2. Geometry Lesson 2.1A Conditional Statements

3. Objectives Students will be able to: Define: conditional statement, hypothesis, conclusion Write a statement in if-then form Write the converse, inverse, and contrapositive of an if-then statement Determine if a conditional statement is true or false; if false, write a counterexample

4. Warm-Up: Conditions What are some things in your life that are conditional? “If X happens, then Y will happen” How do you know if the conditional statement is true?

5. 1. Conditional Statements Conditional statement Conditional statement: A statement usually written in “If-Then” form that has two parts: Hypothesis Hypothesis: The condition Conclusion 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: _____________________________ Conclusion: _____________________________ it is noon in Florida it is 9:00 AM in California

6. Practice 1: Hypothesis & Conclusion If the weather is warm, then we should go swimming. Hypothesis: _________________________ Conclusion: _________________________ the weather is warm we should go swimming

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

8. 3. Counterexamples Conditional statements can be TRUE or FALSE TRUE allcases To show a statement is TRUE, you must prove it for all cases. FALSE one To show a statement is FALSE, you must show only one counterexample.

9. Example 3: True or False 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? _____________________________ Counterexample: ________________________ FALSE For x = -4, x² = 16 TRUE: always Feb 14 None needed

10. Practice 3: True or False Decide if the statement is true or false. If false, provide a counterexample. (a) If you visited New York, then you visited the Statue of Liberty. True / False? ___________ Counterexample: ____________________________ (b) If a number is odd, then it is divisible by 3. True / False? ____________ Counterexample: ____________________________ (c) If it is Saturday, then we are not in school. True / False? ____________ Counterexample: ____________________________ FALSE You could have skipped it FALSE 7 is odd but NOT divisible by 3 TRUE None needed

11. Conditional Statements To fully analyze this conditional statement, we need to find three new conditionals: Inverse Converse Contrapositive

12. 4. Other Forms of If-Then Statements Statement: If m  P = 45°, then  P is acute Converse Converse: SWITCH hypothesis & conclusion If  P is acute, then m  P = 45° Inverse Inverse: NEGATE hypothesis & conclusion If m  P ≠ 45°, then  P is not acute Contrapositive Contrapositive: SWITCH and NEGATE!!! If  P is not acute, then m  P ≠ 45°

13. When two statements are both true or both false, we say that they are equivalent statements. The original conditional statement is equivalent to its contrapositive The converse & inverse of a conditional statement are equivalent

14. Practice 4: Converse, Inverse, Contrapositive (a) If x is odd, then 2x is even Converse: If 2x is even then x is odd Inverse: If x is not odd, then 2x is not even Contrapositive: If 2x is not even then x is not odd (b) If there is snow, then flowers are not in bloom Converse: If flowers are not in bloom then there is snow Inverse: Is there is not snow then flowers are in bloom Contrapositive: If flowers are in bloom then there is no snow

15. Closure If you do your homework, then you will understand Geometry Hypothesis: _______________________ Conclusion: _______________________ Is the following true or false? If false, write a counterexample. If it is snowing, then it is winter T/F? _______ Counterexample: ______________________ you do your homework you will understand Geo. FALSE It can snow in spring

16. USING POINT, LINE, AND PLANE POSTULATES Postulate 5: Point Existence Postulate : Through any two points there exists exactly one line. Postulate 6: Converse of Point Existence Postulate : A line contains at least two points. Postulate 7: Line Intersection Postulate : If two lines intersect, they intersect at exactly one point. Postulate 8: Plane Existence Postulate : Through any 3 non- collinear points, there exists exactly one plane. Postulate 9: Converse of Plane Existence Postulate : A plane contains at least 3 non-collinear points Postulate 10: Plane-Line Postulate : If two points lie in a plane, the line containing the points also lies in the plane. Postulate 11: Plane Intersection Postulate : When two planes intersect, they intersect at a line.

17. exactly P5: Through any 2 points there is exactly one line P6: A line contains at least 2 points SINGLE P7: The intersection of two lines is a SINGLE point 1. Point, Line, and Plane Postulates AB m Line m is the ONLY line that can go through A and B Meaning: You can’t have a line with only one point…that would just be a point!!! P

18. exactly P8: Through any 3 non-collinear points there is exactly one plane P9: A plane contains at least 3 non- collinear points Point, Line, and Plane Postulates A B Plane P is the ONLY plane that can go through A, B and C Meaning: You can’t have a plane with only two points…that would just be a line!!! C P

19. P10: If two points are in a plane, then the line containing them is in it also Point, Line, and Plane Postulates A B Line n goes through A & B, so it is also in plane P P n

20. P11: The intersection of two planes is a line (remember the index cards…) Point, Line, and Plane Postulates P Q m Line m is the intersection of planes P and Q

21. 2. Identifying Postulates Example 1: Which postulate verifies the truth of the statement? Points X, Y, and Z lie in plane B Points X, and Y lie on line m Postulate 8: Any 3 non- collinear points define a plane Postulate 6: A line contains at least 2 points

22. Practice 1: Identifying Postulates Which postulate verifies the truth of the statement? (a) Planes A and B intersect in line l (b) Points X and Y are in plane B; therefore XY lies in plane B Postulate 11 Postulate 10

23. Practice 1, continued Which postulate verifies the truth of the statement? (c) Points E, F, and H are coplanar in plane R (d) Planes Q and R intersect in line l Postulate 9 Postulate 11

24. 3. Counterexamples Is the statement true or false? If false, find a counterexample (a) A line can be in more than one plane (b) Four non-collinear points are always coplanar True…postulate 11 says that two planes intersect in a line, so that line is in both planes at the same time FALSE…postulate 8 says that 3 points define exactly one plane; 4 points define more than one plane

25. Practice 2: Counterexamples Is the statement true or false? If false, find a counterexample (a) Through any 3 points, there exists only one line (b) Three planes can intersect in a point False: Postulate 5 True: Three planes intersect in 2 lines; those 2 lines intersect in a point

26. Practice 2, continued (c) For any one point in a plane, there exists exactly one line through that point (d) For non-collinear points A, B, and C, point C and AB are in different planes False: Postulate 5 says two points make a line False: Postulate 8 says three points make a plane

27. Assignment Ch 2.1 (Pg. 75-76) #10-38 EVEN