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Geometry 1

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  1. Geometry 1 Unit 2: Reasoning and Proof

  2. Geometry 1 Unit 2 2.1 Conditional Statements

  3. Conditional Statements • Conditional Statement- • A statement with two parts • If-then form • A way of writing a conditional statement that clearly showcases the hypothesis and conclusion • Hypothesis- • The “if” part of a conditional Statement • Conclusion • The “then” part of a conditional Statement

  4. Conditional Statements • Examples of Conditional Statements • If today is Saturday, then tomorrow is Sunday. • If it’s a triangle, then it has a right angle. • If x2 = 4, then x = 2. • If you clean your room, then you can go to the mall. • If p, then q.

  5. Conditional Statements • Example 1 • Circle the hypothesis and underline the conclusion in each conditional statement • If you are in Geometry 1, then you will learn about the building blocks of geometry • If two points lie on the same line, then they are collinear • If a figure is a plane, then it is defined by 3 distinct points

  6. Conditional Statements • Example 2 • Rewrite each statement in if…then form • A line contains at least two points • When two planes intersect their intersection is a line • Two angles that add to 90° are complementary If a figure is a line, then it contains at least two points If two planes intersect, then their intersection is a line. If two angles add to equal 90°, then they are complementary.

  7. Conditional Statements • Counterexample • An example that proves that a given statement is false • Write a counterexample • If x2 = 9, then x = 3

  8. Conditional Statements • 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.

  9. Conditional Statements • Negation • In most cases you can form the negation of a statement by either adding or deleting the word “not”.

  10. Conditional Statements • Examples of Negations • Statement: • Negation : • Statement: John is not more than 6 feet tall. • Negation: John is more than 6 feet tall

  11. Conditional Statements • 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.

  12. Conditional Statements • Converse • Formed by switching the if and the then part. • Original • If you like green, then you will love my new shirt. • Converse • If you love my new shirt, then you like green.

  13. Conditional Statements • Inverse • Formed by negating both the if and the then part. • Original • If you like green, then you will love my new shirt. • Inverse • If you do not like green, then you will not love my new shirt.

  14. Conditional Statements • Contrapositive • Formed by switching and negating both the if and then part. • Original • If you like green, then you will love my new shirt. • Contrapositive • If you do not love my new shirt, then you do not like green.

  15. Conditional Statements • Write in if…then form. • Write the converse, inverse and contrapositive of each statement. • I will wash the dishes, if you dry them. • A square with side length 2 cm has an area of 4 cm2.

  16. Conditional Statements • Point-line postulate: • Through any two points, there exists exactly one line • Point-line converse: • A line contains at least two points • Intersecting lines postulate: • If two lines intersect, then their intersection is exactly one point

  17. Conditional Statements • Point-plane postulate: • Through any three noncollinear points there exists one plane • Point-plane converse: • A plane contains at least three noncollinear points • Line-plane postulate: • If two points lie in a plane, then the line containing them lies in the plane • Intersecting planes postulate: • If two planes intersect, then their intersection is a line

  18. Geometry 1 Unit 2 2.2: Definitions and Biconditional Statements

  19. Definitions and Biconditional Statements • Can be rewritten with “If and only if” • Only occurs when the statement and the converse of the statement are both true. • A biconditional can be split into a conditional and its converse.

  20. Definitions and Biconditional Statements Example 1 An angle is right if and only if its measure is 90º A number is even if and only if it is divisible by two. A point on a segment is the midpoint of the segment if and only if it bisects the segment. You attend school if and only if it is a weekday.

  21. Definitions and Biconditional Statements • Perpendicular lines • Two lines are perpendicular if they intersect to form a right angle • A line perpendicular to a plane • A line that intersects the plane in a point and is perpendicular to every line in the plane that intersects it • The symbol is read, “is perpendicular to.

  22. Definitions and Biconditional Statements • Example 2 • Write the definition of perpendicular as biconditional statement.

  23. Definitions and Biconditional Statements • Example 3 • Give a counterexample that demonstrates that the converse is false. • If two lines are perpendicular, then they intersect.

