1. Unit 2 Reasoning and Proof “One meets his destiny often in the road he takes to avoid it.” ~ French Proverb

2. What You’ll Learn Next in this Chapter… (2.1/2.2) How to write special types of statements…conditionals, biconditionals, definitions, etc...

3. Conditional is an if-then statement that contains two parts. The part following the if is the Hypothesis. The part following then is the Conclusion. Example 1: If you live in a country that borders the United States, then you live in Canada. Hypothesis: You live in a country that borders the United States. Conclusion: You live in Canada. (2.1) Conditional Statements

4. Conditional is an if-then statement that contains two parts. The part following the if is the Hypothesis. The part following then is the Conclusion. Example 2: If two lines are parallel, then the lines are coplanar. (2.1) Conditional Statements Hypothesis: Two lines are parallel. Conclusion: The lines are coplanar.

5. Conditional is an if-then statement that contains two parts. The part following the if is the Hypothesis. The part following then is the Conclusion. Example 3: Write the statement as a conditional. An acute angle measures less than 90 degrees. (2.1) Conditional Statements Hypothesis : An angle is acute First part of the Conditional: “If an angle is acute” Conclusion : It measures less than 90 degrees Second part of the Conditional: “then it measures less than 90 degrees.” Conditional Statement: If an angle is acute, then it measures less than 90 degrees.

6. Conditional is an if-then statement that contains two parts. The part following the if is the Hypothesis. The part following then is the Conclusion. Example 4: Write a counterexample to make the conditional false. If x 2 > 0, then x > 0. The counterexample for the conditional is when the hypothesis is true and the conclusion is false. (2.1) Conditional Statements

7. Converse of a conditional switches the hypothesis and the conclusion. Example 1: Writing the converse of the conditional. Conditional: If two lines intersect to form right angles, then they are perpendicular. Converse: If two lines are perpendicular, then they intersect to form right angles. (2.1) Converse

8. Converse of a conditional switches the hypothesis and the conclusion. Example 2: Writing the converse of the conditional. Conditional: If x = 9, then x + 3 = 12 Converse: (2.1) Converse If x + 3 = 12, then x = 9

9. Converse of a conditional switches the hypothesis and the conclusion. Example 3: Write the converse of the conditional, determine the truth value of each. Conditional: If a 2 = 25, then a = 5 Converse: (2.1) Converse If a = 5, then a 2 = 25

10. Biconditionals When a conditional and its converse are true, you can combine them as a true biconditional. You can combine them by using the phrase if and only if. Example 1: Consider the conditional. Write its converse. If they are both true, combine the statements as a biconditional. Conditional: If two angles have the same measure, then the angles are congruent. Converse: Biconditional: (2.2) Biconditionals If two angles are congruent, then the angles have the same measure. Two angles have the same measure if and only if the angles are congruent.

11. Biconditionals When a conditional and its converse are true, you can combine them as a true biconditional. You can combine them by using the phrase if and only if. Example 2: Write the two statements that form each biconditional. You live in Washington, D.C. if and only if you live in the capital of the United States. (2.2) Biconditionals If you live in Washington, D.C., then you live in the capital of the United States. If you live in the capital of the United States, then you live in Washington, D.C.

12. Homework: Pgs. 83-85; 6-32 evens Pgs. 90-91; 2-16 evens, 27-30 all