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Proving Statements in Geometry

Proving Statements in Geometry. Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin. ERHS Math Geometry. Inductive Reasoning. Mr. Chin-Sung Lin. ERHS Math Geometry. Describe and sketch the fourth figure in the pattern:. Visual Pattern. ?. Fig. 1. Fig. 2. Fig. 3. Fig. 4.

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Proving Statements in Geometry

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  1. Proving Statements in Geometry Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin

  2. ERHS Math Geometry Inductive Reasoning Mr. Chin-Sung Lin

  3. ERHS Math Geometry Describe and sketch the fourth figure in the pattern: Visual Pattern ? Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin

  4. ERHS Math Geometry Describe and sketch the fourth figure in the pattern: Visual Pattern Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin

  5. ERHS Math Geometry Describe and sketch the fourth figure in the pattern: Visual Pattern ? Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin

  6. ERHS Math Geometry Describe and sketch the fourth figure in the pattern: Visual Pattern Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin

  7. ERHS Math Geometry Describe and sketch the fourth figure in the pattern: Visual Pattern ? Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin

  8. ERHS Math Geometry Describe and sketch the fourth figure in the pattern: Visual Pattern Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin

  9. ERHS Math Geometry Describe and sketch the fourth figure in the pattern: Visual Pattern ? Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin

  10. ERHS Math Geometry Describe and sketch the fourth figure in the pattern: Visual Pattern Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin

  11. ERHS Math Geometry Describe the pattern in the numbers and write the next three numbers: Number Pattern ? ? ? 1 4 7 10 Mr. Chin-Sung Lin

  12. ERHS Math Geometry Describe the pattern in the numbers and write the next three numbers: Number Pattern 1 4 7 16 19 10 13 3 3 3 3 3 3 Mr. Chin-Sung Lin

  13. ERHS Math Geometry Describe the pattern in the numbers and write the next three numbers: Number Pattern ? ? ? 1 4 9 16 Mr. Chin-Sung Lin

  14. ERHS Math Geometry Describe the pattern in the numbers and write the next three numbers: Number Pattern 1 4 9 36 49 16 25 11 13 7 9 3 5 2 2 2 2 2 Mr. Chin-Sung Lin

  15. ERHS Math Geometry Conjecture An unproven statement that is based on observation Mr. Chin-Sung Lin

  16. ERHS Math Geometry Inductive Reasoning Inductive reasoning, or induction, is reasoning from a specific case or cases and deriving a general rule You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case Mr. Chin-Sung Lin

  17. ERHS Math Geometry Weakness of Inductive Reasoning Direct measurement results can be only approximate We arrive at a generalization before we have examined every possible example When we conduct an experiment we do not give explanations for why things are true Mr. Chin-Sung Lin

  18. ERHS Math Geometry Strength of Inductive Reasoning A powerful tool in discovering new mathematical facts (making conjectures) Inductive reasoning does not prove or explain conjectures Mr. Chin-Sung Lin

  19. ERHS Math Geometry Give five colinear points, make a conjecture about the number of ways to connect different pairs of the points Make a Conjecture Mr. Chin-Sung Lin

  20. ERHS Math Geometry Give five colinear points, make a conjecture about the number of ways to connect different pairs of the points Make a Conjecture Mr. Chin-Sung Lin

  21. ERHS Math Geometry Make a Conjecture ? 3 2 1 Mr. Chin-Sung Lin

  22. ERHS Math Geometry Make a Conjecture 4 3 2 1 Conjecture: You can connect five colinear points 6 + 4 = 10 different ways Mr. Chin-Sung Lin

  23. ERHS Math Geometry Prove a Conjecture 4 3 2 1 Conjecture: You can connect five colinear points 6 + 4 = 10 different ways Mr. Chin-Sung Lin

  24. ERHS Math Geometry To show that a conjecture is true, you must show that it is true for all cases Prove a Conjecture Mr. Chin-Sung Lin

  25. ERHS Math Geometry To show that a conjecture is false, you just need to find one counterexample A counterexample is a specific case for which the conjecture is false Disprove a Conjecture Mr. Chin-Sung Lin

  26. ERHS Math Geometry Conjecture: the sum of two number is always greater than the larger number Exercise: Disprove a Conjecture Mr. Chin-Sung Lin

  27. ERHS Math Geometry Conjecture: the value of x2 always greater than the value of x Exercise: Disprove a Conjecture Mr. Chin-Sung Lin

  28. ERHS Math Geometry Conjecture: the product of two numbers is even, then the two numbers must both be even Exercise: Disprove a Conjecture Mr. Chin-Sung Lin

  29. ERHS Math Geometry Analyzing Reasoning Mr. Chin-Sung Lin

  30. ERHS Math Geometry Analyzing Reasoning Use inductive reasoning to make conjectures Use deductive reasoning to show that conjectures are true or false Mr. Chin-Sung Lin

  31. ERHS Math Geometry What conclusion can you make about the product of an even integer and any other integer 2 * 5 = 10 (-4) * (-7) = 28 2 * 6 = 12 6 * 15 = 90 use inductive reasoning to make a conjecture Analyzing Reasoning Example Mr. Chin-Sung Lin

  32. ERHS Math Geometry What conclusion can you make about the product of an even integer and any other integer 2 * 5 = 10 (-4) * (-7) = 28 2 * 6 = 12 6 * 15 = 90 use inductive reasoning to make a conjecture Conjecture: Even integer * Any integer = Even integer Analyzing Reasoning Example Mr. Chin-Sung Lin