  24. Definitions and Biconditional Statements • Example 4 • The following statement is true. Write the converse and decide if it is true or false. If the converse is true, combine it with its original to form a biconditional. • If x2 = 4, then x = 2 or x = -2

  25. Definitions and Biconditional Statements • Example 5 • Consider the statement x2 < 49 if and only if x < 7. • Is this a biconditional? • Is the statement true?

  26. Geometry 1 Unit 2 2.3 Deductive Reasoning

  27. Deductive Reasoning • Symbolic Logic • Statements are replaced with variables, such as p, q, r. • Symbols are used to connect the statements.

  28. Deductive Reasoning Symbol Meaning ~ not Λ and V or → if…then ↔ if and only if

  29. Deductive Reasoning Example 1 Let p be “the measure of two angles is 180º” and Let q be “two angles are supplementary”. What does p → q mean? What does q → p mean?

  30. Deductive Reasoning 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 Λ q mean?

  31. Deductive Reasoning Example 3 Given t and s, determine the meaning of the statements below. t: Jeff has a math test today s: Jeff studied t V s s → t ~s → t

  32. Deductive Reasoning • Deductive Reasoning • Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical argument.

  33. Deductive Reasoning • Law of Detachment • When you have a true conditional statement and you know the hypothesis is true, you can conclude the conclusion is true. Given: p → q Given: p Conclusion: q

  34. Deductive Reasoning Example 4 Determine if the argument is valid. If Jasmyn studies then she will ace her test. Jasmyn studied. Jasmyn aced her test.

  35. Deductive Reasoning Example 5 Determine if the argument is valid. If Mike goes to work, then he will get home late. Mike got home late. Mike went to work

  36. Deductive Reasoning • Law of Syllogism • Given two linked conditional statements you can form one conditional statement. Given: p → q Given: q → r Conclusion: p → r

  37. Deductive Reasoning Example 6 Determine if the argument is valid. If today is your birthday, then Joe will bake a cake. If Joe bakes a cake, then everyone will celebrate. If today is your birthday, then everyone will celebrate.

  38. Deductive Reasoning Example 7 Determine if the argument is valid. If it is a square, then it has four sides. If it has four sides, then it is a quadrilateral. If it is a square, then it is a quadrilateral.

  39. Geometry 1 Unit 2 2.4 Reasoning with Properties from Algebra

  40. Reasoning with Properties from Algebra • Objectives • Review of algebraic properties • Reasoning • Applications of properties in real life

  41. Reasoning with Properties from Algebra • Addition property • If a = b, then a + c = b + c • Subtraction property • If a = b, then a – c = b – c • Multiplication property • If a = b, then ac = bc • Division property • If a = b, then

  42. Reasoning with Properties from Algebra • Reflexive property • For any real number a, 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, then a can be substituted for b in any equation or expression • Distributive property • 2(x + y) = 2x + 2y

  43. Reasoning with Properties from Algebra • Example 1 • Solve 6x – 5 = 2x + 3 and write a reason for each step

  44. Example 2 2(x – 3) = 6x + 6 Given Reasoning with Properties from Algebra

  45. Reasoning with Properties from Algebra • Determine if the equations are valid or invalid. • (x + 2)(x + 2) = x2 + 4 • x3x3 = x6 • -(x + y) = x – y

  46. Reasoning with Properties from Algebra • Geometric Properties of Equality • Reflexive property of equality • For any segment AB, AB = AB • Symmetric property of equality • If then • Transitive property of equality • If AB = CD and CD = EF, then, AB = EF

  47. A B C D Reasoning with Properties from Algebra Example 3 In the diagram, AB = CD. Show that AC = BD

  48. Geometry 1 Unit 2 2.5: Proving Statements about Segments

  49. Proving Statements about Segments • Key Terms: • 2-column proof • A way of proving a statement using a numbered column of statements and a numbered column of reasons for the statements • Theorem • A true statement that is proven by other true statements

  50. Proving Statements about Segments • Properties of Segment Congruence • Reflexive • For any segment AB, • Symmetric • If , then • Transitive • If and ,then