1. Lesson 2-3 Conditional Statements

2. 5-Minute Check on Lesson 2-2 Transparency 2-3 Use the following statements to write a compound statement for each conjunction or disjunction. Then find its truth value. p: 12 + –4 = 8 q: A right angle measures 90 degrees. r: A triangle has four sides. 1. p and r 2. q or r 3. ~p  r 4. q  ~r 5. ~p  ~q 6. Given the following statements, which compound statement is false? s: Triangles have three sides.q: 5 + 3 = 8 Standardized Test Practice: ACB D s  qs  q s  qs  q ~s  ~q ~s  q

3. 5-Minute Check on Lesson 2-2 Transparency 2-3 Use the following statements to write a compound statement for each conjunction or disjunction. Then find its truth value. p: 12 + –4 = 8 q: A right angle measures 90 degrees. r: A triangle has four sides. 1. p and r12 + –4 = 8 and a triangle has four sides; FALSE 2. q or rA right angle measures 90 degrees or a triangle has four sides; TRUE 3. ~p  r12 + –4  8 or a triangle has four sides; FALSE 4. q  ~rA right angle measures 90 degrees and a triangle does not have four sides; TRUE 5. ~p  ~q12 + –4  8 or a right angle does not measure 90 degrees; FALSE 6. Given the following statements, which compound statement is false? s: Triangles have three sides.q: 5 + 3 = 8 Standardized Test Practice: ACB D s  qs  q s  qs  q ~s  ~q ~s  q

4. Objectives Analyze statements in if-then form Write the converse, inverse and contrapositive of if-then statements

5. Vocabulary Implies symbol (→) Conditional statement – a statement written in if-then form Hypothesis – phrase immediately following the word “if” in a conditional statement Conclusion – phrase immediately following the word “then” in a conditional statement Converse – exchanges the hypothesis and conclusion of the conditional statement Inverse – negates both the hypothesis and conclusion of the conditional statement Contrapositive – negates both the hypothesis and conclusion of the converse statement Logically equivalent – multiple statements with the same truth values Biconditional – conjunction of the conditional and its converse

6. If-then Statement: if, then or p implies q or in symbols p → q

7. Example: If two segments have the same measure, then they are congruent Hypothesis p two segments have the same measure Conclusion q they are congruent StatementFormed bySymbolsExamples Conditional Given hypothesis and conclusion p → q If two segments have the same measure, then they are congruent Converse Exchanging the hypothesis and conclusion of the conditional q → p If two segments are congruent, then they have the same measure Inverse Negating both the hypothesis and conclusion of the conditional ~p → ~q If two segments do not have the same measure, then they are not congruent Contra- positive Negating both the hypothesis and conclusion of the converse ~q → ~p If two segments are not congruent, then they do not have the same measure Related Conditionals:

8. Identify the hypothesis and conclusion of the following statement. If a polygon has 6 sides, then it is a hexagon. Answer: Hypothesis: a polygon has 6 sides Conclusion: it is a hexagon hypothesis conclusion

9. Answer: Hypothesis: Tamika completes the maze in her computer game Conclusion: she will advance to the next level of play Identify the hypothesis and conclusion of the following statement. Tamika will advance to the next level of play if she completes the maze in her computer game.

10. Determine the truth value of the following statements for each set of conditions. If it rains today, then Michael will not go skiing. a. It does not rain today; Michael does not go skiing. b. It rains today; Michael does not go skiing. c. It snows today; Michael does not go skiing. d. It rains today; Michael goes skiing. Answer: true Answer: false

11. Write the converse, inverse, and contrapositive of the statement All squares are rectangles. Determine whether each statement is true or false. If a statement is false, give a counterexample. Conditional: If a shape is a square, then it is a rectangle. The conditional statement is true. First, write the conditional in if-then form. Write the converse by switching the hypothesis and conclusion of the conditional. Converse: If a shape is a rectangle, then it is a square. The converse is false. A rectangle with ℓ = 2 and w = 4 is not a square.

12. Inverse: If a shape is not a square, then it is not a rectangle. The inverse is false. A 4-sided polygon with side lengths 2, 2, 4, and 4 is not a square, but it is a rectangle. The contrapositive is the negation of the hypothesis and conclusion of the converse. Contrapositive: If a shape is not a rectangle, then it is not a square. The contrapositive is true.

13. Write the converse, inverse, and contrapositive of the statement The sum of the measures of two complementary angles is 90 . Determine whether each statement is true or false. If a statement is false, give a counterexample. Answer: Conditional: If two angles are complementary, then the sum of their measures is 90  ; true. Converse: If the sum of the measures of two angles is 90 , then they are complementary; true. Inverse: If two angles are not complementary, then the sum of their measures is not 90  ; true. Contrapositive: If the sum of the measures of two angles is not 90 , then they are not complementary; true.

14. Conditional If two lines are perpendicular, then their angle is right. Converse Inverse Contrapositive Conditional Converse If two angles are supplementary, then they sum to 180º Inverse Contrapositive Conditional Converse Inverse If today is not Friday, then we do not have a quiz. Contrapositive Conditional Converse Inverse Contrapositive If two angles aren’t a linear pair, then they aren’t supplementary.

15. Conditionals in Symbols StatementsSymbology ConditionalP →Q ConverseQ→P Inverse~P →~Q Contrapositive~Q→~P

16. Summary & Homework Summary: –Conditional statements are written in if-then form –Form the converse, inverse and contrapositive of an if-then statement by using negations and by exchanging the hypothesis and conclusion Homework: Day 1: pg 78, 5, 6, 8, 9, 13, 17, 21, 23, 25, 27 Day 2: pg 79-81, 43, 45, 53, 55, 57, (pg 81 1, 3)