!
  29. 29. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions 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° Example 4: Identifying Property of Equality and Congruence Symm. Prop. of = Trans. Prop of Reflex. Prop. of = Reflex. Prop. of .
  30. 30. Holt McDougal Geometry 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 Sym. Prop. of = Reflex. Prop. of = Trans. Prop. of = Sym. Prop. of
  31. 31. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions Lesson Quiz: Part I Solve each equation. Write a justification for each step. 1. z – 5 = –12 Mult. Prop. of = z = –7 Add. Prop. of = Given
  32. 32. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions 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. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions 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. Holt McDougal Geometry 2-4 Biconditional Statements and Definitions 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° Converse: If an measures 90°, then the is right. Biconditional: An is right iff its measure is 90°. 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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