  33. ERHS Math Geometry Use deductive reasoning to show that a conjecture is true Conjecture: Even integer * Any integer = Even integer Let n and m be any integer 2n is an even integer since any integer multiplied by 2 is even (2n)m represents the product of an even interger and any integer (2n)m = 2(nm) is the product of 2 and an integer nm. So, 2nm is an even integer Analyzing Reasoning Example Mr. Chin-Sung Lin

  34. ERHS Math Geometry Deductive Reasoning Deductive reasoning, or deduction, is using facts, definitions, accepted properties, and the laws of logic to form a logical argument While inductive reasoning is using specific examples and patterns to form a conjecture Mr. Chin-Sung Lin

  35. ERHS Math Geometry Definitions as Biconditionals Mr. Chin-Sung Lin

  36. ERHS Math Geometry • Right angles are angles with measure of 90 • Angles with measure of 90 are right angles • When a conditional and its converse are both true: Definitions as Biconditionals Mr. Chin-Sung Lin

  37. ERHS Math Geometry • Right angles are angles with measure of 90 • If angles are right angles, then their measure is 90 • p  q (T) • Angles with measure of 90 are right angles • When a conditional and its converse are both true: Definitions as Biconditionals Mr. Chin-Sung Lin

  38. ERHS Math Geometry • Right angles are angles with measure of 90 • If angles are right angles, then their measure is 90 • p  q (T) • Angles with measure of 90 are right angles • If measure of angles is 90, then their are right angles • q  p (T) • When a conditional and its converse are both true: Definitions as Biconditionals Mr. Chin-Sung Lin

  39. ERHS Math Geometry • Right angles are angles with measure of 90 • If angles are right angles, then their measure is 90 • p  q (T) • Angles with measure of 90 are right angles • If measure of angles is 90, then their are right angles • q  p (T) • When a conditional and its converse are both true: • Angles are right angles if and only if their measure is 90 • q  p (T) Definitions as Biconditionals Mr. Chin-Sung Lin

  40. ERHS Math Geometry Deductive Reasoning Mr. Chin-Sung Lin

  41. ERHS Math Geometry • A proof is a valid argument that establishes the truth of a statement • Proofs are based on a series of statements that are assume to be true • Definitions are true statements and are used in geometric proofs • Deductive reasoning uses the laws of logic to link together true statements to arrive at a true conclusion Proofs Mr. Chin-Sung Lin

  42. ERHS Math Geometry • given: The information known to be true • prove: Statements and conclusion to be proved • two-column proof: • In the left column, we write statements that we known to be true • In the right column, we write the reasons why each statement is true • * The laws of logic are used to deduce the conclusion but the laws are not listed among the reasons Proofs of Euclidean Geometry Mr. Chin-Sung Lin

  43. ERHS Math Geometry • Given: In ΔABC, AB  BC • Prove: ΔABC is a right triangle • Proof: • Statements Reasons • AB  BC 1. Given. Two Column Proof Example Mr. Chin-Sung Lin

  44. ERHS Math Geometry • Given: In ΔABC, AB  BC • Prove: ΔABC is a right triangle • Proof: • Statements Reasons • AB  BC 1. Given. • ABC is a right angle. 2. If two lines are perpendicular, then they intersect to form right angles. Two Column Proof Example Mr. Chin-Sung Lin

  45. ERHS Math Geometry • Given: In ΔABC, AB  BC • Prove: ΔABC is a right triangle • Proof: • Statements Reasons • AB  BC 1. Given. • ABC is a right angle. 2. If two lines are perpendicular, then they intersect to form right angles. • ΔABC is a right triangle. 3. If a triangle has a right angle then it is a right triangle. Two Column Proof Example Mr. Chin-Sung Lin

  46. ERHS Math Geometry Given: In ΔABC, AB  BC Prove: ΔABC is a right triangle Proof: We are given that AB  BC. If two lines are perpendicular, then they intersect to form right angles. Therefore, ABC is a right angle. A right triangle is a triangle that has a right angle. Since ABC is an angle of ΔABC, ΔABC is a right triangle. Paragraph Proof Example Mr. Chin-Sung Lin

  47. ERHS Math Geometry Given: BD is the bisector of ABC. Prove: mABD = mDBC Proof: Statements Reasons Two Column Proof Example Mr. Chin-Sung Lin

  48. ERHS Math Geometry • Given: BD is the bisector of ABC. • Prove: mABD = mDBC • Proof: • Statements Reasons • BD is the bisector of ABC. 1. Given. Two Column Proof Example Mr. Chin-Sung Lin

  49. ERHS Math Geometry • Given: BD is the bisector of ABC. • Prove: mABD = mDBC • Proof: • Statements Reasons • BD is the bisector of ABC. 1. Given. • ABD ≅DBC 2. The bisector of an angle is a ray whose endpoint is the vertex of the angle and that divides the angle into two congruent angles. Two Column Proof Example Mr. Chin-Sung Lin

  50. ERHS Math Geometry • Given: BD is the bisector of ABC. • Prove: mABD = mDBC • Proof: • Statements Reasons • BD is the bisector of ABC. 1. Given. • ABD ≅DBC 2. The bisector of an angle is a ray whose endpoint is the vertex of the angle and that divides the angle into two congruent angles. • mABD = mDBC 3. Congruent angles are angles that have the same measure. Two Column Proof Example Mr. Chin-Sung Lin

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