1. Reasoning and Proof Chapter 2

2. 2.1 – Conditional Statements Conditional statements – If, then form If – hypothesis Then – conclusion Negation of a statement- the opposite of a statement Biconditional statement – A statement in the form of if and only if (iff)

3. The Forms of Conditional Statements Conditional – (original) If, then. If p, then q. Converse – Switch the hypothesis and conclusion. If q, then p. Inverse – The negation of the conditional statement. If  p, then  q. Contrapositive – The negation of the converse. If  q, then  p.

4. Examples Con – If two angles are vertical, then they are congruent. True or False? Converse Inverse Contrapositive

5. Example Write the statement in If, then form: All monkeys have tails. Converse Inverse Contrapositive

6. Example Conditional – If it is cloudy, then it is raining. Converse Inverse Contrapositive

7. Example Conditional – If two angles are congruent, then they have equal measure. Converse Inverse Contrapositive

8. Logically Equivalent A pair of statements with the same truth value. CONDTIONAL  CONTRAPOSITIVE – Both have the same truth value. CONVERSE  INVERSE – Both have the same truth value.

9. Point, Line, and Plane Postulates Through any two points there exists exactly one line. A line contains at least two points. If two lines intersect, then their intersection is exactly one point. Through any three noncollinear points there is exactly one plane.

10. More Postulates A plane contains at least three noncollinear points. If two points lie in a plane, then the line containing them lies in the plane. If two planes intersect, then their intersection is a line. If there is a line and a point not on the line, then exactly one plane contains them.

11. 2.2 Definitions and Biconditional Statements Definition of Perpendicular Lines – Two lines are perpendicular  iff they intersect to form a right angle. Definition of a line  to a plane – If a line is  to a plane, then it is  to every line in that plane that intersects it.

12. Biconditional statements All biconditional statements are written using if and only if (iff). Writing a biconditional statement is equivalent to writing a conditional statement and its converse. A biconditional statement is true ONLY IF the conditional and the converse are both true.

13. Examples of Biconditionals Conditional statement – If x² = 4, then x = 2 or x = -2. Is the statement true? What is the biconditional of that statement? Is the biconditional true?

14. 2.3 Deductive Reasoning Conditional – If p, then q. p  q Converse – If q, then p. q  p Inverse – If  p, then  q.  p   q Contrapositive – If  q, then  p.  q   p Biconditional – p iff q. p q

15. Laws of Logic Law of Detachment – If p  q is a true conditional statement and p is true, then q is true. Law of Syllogism - If p  q and q  r are true conditional statements, then p  r is true.

16. Example – Law of Detachment Is the argument valid? Michael knows that if he does not do his chores in the morning, he will not be allowed to play video games later that same day. Michael does not play video games Friday afternoon. So Michael did not do his chores on Friday morning.

17. Example – Law of Detachment Is the argument valid? If two angles are vertical then they are congruent.  ABC and  DBE are vertical. So  ABC and  DBE are congruent.

18. Example – Law of Syllogism What can you conclude? If a fish swims at 68 mi/h, then it swims at 110 km/h. If a fish can swim at 110 km/h, then it is a sailfish. Therefore,

19. Example – Law of Syllogism What can you conclude? If the stereo is on, then the volume is loud. If the volume is loud, then the neighbors will complain. Therefore,

20. 2.4 Reasoning with Properties from Algebra Algebraic Properties of Equality Algebraic Properties of Equality and Congruence

21. Properties Of Equality ADDITION PROPERTY SUBTRACTION PROPERTY MULTIPLICATION PROPERTY DIVISION PROPERTY SUBSTITUTION PROPERTY If a = b, then a+c = b +c. If a = b, then a-c = b-c. If a = b, then ac = bc. If a = b and c  0, then a/c = b/c. If a = b, then a can be substituted for b in any equation or expression.

22. Properties of Equality and Congruence REFLEXIVE PROPERTY SYMMETRIC PROPERTY TRANSITIVE PROPERTY For any real number a, a = a. or AB  AB. If a = b, then, b = a. If AB  CD, then CD  AB. If a = b and b = c, then a = c. If AB  CD and CD  EF, then AB  EF.

23. 2.5 – Proving Statements About Segments Q is the MidQ is the Mid Q is midpoint PR PQ = ½PR and QR = ½PR

24. 2.6 – Proving Statements About Angles  1 supp  2  3 supp  4  1   4  2   3

25. Angle Theorems If two angles are supplementary to the same angle or congruent angles, then they are congruent. (Supp of   s are .) If two angles are complementary to the same angle or congruent angles, then they are congruent. (Comp of   s are .)

26. More Theorems All right angles are congruent. All vertical angles are congruent. Linear Pair Postulate – If two angles form a linear pair, then they are supplementary.