Geometry Lesson 2.1 Conditional Statements. September 4, 2007

Transcription

1 Geometry Lesson 2.1 Conditional Statements September 4, 2007

2 Objectives Students will be able to: Define: conditional statement, hypothesis, conclusion, converse, inverse, contrapositive, equivalent statements Identify the hypothesis and conclusion in an If- Then conditional statement Rewrite conditional statements in If-Then form Write a counterexample to an If-Then Write the converse, inverse, and contrapositive of an If-Then statement Identify equivalent statements

3 Warm-Up: Conditions What are some things in your life that are conditional, such as If X occurs, then Y will occur Suppose I say If it is August, then it is summer is this always true (in AZ)? Suppose I say If it is summer, then it is August...is this always true?

4 1. Conditional Statements Conditional statement: A statement usually written in If-Then form that has two parts: Hypothesis: The condition Conclusion: What follows when the condition is met Example 1: If it is noon in Florida, then it is 9:00 AM in California. Hypothesis: it is noon in Florida Conclusion: it is 9:00 AM in CA

5 Practice 1: Identify Hypothesis / Conclusion If the weather is warm, then we should go swimming. Hypothesis: the weather is warm Conclusion: we should go swimming If the sun is shining, then we should go to the beach. Hypothesis: the sun is shining Conclusion: we should go to the beach

6 2. Rewrite Into If-Then Form Example 2: Today is Monday if yesterday was Sunday. If-Then form: If yesterday was Sunday, then today is Monday Practice 2: An angle is acute if it measures less than 90. If-Then form:

7 3. Counterexamples Conditional statements can be TRUE or FALSE To show a statement is TRUE, you must prove it for ALL cases To show a statement is FALSE, you must show only ONE counterexample

8 Example 3: Provide a Counterexample Decide if the statement is true or false If false, provide a counterexample (a) Statement: If x² = 16, then x = 4. True / False? Counterexample: (b) Statement: If it is February 14, then it is Valentine s Day. True / False? FALSE Counterexample: x = -4 (-4)² = 16 TRUE it s always Feb 14 None is needed

9 Practice 3: Provide a Counterexample Decide if the statement is true or false If false, provide a counterexample (a) Statement: If you visited New York, then you visited the Statue of Liberty. True / False? Counterexample: (b) Statement: If a number is odd, then it is divisible by 3. True / False? Counterexample:

10 Practice 3, cont. (c) Statement: If it is Saturday, then we are not in school. True / False? Counterexample:

11 4. Other Forms of If-Then Statements Statement: If m P = 45, then P is acute. Converse: SWITCH hypothesis & conclusion If P is acute, then m P = 45. Inverse: NEGATE hypothesis & conclusion If m P 45, then P is not acute. Contrapositive: SWITCH and NEGATE!!! If P is not acute, then m P 45. Note: The symbol means not equal

12 Practice 4: Other Forms of Statements Write the converse, inverse, and contrapositive (a) If x is odd, then 2x is even. Converse: Inverse: Contrapositive: (b) If there is snow, then flowers are not in bloom Converse: If flowers not in bloom, then no snow Inverse: If not snow, then flowers in bloom Contrapositive: If flowers in bloom, then no snow

13 Practice 4: Other Forms of Statements Write the converse, inverse, and contrapositive (c) If a student is in Geometry, then she is in 10 th grade. Converse: If in 10 th grade, then in Geometry Inverse: If not in Geo, then not in 10 th grade If not in 10 th grade, then not in Geo Contrapositive: (d) If an angle measures 90, then it is a right angle. If right, then m = 90 Converse: If m 90, then not right If not right, then m 90 Inverse: Contrapositive:

14 5. Equivalent Statements Two conditional statements are equivalent if they are BOTH true or BOTH false A conditional statement and its contrapositive are equivalent statements The inverse and converse are equivalent The statements themselves are not the same, but their logic is the same

15 Example of Equivalent Statements If m A = 30, then A is acute Converse If A is acute, then m A = 30 Inverse FALSE FALSE If m A 30, then A is not acute Contrapositive TRUE TRUE If A is not acute, then m A 30

16 Closure If you do homework, then you will succeed. Hypothesis: Conclusion: Converse: Inverse: Contrapositive: