2.2 Conditional Statements Goal: Students will be able:  To recognize conditional statements and their parts.  To write converses, inverses, and contrapositives.

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2.2 Conditional Statements Goal: Students will be able:  To recognize conditional statements and their parts.  To write converses, inverses, and contrapositives of conditional statements.

Conditional Statement  A statement that can be written in “if – then” form.  Symbol: p → q, read if p then q, or p implies q.  Example: If it rains on Thursday, then the baseball game will be canceled. p p q

Hypothesis  The phrase immediately following the word if in a conditional statement  The p part following if.  The phrase immediately following the word then in a conditional statement. Conclusion If p, then q. Hypothesis Conclusion

Examples 1: Determine Hypothesis and conclusion If trout are fish, then trout live in a pond. If you buy a car, then you get $1500 cash back. Hypothesis: trout are fish Conclusion: trout live in a pond Hypothesis: you buy a car Conclusion: you get $1500 cash back

Got it 1? Identify the hypothesis and conclusion of each statement. If an animal is a robin, then the animal is a bird. If an angle measures 180°, then the angle is obtuse. If a polygon has 6 sides, then it is a hexagon. Hypothesis: an animal is a robin Conclusion: the animal is a bird. Hypothesis: an angle measures 180° Conclusion: the angle is obtuse Hypothesis: a polygon has 6 sides Conclusion: it is a hexagon

Example 2: Writing a Conditional How can you write the following statement as a conditional? Vertical angles share a vertex. Step 1: Identify the hypothesis and conclusion. In order for two angles to be vertical, they must share a vertex. So the set of vertical angles is inside the set of angles that share a vertex. Hypothesis: Vertical anglesConclusion: share a vertex. If two angles are vertical angles, then they share a vertex.

Got it 2? How can you write “Dolphins are mammals” as a conditional? Mammals Dolphins If an animal is a dolphin, then it is a mammal.

Truth Value  Is either true or false  To show that a conditional is true, show that every time the hypothesis is true, the conclusion is also true.  To show that a conditional is false, find only one counterexample, where the hypothesis is true, and the conclusion is false.

Example 3: Finding the truth value of a conditional. Is the conditional true or false? If it is false, find a counterexample. If a number is divisible by 3, then it is odd. The conclusion is false. The number 12 is divisible by 3, and 12 is even.

Got it? 3: Finding the truth value of a conditional. Is the conditional true or false? If it is false, find a counterexample. If a month has 28 days, then it is February. If two angles form a linear pair, then they are supplementary. False, January has 28 days plus 3 more. True

Negation  Is the opposite of the original statement ~p: The sky is not blue. p: The sky is blue. Examples:  ~p, read not p  Symbol: ~ ~q: A triangle does not have 4 sides. q: A triangle has 4 sides.

Converse, Inverse, and Contrapositive StatementFormed bySymbolsExamples Conditional Converse Inverse Contrapositive Given hypothesis and conclusion p → q If two angles have the same measure, then they are congruent. Exchange the hypothesis and conclusion of the conditional q → p If two angles are congruent, then they have the same Measure. Negate both the hypothesis and conclusion of the conditional ~p → ~ q If two angles do not have the same measure, then they are not congruent. Exchange and Negating both the hypothesis and conclusion of the conditional ~q → ~ p If two angles are not congruent, then they do not have the same measure.

Equivalent Statements  Have the same truth value The conditional and the contrapositive are equivalent statements. The converse and the inverse are equivalent statements.

Biconditional Statements:  When a conditional statement and its converse are both true, you can write them as a single biconditional statement.  A biconditional statement is a statement that contains the phrase “if and only if.” Any valid definition can be written as a biconditional statement.

Example 1: Rewrite the conditional statement in if-then form. All birds have feathers. Conditional: If it is a bird, then it has feathers. Conditional: If two angles are a linear pair, then they are supplementary. Two angles are supplementary if they are a linear pair.

Example 2: Write the converse, inverse, and contrapositive of the following conditional? If a dog is a Great Dane, then it is large. Converse: If the dog is large, then it is a Great Dane. Inverse: If the dog is not a Great Dane, then it is not large. Contrapositive: If the dog is not large, then it is not a Great Dane.

Example 4. Write the definition of perpendicular lines as a biconditional. Definition: If two lines intersect to form a right angle, then they are perpendicular. Converse: If two lines are perpendicular, then they intersect to form right angles. Biconditional: Lines intersect to form right angles iff they are perpendicular lines.