1. A.A B.B C.C D.D A.Distributive Property B.Substitution Property C.Addition Property D.Transitive Property State the property that justifies the statement. If m  R = m  S, then m  R + m  T = m  S + m  T. Bellwork

2. A.A B.B C.C D.D A.Reflexive Property B.Symmetric Property C.Substitution Property D.Transitive Property State the property that justifies the statement. If BC = CD and CD = EF, then BC = EF. Bellwork

3. A.A B.B C.C D.D A.x = x B.If x = 3, then x + 4 = 7. C.If x = 3, then 3 = x. D.If x = 3 and x = y, then y = 3. Which statement shows an example of the Symmetric Property? Bellwork

4. quod erat demonstrandum Unit 3 Lesson 5 Proof Honors Geometry

5. Objectives I can prove theorems about lines I can make conclusions and inferences about line theorems I can use theorems, etc to complete geometric proofs

6. What is a Proof? Proof – a logical argument, with a clear goal, using given information in which each statement made is supported by a reason that is accepted as true We will use a two-column proof format – One column with statements – One column with reasons (evidence)

7. Famous Proofs According to the Guinness Book of World Records, the Pythagorean theorem has the most known proofs: 520 different proofs. Known as the father of geometry, Euclid proved most of our fundamental theorems in his collected works, called the Elements – The elements consist of 13 volumes

8. Euclid’s fifth proposition states that the base angles of an isosceles triangle are equal. This proposition was also called “the bridge of fools” because the diagram in Euclid’s proof resembled a bridge and because many weaker geometry students could not follow the logic of the proof and thus could not cross over to the rest of the Elements. A Mathematical Rite of Passage

9. Proof writing = $$$$ Prove one of the Clay Institute’s seven Millenium Prize problems… win $1 million! http://www.claymath.org/millennium/

10. Why is proof writing so important? Proofs are powerful Mathematicians seek pattern in the world, and then strive to generalize it (can you say that this will happen every time, no matter what) Theorems are therefore unbreakable, no exceptions – very powerful!

11. Proof on TV Keeler’s theorem from Futurama http://www.youtube.com/watch?v=8M4dUj7vZJ c http://www.youtube.com/watch?v=8M4dUj7vZJ c (4:51)

12. Proof in Geometry Class… We will look at the progression of proof – Follow the path of logic – Fill in any “blanks” in a proof We will look at proofs involving – Algebra – Line segments – Angles

13. First, Proof with Algebra Using a two-column proof format to illustrate the process of solving algebraic equations Put steps in the left columns Put the reasons for those steps in the right column

14. Example Solve 2(5 – 3a) – 4(a + 7) = 92. StatementsReasons 2(5 – 3a) – 4(a + 7)=92Given 10 – 6a – 4a – 28=92Distributive Property –18 – 10a=92Substitution Property –18 – 10a + 18 =92 + 18Addition Property -10a = 110 Substitution Property Division Property a=–11 Substitution Property

15. Write an Algebraic Proof ALWAYS Begin by stating what is given and what you are to prove.

16. Write an Algebraic Proof 2. d – 5 = 20t + 5 - 52. Subtraction Property of Equality StatementsReasons Proof: 1. Given 1. d = 20t + 5 6.6. Symmetric Property of Equality 4.4. Division Property of Equality = 3. d – 5 = 20t3. Substitution Property of Equality 5.5. Substitution Property of Equality

18. StatementsReasons Proof: 1. Given 1. 2.2. _______________ ? 3. AB = RS3Definition of congruent segments 4. AB = 124. Given 5. RS = 125. Substitution A. Reflexive Property of Equality B. Symmetric Property of Equality C.Transitive Property of Equality D. Substitution Property of Equality

19. Use the Segment Addition Postulate 2. Definition of congruent segments AB = CD 2. 3. Reflexive Property of Equality BC = BC 3. 4. Segment Addition Postulate AB + BC = AC 4. StatementsReasons 1. 1. Given AB ≈ CD ___ 5. Substitution Property of Equality 5. CD + BC = AC 6. Segment Addition Postulate CD + BC = BD 6. 7. Transitive Property of Equality AC = BD 7. 8. Definition of congruent segments8. AC ≈ BD ___

20. 1. Given AC = AB, AB = BX 1. 2. Transitive Property AC = BX 2. 3. Given CY = XD 3. 4. Addition PropertyAC + CY = BX + XD4. AY = BD 6. Substitution6. Proof: StatementsReasons Which reason correctly completes the proof? 5. ________________ AC + CY = AY; BX + XD = BD 5. ?

21. Proofs Using Congruent Complements or Supplements Theorems Given: Prove:

22. Proofs Using Congruent Comp. or Suppl. Theorems 1. Given 1.m  3 + m  1 = 180  1 and  4 form a linear pair. 4.  s suppl. to same  are . 4.  3   4 Proof: StatementsReasons 2. Linear pairs are supplementary. 2.  1 and  4 are supplementary. 3. Definition of supplementary angles 3.  3 and  1 are supplementary.

23. In the figure,  NYR and  RYA form a linear pair,  AXY and  AXZ form a linear pair, and  RYA and  AXZ are congruent. Prove that  NYR and  AXY are congruent. StatementsReasons 1. Given 1. linear pairs. 2.If two  s form a linear pair, then they are suppl.  s. 2. 3. Given 3. 4.  NYR   AXY 4. ____________ ?