Chapter 6 Inequalities in Geometry

1. Chapter 6Inequalities in Geometry

2. 6-1 Inequalities Objectives • Apply properties of inequality to positive numbers, lengths of segments, and measures of angles • State and use the Exterior Angle Inequality Theorem.

3. Law of Trichotomy • The "Law of Trichotomy" says that only one of the following is true

4. Alex Has Less Money Than Billy or • Alex Has the same amount of money that Billy has or • Alex Has More Money Than Billy Makes Sense Right !

5. Equalities vs Inequalities • To this point we have dealt with congruent • Segments • Angles • Triangles • Polygons

6. Equalities vs Inequalities • In this chapter we will work with • segments having unequal lengths • Angles having unequal measures

7. The 4 Inequalities

8. The symbol "points at" the smaller value

9. A review of some properties of inequalities • When you use any of these in a proof, you can write as your reason, A property of Inequality

10. 1. If a < b, then a + c < b + c

11. Example If a < b, then a + c < b + c Alex has less coins than Billy. • If both Alex and Billy get 3 more coins each, Alex will still have less coins than Billy.

12. Likewise • If a < b, then a − c < b − c • If a > b, then a + c > b + c, and • If a > b, then a − c > b − c So adding (or subtracting) the same value to both a and b will not change the inequality

13. 2. If a < b, and c is positive, then ac < bc

14. Likewise • If a < b, and c is positive, then a < b c c

15. So multiplying (or dividing) the same value to both a and b will no change the inequality if c is POSITIVE !

16. 3. If a < b, and c is negative, then ac > bc (inequality swaps over!)

17. Likewise • If a < b, and c is negative, then a > b c c

18. So multiplying (or dividing) the same value to both a and b will change the inequality if c is NEGATIVE !

19. 4. If a < b and b < c, then a < c

20. Example If a < b and b < c, then a < c 1.) If Alex is younger than Billy and 2.) Billy is younger than Carol, Then Alex must be younger than Carol also!

21. 5. If a = b + c and c is > 0, then a > b and a > c

22. The Exterior Angle Inequality Theorem Why ? • The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. 2 m  4 > m  1 m  4 > m  2 1 3 4

23. Remote time

24. If a and b are real numbers and a < b, which one of the following must be true? A. -a < -b B. -a > -b C. a < -b • -a > b • I don’t know

25. Remote Time • True or False

26. True or False • If XY = YZ + 15, then XY > YZ

27. True or False • If m  A = m  B + m  C, then m  B > m  C

28. True or False • If m  H = m  J+ m  K, then m  K > m  H

29. True or False • If 10 = y + 2, then y > 10

30. White Board Practice Given: RS < ST; ST< RT Conclusion: RS ___ RT R S T

31. White Board Practice Given: RS < ST; ST< RT Conclusion: RS < RT R S T

32. White Board Practice Given: m  PQU = m PQT + m TQU Conclusion: m  PQU ____ m TQU m  PQU ____ m PQT U T R Q P

33. White Board Practice Given: m  PQU = m PQT + m TQU Conclusion: m  PQU >m TQU m  PQU >m PQT U T R Q P

34. 6-2: Inverses and Contrapositives • State the converse and inverse of an if-then statement. • Understand the relationship between logically equivalent statements. • Draw correct conclusions from given statements.

35. Review • Identify the hypothesis and the conclusion of each statements. • If Maria gets home from the football game late, then she will be grounded. • If Mike eats three happy meals, then he will have a major stomach ache.

36. If you are in your room, then you are in your house. What can you conclude if a) You are in your house b) You are in your room c) You are not in your room d) You are not in your house

37. She will be grounded Maria gets home from the game late Venn Diagrams A conditional statement can also be illustrated with a Venn Diagram. If Maria gets home from the football game late, then she will be grounded..

38. He will have a major stomach ache Mike eats three happy meals Venn Diagrams A conditional statement can also be illustrated with a Venn Diagram. If Mike eats three happy meals, then he will have a major stomach ache

39. THEN IF Venn Diagrams

40. Venn Diagrams THEN IF

41. Aren’t there other reasons why Maria might get grounded? Then she is grounded Late from football game

42. Aren’t there other reasons why Mike might get a stomach ache? Has a major stomach ache Eats three happy meals

43. Summary of If-Then Statements

44. Logically Equivalent These statements are either both true or both false

45. Summary of If-Then Statements These statements are either both true or both false

46. It’s a funny thing • This part of geometry is called LOGIC, however, if you try and “think logically” you will usually get the question wrong. • Let me show you

47. Example 1 If it is snowing, then the game is canceled. What can you conclude if I say, the game was cancelled?

48. Example 1 Nothing ! If it is snowing, then the game is canceled. What can you conclude if I say, the game was cancelled?

49. There are other reasons that the game would be cancelled Game cancelled D B Snowing A C

50. All you can conclude it that it MIGHT be snowing and that isn’t much of a conclusion.