2.3

1. Check your skills 2. Homework Check Notes 4. Practice 5. Homework/work on Projects

If a point is in the third quadrant then the coordinates are negative. 1. Write the converse: 2. Write the inverse: 3. Write the contrapositive

1. If today is Friday, then tomorrow is Saturday. a. Today is Fridayb. Tomorrow is Saturday 2.The car will not start, if the battery is discharged. (hint: rearrange the order) a. 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.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 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

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. 9. 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.

 Objectives ◦ To write biconditionals ◦ To recognize good definitons

 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 A biconditional combines p → q and q → p as p ↔ q.

 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.

 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.

 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

 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.

 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!

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

1. 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.

2. 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.

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.

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.

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