Chapter 2 Reasoning and Proof 2 1 Conditional

Chapter 2 Reasoning and Proof

2 -1 Conditional Statements EQ: 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

2 -1 Conditional Statements EQ: Identify parts of conditional statements • Write each statement as a conditional 1. 2. 3. 4. A bicycle has two wheels My birthday cake is chocolate i. Phones cost too much money Geometry students always do their homework

2 -1 Conditional Statements EQ: 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.

2 -1 Conditional Statements EQ: 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

2 -1 Conditional Statements EQ: 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.

2 -1 Conditional Statements EQ: 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.

2 -1 Conditional Statements EQ: Identify parts of conditional statements

2 -2 Biconditionals EQ: 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.

2 -2 Biconditionals EQ: What makes a biconditional statement?

2 -2 Biconditionals EQ: 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.

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

• Warm Up: • Write the converse of each statement 1. If you don’t sleep much, then your grades will suffer. 2. If you want to arrive on time, then you must start early. • Write each statement as a conditional. 3. Leap years have 366 days 4. Two lines that are perpendicular meet to form right angles 5. Every sixteen year old is a teenager. Are any of the above statements biconditionals?

2 -3 Deductive Reasoning EQ: 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?

2 -3 Deductive Reasoning EQ: 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.

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

2 -3 Deductive Reasoning EQ: 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.

2 -3 Deductive Reasoning EQ: 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.

2 -3 Deductive Reasoning EQ: 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

• 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!

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

warm up

2 -5 Proving Angles Congruent EQ: 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.

2 -5 Proving Angles Congruent EQ: What is the difference between a postulate and a theorem?

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

2 -5 Proving Angles Congruent EQ: What is the difference between a postulate and a theorem?

2 -5 Proving Angles Congruent EQ: What is the difference between a postulate and a theorem?

• More Angle Theorems • homework: p 112 (1 -7, 12 -18) all

chapter 2 vocabulary review 1. _________is the process of reasoning logically from given facts to a conclusion. 2. The ______is the part of a conditional statement that follows the “then. ” 3. The _______of the conditional statement “if p then q” is “if q then p. ” 4. A ______ is the combination of a conditional statement and its converse. It contains the words “if and only if. ”

chapter 2 vocabulary review 5. The ______means that AB = AB. 6. The ________states that if the conditional “if p then q” is true, and p is true, then q is true. 7. A ________ is an if-then statement. 8. The ________of a conditional is true or false, depending on whether the statement is true or false. 9. The ________means that if AB = CD then CD = AB.

chapter 2 vocabulary review 10. The _______ is the part that follows “If” in a conditional statement. 11. A _______ is a conjecture that has been proven. 12. The ________states that if A=B and B=C then A=C. 13. The _______ states that “if p then q” is true, and “If q then r” is true, then “If p then r” is true.

Chapter 2 Review