M2 Geometry Journal Chapter 2

1. M2 Geometry Journal Chapter 2 By: Jose Antonio Weymann

2. Conditional If-Then Statement • A conditional if-then statement is a statement that starts with “if” hypothesis (P) a then ( ) and a “this happens” conclusion (Q) P Q If you do a Rotationthen you are doing a Transformation P Q Transformation Rotation

3. Conditional If-Then Statement • CONVERSE: Switch the hypothesis and conclusion in a conditional statement. Q P (not always true) If You are doing a transformation, thenyou are doing a Rotation Q P Transformation Rotation

4. Conditional If-Then Statement • INVERSE: Same as conditional statement, but hypothesis and Conclusion are NOT. ˜P˜Q If you are not doing a Rotation, thenyou are not doing a Transformation ˜P ˜Q Transformation Rotation

5. Conditional If-Then Statement • CONTRAPOSITIVE: Same as converse conditional statement, but hypothesis and conclusion are NOT. ˜Q ˜P If you are not doing a Transformation, then you are not doing a Rotation ˜Q ˜P Transformation Rotation

6. Counter-Example • It is one example that disproves a hypothesis, proposition or theorem (inductive reasoning). • Hypothesis: All prime numbers are odd Counter example: number 2 • Hypothesis: All right triangles are isosceles Counterexample: isosceles have to have all angles the same • Hypothesis: All water Animals are fish Counterexample: whales and dolphins are mammals.

7. Definitions • Definitions are Bi-conditionals statements that define what an object or subject is always. • Definition of perpendicular lines: Lines are perpendicular to each other if they cross at right angles.

8. Definitions • Definition of lines perpendicular to a plane: Lines are perpendicular to a plane in some point if those lines are perpendicular to any line in the plane that pass through that point.

9. Bi-Conditional Statements • It is when BOTH a conditional statement and the converse are true. If and only If *IFF: if and only if. • They are used to show how hypothesis depends on conclusion; in real life to define objects and subjects. • They are important because we use the in things like proofs, and daily life logic and planning, how specific they are is very important.

10. Bi-Conditional Statements • Examples: • A shape is a triangle iff it has 3 sides. • You do homework iff you are student. • You get grounded iff you behave badly.

11. Deductive Reasoning • It is the type of reasoning in which you look at facts and data to make conclusions. • Collect Data • Look at Facts • USE LOGIC Symbolic Notation: When writing simple expressions; writing a symbol instead of a word. E.g. Equal, T = V symbolic notation * LOGIC-MAKE CONCLUSIONS

12. Deductive Reasoning: examples • Research shows one in two girls in Colegio Americano High School have a blackberry Smartphone, meaning that if you see ten Colegio Americano High school girls at least 5 of the will have a blackberry. • If research shows that antibiotics don’t kill viruses; and you are given one for the flu it can be stated that it won’t make an effect on you. • If investigation says that 1 in 3 boys of Harlan High school have brown hair, then if you encounter 3 boys of Harlan High school at least one of the will have brown hair.

13. LOGIC • LAW OF DETACHMENT: If P Q, then if “P” is true the “Q” must also be true. • E.g. #1: If you have below a 65 in any class during a trimester ;then you have to take remedial exams, Willy got a 56 in science. Willyneeds to take a remedial exam. • E.g.#2 If you want to solicit for a mortgage in the Bank of Weymann; you need a good credit record, John Smith does not have a good credit record. John Smith does not get to solicit a mortgage. • E.g.#3: Uranium is very unstable, when thrown neutrons at it separates making fission. A unstable atom of uranium is thrown neutrons at it. The particle reacted in fission.

14. LOGIC • LAW OF SYLLOGISM: If P Q and Q R are both true statements, the if “P” is true then “R” is true. • E.g.#1: If you live on Manhattan, then you live in New York. If you live in New York you live in the east coast. Adrian lives In Manhattan therefore he lives in the west coast. • E.g.#2: If you buy a Ferrari, then you are buying a very expensive car. If you buy a very expensive you make a lot of money. Warren Buffet bought a Ferrari , therefore he makes a lot of money. • E.g.#3: If you are an organized worker you have a labor union contract. If you have a labor union contract you have better working benefits and conditions. If Jimmy is an organized worker he has better working conditions and benefits.

15. Algebraic Proofs • They are logical step by step arguments that validate your conclusion. • Be given an equation • Solve it - showing all work and explain why each step.

16. Algebraic Properties • Addition Property If a=b, then a+b=c • Subtraction Property If a=b, then a – c is b – c • Multiplication Property If a=b, then ac=bc • Division Property If a=b & c don’t equal zero, then a/c = b/c • Reflexive Property a=a • Symmetric Property If a=b then b=a • Transitive Property if a=b and b=c, then a=c • Substitution If a=b, then b can be replaced by a in any situation

17. Algebraic Proofs, examples statement reason 3X-8=19 Given +8 +8 Addition property 3X=27 simplification /3 /3 Division property X=9 simplification Q.E.D

18. Algebraic Proofs, examples statement reason 3X-6=2X+4 Given +6 +6 addition property 3X=2X+10 simplification -2x -2x subtraction property X= 10 simplification Q.E.D

