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Lesson 2.1

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Lesson 2.1

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  1. Lesson 2.1 Conditional Statements You will learn to… * recognize and analyze a conditional statement * write postulates about points, lines, an planes using conditional statements

  2. A conditional statement has two parts, a hypothesis and a conclusion. p q If p, then q.

  3. hypothesis (p) If the team wins the game, then they will win the tournament. conclusion (q)

  4. Write an if-then statement. 1. The intersection of two planes is a line. If two planes intersect, then their intersection is a line.

  5. Write an if-then statement. 2. A line containing two given points lies in a plane if the two points lie in the plane. If two points lie in a plane, then the line containing them lies in the plane.

  6. The converse is formed by switching the hypothesis and conclusion. The converse is q  p. If q, then p.

  7. Write the converse of this if-then statement. Is it true or false? 3. If mA = 125°, then A is obtuse. If A is obtuse, then mA = 125°. False

  8. The negation of a statement is formed by negating the statement. The negation is written ~ p.

  9. Write the negation of this statement. 4. mA = 125° mA  125° 5.A is not obtuse A is obtuse

  10. The inverse is formed by negating the hypothesis and the conclusion. The inverse is ~ p  ~ q. If ~ p, then ~ q.

  11. Write the inverse of this if-then statement. Is it true or false? 6. If mA = 125°, then A is obtuse. If mA  125°, then A is not obtuse. False

  12. The contrapositive is formed by negating the hypothesis and conclusion of the converse. The contrapositive is ~ q  ~ p. If ~ q, then ~ p.

  13. Write the contrapositive of this if-then statement. Is it true or false? 7. If mA = 125°, then A is obtuse. If A is not obtuse, then mA  125°. True

  14. Postulate 5 Through any two points there exists exactly one line.

  15. Postulate 6 A line contains at least two points.

  16. Postulate 7 If two lines intersect, then their intersection is exactly one point.

  17. B A C T Postulate 8 Through any three noncollinear points there exists exactly one plane.

  18. Postulate 9 A plane contains at least three noncollinear points.

  19. Postulate 10 If two points lie in a plane, then the line containing them lies in the plane.

  20. Postulate 11 If 2 planes intersect, then their intersection is ___________. a line

  21. Workbook Page 23 (1-5)

  22. Lesson 2.2 Biconditional Statements You will learn to… * recognize and use definitions * recognize and use biconditional statements

  23. All definitions are biconditional. All definitions can be interpreted “forward” and “backward.”

  24. For example, perpendicular lines are defined as two lines that intersect to form one right angle.

  25. If two lines are perpendicular, then they intersect to form one right angle. If two lines intersect to form one right angle, then they are perpendicular.

  26. Two lines are perpendicular if and only if they intersect to form one right angle. A biconditional statement contains the phrase “if and only if.”

  27. A biconditional statement is true when the original if-then statement AND its converse are both true.

  28. 1.Two angles are supplementary if and only if the sum of their measures is 180°. If two angles are supplementary, then the sum of their measures is 180°. if-then statement: converse: If the sum of the measures of two angles is 180°, then they are supplementary.

  29. 2.If an angle is 135˚, then it is an obtuse angle. converse: If an angle is obtuse, then its measure is 135°. Can we write a biconditional statement? counterexample?

  30. 3.If two angle measures add up to 90˚, then they are complementary angles. converse: If two angles are complementary, then the sum of their measures is 90°. Can we write a biconditional statement? Two angles are complementary if and only if the sum of their measures is 90°.

  31. Workbook Page 25 (1-7)

  32. Lesson 2.3 You will learn to… * use symbolic notation to represent logical statements * form conclusions by applying laws of logic Deductive Reasoning

  33. Using these phrases, write the conditional statement. If mB = 90˚, then  B is a right angle. 1. p  q If  B is a right angle, then mB = 90˚ If mB ≠ 90˚, then  B is not a right angle. 2. q  p p: mB = 90˚ q: B is a right angle If  B is not a right angle, then mB ≠ 90˚ 3. ~p  ~ q mB = 90˚ if and only if  B is a right angle. 4. ~ q  ~ p 5. p  q

  34. facts Deductive Reasoning uses facts to make a logical argument. definitions, properties, postulates, theorems, and laws of logic

  35. p  q p conclusion must be true hypothesis is true Given facts q Law of Detachment Therefore: You can use these symbols when asked to explain your reasoning.

  36. q Therefore, I passed geometry. q p p Law of Detachment If I learn my facts, then I will pass geometry. I learned my facts.

  37. p  q q  r Given facts p  r Law of Syllogism Therefore: You can use these symbols when asked to explain your reasoning.

  38. p Therefore, if I pass geometry, then I will get a cell phone. r p q q r Law of Syllogism If I pass geometry, then my dad will be happy. If my dad is happy, then I will get a cell phone.

  39. 6. Is this argument valid? If 2 lines in a plane are parallel, then they do not intersect. p  q Coplanar lines n and m are parallel. p Therefore, lines n and m do not intersect. q VALID – Law of Detachment

  40. r  p p  q 7. Is this argument valid? If 2 angles are supplementary, then the sum of their measures is 180˚. p  q If 2 angles form a linear pair, then they are supplementary. r  p Therefore, if 2 angles form a linear pair, then the sum of their measures is 180˚ r  q VALID – Law of Syllogism

  41. 8. Is this argument valid? If 2 angles are a linear pair, then the sum of their measures is 180˚. p  q m1 + m2 = 180˚ q Therefore, 1 and 2 are a linear pair. p INVALID

  42. r  q p  q 9. Is this argument valid? If you live in Canada, then you live in North America. p  q If you live in South Carolina, then you live in North America. r  q Therefore, if you live in Canada, then you live in South Carolina p  r INVALID

  43. If you use this product, then you will have even-toned skin. If you have even-toned skin, then If you use this product, then you will be beautiful. you will be beautiful.

  44. Lesson 2.4 Properties of Equality and Congruence You will learn to… * use properties from algebra * use properties of length and measure to justify segment and angle relationships

  45. Equality Properties Addition Property Subtraction Property Multiplication Property Division Property Distributive Property Substitution Property Reflexive Property Symmetric Property Transitive Property

  46. Reflexive Property

  47. Symmetric Property

  48. Transitive Property If mA=mB and mB=10°, then mA=10° If XY = ST and ST = 10, then XY = 10

  49. If 8x=16, then x=2. Division Property