2.2 Proof Properties

1. 2.2 Proof Properties Memorize these ASAP!

2. Properties of Equality Addition Property of Equality: _________________________________ Subtraction Property of Equality: ________________________________ Multiplication Prop. of Equality: ________________________________

3. Division Property of Equality: ________________________________ Substitution Property of Equality: _______________________________ Reflexive Prop. of Equality: _______________________________ Symetric Prop. of Equality: _______________________________

4. Transitive Property of Equality: _________________________________ Definition of Congruence: ______________________________________________ ______________________________________________ ______________________________________________ ______________________________________________

5. Definition of a Midpoint: ______________________________________ Midpoint Theorem: ________________________________________________ ________________________________________________ ________________________________________________ A X B

6. Definition of Angle Bisector: ______________________________________ Angle Bisector Theorem: ______________________________________________ ______________________________________________ A Y B C

7. Vertal Angle Theorem (VAT): Segment Addition Postulate (SAP): Angle Addition Postulate (AAP): 1 2 A B C A D B C

8. Proof- ____________________________ __________________________________ We will now use all the theorems, postulates and properties we just defined to solve proofs. Lets try a Algebra Proof first.

9. Given: 4x+3 = 2x -9 Prove: x= -6 Given always first statement and reason! Statements Reasons

10. A B C D E F

11. A B C D

12. A B C D

13. Have we noticed a pattern? • Parts to Whole (given pieces, asked to prove whole) • ______________________________________ • ______________________________________ • ______________________________________ • Whole to Parts (given Whole, asked to prove parts) • ______________________________________ • ______________________________________ • ______________________________________

14. 2.2 Proofs Continued

15. K G H J K

16. E F H G

17. A B C D E F

18. S Q P 3 4 1 2 R T

19. S Q P Z 3 4 1 2 R T

20. S Q P Z 3 4 1 2 R T

21. S Q P Z 3 4 1 2 R T

22. 3 1 2

23. C B D 3 1 2 4 A E F

24. 2.4 Vertical Angle Theorem

25. Lets Prove VAT 3 1 2

26. VAT- ___________________________ ________________________________________________________________________________________________ Supplement and Complement:

27. Complete with Always, Sometimes or Never • Vertical angles____________ have a common vertex? • 2 right angles are _________complementary. • Right angles are _________ vertical angles. • Vertical angles___________ have a common supplement.

28. Lets Prove VAT 1 2 3 4

29. 2.5 Perpendicular Lines

30. Definition of Perpendicular Lines: • _____________________________________________________________________ • ________________________________________________________________________________________________________

31. Theorem 2-4:___________________________ ____________________________________________________________________________ s 1 2 t

32. Definition of Complementary Angles ________________________________________ ________________________________________ Definition of Supplementary Angles _________________________________________ _________________________________________ _________________________________________

33. Exterior Sides Theorem (EST): ___________________________________________________________________________________________________________________________ S 2 1 T ___________________________________________________________________________________________________________________________

34. E C A G • True or False: • Angle CGB is a right angle? • Angle CGA is a right angle? • Angle DGB is 90 degrees? • EGC and EGA are compliments? • DGF is complementary to DGA? • EGA is complementary to DGF B D F

35. 2.6 SAT and CAT

36. Supplementary Angle Theorem (SAT) ________________________________________ ________________________________________ ________________________________________ ________________________________________ ________________________________________ Example: If 1 is supp to 2 and 2 is sup to 3 Then SAT says

37. Complementary Angle Theorem (CAT) ____________________________________________________________________________________________________________________________________________________________________________________ Example: If 1 is comp to 2 and 2 is comp to 3 Then CAT says

38. When to use? • ___________________________________________________________________________________ • ___________________________________________________________________________________ • ___________________________________________________________________________________ • ___________________________________________________________________________________

39. 1 2 4 3

40. 3 4 1 2

41. 2 1 3 4

42. A 3 2 1 C B

43. 1 2 3 4

44. 1 2 3 4

45. 1 2 3 4