19. Algebraic Proofs, examples statement reason 5X-4=2X+8 Given +4 +4 Addition property 5X=2X+2 simplification -2x -2x Subtraction property 3X=12 simplification /3 /3 Division property X=4 simplification Q.E.D

20. segment and angle properties of equality and congruence • SEGMENTS • Transitive: If AB = CD, and CD = EF, then AB = EF If AB congruent to CD and CD congruent to EF then AB congruent to EF. EXAMPLES: N is the midpoint of segment MP. And Pfrom NQ m n p q MN= PQ

21. segment and angle properties of equality and congruence • SEGMENTS • Symmetric: If AB = CD, then CD = AB. If AB congruent CD, then CD congruent AB EXAMPLES: HG=XY CD congruent to EF XY=HG EF congruent to CD

22. segment and angle properties of equality and congruence • SEGMENTS • Reflexive: AB = AB, AB congruent to AB • EXAMPLES: EF= EF EF congruent to EF

23. segment and angle properties of equality and congruence • ANGLES • Symmetric: If m∠A = m∠B, then m∠B = m∠A, If ∠A congruent ∠B, then ∠B congruent ∠A EXAMPLES: m∠ 1 = m ∠ 2; m ∠ 2 = m ∠ 1 ∠ 1 congruent to ∠2; ∠2 congruent to ∠1

24. segment and angle properties of equality and congruence • ANGLES • Transitive: If m∠A = m∠B and m∠B = m∠C, then m∠A = m∠C, If ∠A congruent ∠B and ∠B congruent ∠C, then ∠A congruent ∠C • EXAMPLES: • m∠1 = m∠2 and m∠2 = m∠3, then m∠1= m∠3 • If ∠1 congruent ∠2 and ∠2congruent ∠3, then ∠1congruent ∠3

25. segment and angle properties of equality and congruence • ANGLES • Reflexive: m∠A = m∠A, ∠A congruent to ∠A • EXAMPLES: • m∠1 = m∠1 • ∠X congruent to ∠X

26. Two Column Proofs • Is a type of prove in which you put your statement left and reasoning in the right. • Write Down important information • DRAW A PICTURE • Identify and mark given • Look at your pictures and facts, WRITE INFO • Go For It !!!

27. Two Column Proofs • Examples: statement reason Q is the midpoint of PR, R is the Given midpoint of QS PQ congruent to QR and QR Definition of midpoint congruent to RS PQ= QR and QR=RS Definition of congruent PQ=RS TRANSITIVE PROPERTY PQ+QR=RS+QR Segment Addition Postulate PR=QS Common Segments Theorem Q.E.D

28. Two Column Proofs statement reason ∠AOC congruent to ∠BOD Given BOC bisects ∠AOD Definition midpoint AOC= BOD Transitive property COD=AOC Definition of equal measures ∠AOB congruent ∠COD Definiton congruent Q.E.D

29. Two Column Proofs statement reason ∠1 and ∠2 form a linear pair Given BA and BC form a line Definition of linear pair m∠ABC=180º Definition of straight angles m∠1+m∠2= m∠ABC angle addition postulate m∠1+m∠2= 180º substitution ∠1&∠2 are supplementary Definition of supplementary Q.E.D

30. L.P.P. • OH LPP YEAH YOU KNOW ME ! • Linear Pair Postulate: all linear pairs of angles are supplementary .

31. congruent complements and supplements theorems • Congruent Complements theorem: If two angles are complementary to the same angle then they are congruent. • Examples: • If I say that <1 and <2 are complementary then <2 and <3 are complementary. Therefore <1 must be congruent to <3. • m<A + m<B = 90 degrees and m<B + m<C = 90 degrees , So < 1 and <3 are congruent. • M<4= 30 degrees    M<5=60 degrees   M<6= 30 degrees 30 degrees + 60 degrees = 90 degrees and 30 degrees + 60 degrees = 90 degrees  <4 and <6 are congruent.

32. congruent complements and supplements theorems • Congruent Supplements theorem: If two angles are supplementary to the same angle then they are congruent. • Examples: • m<C + m<D = 180 degrees and m<D + m<E = 180 degrees. So, < C and <E are congruent. • M<A = 150 degrees    M<B=30 degrees    M<a=150 degrees 150 degrees + 30 degrees = 180 degrees and 30 degrees + 150 degrees = 180 degrees. So,  <A and <C = congruent. • If I say that <E and <F are supplementary then <F and <G are supplementary. We conclude <E must be congruent to <G.

33. Vertical Angles Theorem • This Says: All vertical angles (non adjacent) are congruent. • Examples: 90º 90º 90º 90º

34. Congruent Segments Theorem • This says: if points A, B, C, and D are all collinear, then segment AB is congruent to segment CD then segment AC is congruent to segment BD. • Examples: • If Chicago to Detroit is the same as Seattle to Los Angeles then Chicago to Seattle is the same as Detroit to Los Angeles. • If San Marcos to Livingston is the same distance as Guatemala city to Flores, Petén. Then from San Marcos to Guatemala city is congruent with from Livingston to Flores, Petén. • From Mathew’s house to John’s house is the same as from Ana’s house to Jenna’s house, meaning that from Mathew’s to Ana’s is the same as from John’s house to Jenna’s house.

35. Conclusion • I hope you enjoyed the journal, and felt I fulfilled the requirements. T.T.J ! Take That Journal