Reasoning and Proof

1. Reasoning and Proof 2.3

2. Agenda • Check your skills • Homework Check • 2-3 Notes • Practice • Homework/work on Projects

3. Check your skills If a point is in the third quadrant then the coordinates are negative. • Write the converse: • Write the inverse: • Write the contrapositive

4. Homework Check 1. If today is Friday, then tomorrow is Saturday. a. Today is Friday b. Tomorrow is Saturday 2. The car will not start, if the battery is discharged. (hint: rearrange the order) • The battery is discharged b. The car will not start 3. Alligators are reptiles. If it is an alligator then it is a reptile. 4. College students work diligently. If a person is a college student then they work diligently.

5. 5. If two angles’ sum is 180, then they are supplementary. converse: If angles’ are supplementary, then the two angles’ sum is 180. inverse: If two angles’ sum is not 180, then the two angles are not supplementary. contrapositive: If two angles are not supplementary then their sum is not 180. 6. If two angles are a vertical, then they are congruent. converse: If two angles are congruent, then the two angles are vertical. inverse: If two angles are not a vertical pair then they are not congruent. contrapositive: If two angles are not congruent, then they are not a vertical pair

6. 7. If today is not Tuesday, then tomorrow is not Wednesday. converse: If tomorrow is not Wednesday, then today is not Tuesday. inverse: If today is Tuesdays, then tomorrow is Wednesday. contrapositive: If tomorrow is Wednesdays, then today is Tuesday. 8. If two lines are parallel, then they do not intersect. Converse: If two lines do not intersect, then they are parallel. False Counterexample: Skew lines – do not intersect and are not parallel. • If two lines skew, then they are noncoplanar. Converse: If two lines are noncoplanar, then they are skew. True Biconditional: Two lines are skew if and only if they are noncoplanar.

7. 2-2 Biconditionals and Definitions • Objectives • To write biconditionals • To recognize good definitons

8. 2-2 Biconditionals and Definitions A biconditional combines p → q and q → p as p ↔ q. When a conditional and its converse are true, you can combine them as a true biconditional. This is a statement you get by connecting the conditional and its converse with the word and. You can also write a biconditional by joining the two parts of each conditional with the phrase if and only if

9. Example of a Biconditional • Conditional • If two angles have the same measure, then the angles are congruent. • True • Converse • If two angles are congruent, then the angles have the same measure. • True • Biconditional • Two angles have the same measure if and only if the angles are congruent.

10. Example • Consider this true conditional statement. Write its converse. If the converse is also true, combine them as a biconditional • If three points are collinear, then they lie on the same line. • If three points lie on the same line, then they are collinear. • Three points are collinear if and only if they lie on the same line.

11. Definitions • A good definition is a statement that can help you identify or classify an object. • A good definition has several important components: • …Uses clearly understood terms. The terms should be commonly understood or already defined. • …Is precise. Good definitions avoid words such as large, sort of, and some. • …is reversible. That means that you can write a good definition as a true biconditional

12. Example • Show that this definition of perpendicular lines is reversible. Then write it as a true biconditional • Definition: Perpendicular lines are two lines that intersect to form right angles. • Conditional: If two lines are perpendicular, then they intersect to form right angles. • Converse: If two lines intersect to form right angles, then they are perpendicular. • Biconditional: Two lines are perpendicular if and only if they intersect to form right angles.

13. Real World Examples • Are the following statements good definitions? Explain • An airplane is a vehicle that flies. • Is it reversible? • NO! A helicopter is a counterexample because it also flies! • A triangle has sharp corners. • Is it precise? • NO! Sharp is an imprecise word!

14. 2.3 Laws of Detachment and Syllogism Law of Detachment: If a conditional is true and its hypothesis is true, then its conclusion is true. If p  q is a true statement and p is true, then q is true **Do not need to know terminology but must be able to apply the laws

15. Use the Law of Detachment to draw a conclusion. If it is not possible to draw a conclusion based on the given information write not possible • Given: • If an angle is obtuse, then its measure is greater than 90⁰. • Angle A is obtuse. Conclusion: The measure of angle A is greater than 90⁰.

16. Given: • If an angle is obtuse, then its measure is greater than 90⁰. • The measure of angle A is greater than 90⁰. Conclusion: Cannot draw a conclusion. You are only told the conclusion of the statement above is true.

17. Law of Syllogism: If p  q and q  r are true statements, then p  r is a true statement. ** Do not need to know terminology but must be able to apply the law.

18. Use the Law of Syllogism to draw a conclusion. 3. If angles are a linear pair, then they are supplementary. If angles are supplementary, then their sum is 180⁰. If angles are a linear pair then their sum is 180⁰.

19. Class Work You may work in Groups but I need one paper per person turned in for a classwork grade! Homework: Worksheet Scrapbook project due Friday Distance/Midpoint mini-project due Tues. 9/18