Chapter 2

1. Chapter 2 Reasoning and Proof

2. 2-1 Conditional StatementsEQ: Identify parts of conditional statements • Give examples of if-then statements • Another name for an if-then statement is a conditional. • Conditionals have two parts: a hypothesis and a conclusion. • Statement: Triangles have three sides. • Conditional: If a shape is a triangle then it has three sides hypothesis conclusion

3. 2-1 Conditional StatementsEQ: Identify parts of conditional statements • Write each statement as a conditional • A bicycle has two wheels • My birthday cake is chocolate • iPhones cost too much money • Geometry students always do their homework

4. 2-1 Conditional StatementsEQ: Identify parts of conditional statements • A conditional can have a truth value of true or false. • To show that a conditional is true, show that every time the hypothesis is true, the conclusion is also true. • To show a conditional is false, you need to find only one counterexample for which the hypothesis is true and the conclusion is false.

5. 2-1 Conditional StatementsEQ: Identify parts of conditional statements • Venn Diagrams can show whether a conditional is true or false. • If you live in Rocklin then you • live in California. • If you live in California then • you live in Rocklin. live in California live in Rocklin

6. 2-1 Conditional StatementsEQ: Identify parts of conditional statements • The converse of a conditional switches the hypothesis and the conclusion. • Conditional: If a figure has three sides then it is a triangle. • Converse: If a figure is a triangle then it has three sides. What is the converse? If I have $10 then I can afford a movie ticket. If I don't wake up on time then I will be late for school.

7. 2-1 Conditional StatementsEQ: Identify parts of conditional statements • A statement and its converse may not have the same truth value. • Come up with a statement that is true for both the statement and its converse. • Come up with a conditional that is true, but its converse is false. • Come up with a conditional that is false, but its converse is true.

8. 2-1 Conditional StatementsEQ: Identify parts of conditional statements

9. 2-2 BiconditionalsEQ: What makes a biconditional statement? • When a conditional and its converse are true you can combine them to form a biconditional • You can combine the two parts of each conditional with if and only if • If two angles have the same measure then they are congruent • If two angles are congruent then they have the same measure. • Two angles have the same measure if and only if they are congruent.

10. 2-2 BiconditionalsEQ: What makes a biconditional statement?

11. 2-2 BiconditionalsEQ: What makes a biconditional statement? • You can separate a biconditional into parts. • Biconditional: Two angles are supplementary if and only if the sum of their angles is 180˚ • If two angles are supplementary then the sum of their angles is 180˚ • If the sum of two angles is 180˚ then the angles are supplementary.

12. Homework: • page 83 (1-33) odd • page 90 (1-23) odd

13. Warm Up: • Write the converse of each statement • If you don’t sleep much, then your grades will suffer. • If you want to arrive on time, then you must start early. • Write each statement as a conditional. • Leap years have 366 days • Two lines that are perpendicular meet to form right angles • Every sixteen year old is a teenager. Are any of the above statements biconditionals?

14. 2-3 Deductive ReasoningEQ: Give examples of deductive reasoning • Deductive reasoning is the process of reasoning logically from given statements to a conclusion. • An auto mechanic knows that if a car has a dead battery it will not start. A mechanic begins to work on a car and finds the battery is dead. What conclusion can the mechanic make? • What if the mechanic begins to work on a car and finds it won't start. Can the mechanic conclude the car has a dead battery?

15. 2-3 Deductive ReasoningEQ: Give examples of deductive reasoning • If a conditional is true, and the hypothesis is true, then the conclusion is true. • Called the Law of Detachment • Given: If it is snowing, then the temperature is below 32 degrees. • It is snowing. • It is 17 degrees.

16. 2-3 Deductive ReasoningEQ: Give examples of deductive reasoning • Given: If the road is icy, then driving conditions are hazardous. • Driving conditions are hazardous. • The road is icy.

17. 2-3 Deductive ReasoningEQ: Give examples of deductive reasoning • If p  q and q  r then p  r • Called the Law of Syllogism • If there is a baseball game at the stadium then people eat sunflower seeds. • If people eat sunflower seeds then there are shells on the ground. • If there is a baseball game there are sunflower seed shells on the ground.

18. 2-3 Deductive ReasoningEQ: Give examples of deductive reasoning • What can you conclude? • If a number ends in 0, it is divisible by 10. • If a number is divisible by 10, then it is divisible by 5. • If a number ends in 6, then it is divisible by 2. • If a number ends in 4, then it is divisible by 2.

19. 2-3 Deductive ReasoningEQ: Give examples of deductive reasoning • Deductive Reasoning game • You will each get an index card. Write your name and one true fun fact about yourself that you think no one else knows. • for example: I have never been to Disneyland • I have ridden on the back of an elephant • I can fly an airplane • I will be reading these facts out loud exactly how you write them

20. As I read the facts, take notes and write them down. • Each student will get to ask one yes or no question of any one student. • You MUST answer the question truthfully. • At the end of the game, we will see how many answers you got right!

21. 2-3 Deductive ReasoningEQ: Give examples of deductive reasoning • Exit pass: worksheet • Homework: • page 96 (1-32, 38-44) all • Review packet for make up test on Friday

28. warm up

29. 2-5 Proving Angles CongruentEQ: What is the difference between a postulate and a theorem? • You can use deductive reasoning to show that a conjecture is true. • The set of steps you take is called a proof. • The statement you prove true is a theorem.

30. 2-5 Proving Angles CongruentEQ: What is the difference between a postulate and a theorem?

31. 2-5 Proving Angles CongruentEQ: What is the difference between a postulate and a theorem? • statement reason • m<1 + m<3 = 180° • m<2 + m<3 = 180° • m<1 + m<3 = m<2 + m<3 • m<1 = m<2

32. 2-5 Proving Angles CongruentEQ: What is the difference between a postulate and a theorem?

33. 2-5 Proving Angles CongruentEQ: What is the difference between a postulate and a theorem?

34. More Angle Theorems • homework: p112 (1-7, 12-18) all

35. chapter 2 vocabulary review • _________________is the process of reasoning logically from given facts to a conclusion. • The ____________is the part of a conditional statement that follows the “then.” • The ______________of the conditional statement “if p then q” is “if q then p.” • A ___________ is the combination of a conditional statement and its converse. It contains the words “if and only if.”

36. chapter 2 vocabulary review • The ____________means that AB = AB. • The ________________states that if the conditional “if p then q” is true, and p is true, then q is true. • A ________________ is an if-then statement. • The _______________of a conditional is true or false, depending on whether the statement is true or false. • The _______________means that if AB = CD then CD = AB.

37. chapter 2 vocabulary review • The _____________ is the part that follows “If” in a conditional statement. • A _____________ is a conjecture that has been proven. • The _______________states that if A=B and B=C then A=C. • The ______________ states that “if p then q” is true, and “If q then r” is true, then “If p then r” is true.

38. Chapter 2 Review