Chapter 2: Reasoning & Proof Conditionals, Biconditionals, & Deductive Reasoning

Vocab conditional: an if-then statement –“if something happens, then something else happens” hypothesis: follows the if –what we think “might” happen conclusion: follows the then –what might happen if the hypothesis is true

Example 1 Identify the hypothesis and the conclusion of the conditional: If today is the first day of fall, then the month is September. H: C:

Example 2 You try: Identify the hypothesis and the conclusion: If y – 3 = 5, then y = 8. H: C:

Example 3 Write each sentence as a conditional: A rectangle has four right angles.

Example 4 Write each sentence as a conditional: A tiger is an animal.

Vocab truth value: tells whether a conditional is true or false –true if you find an example where the hypothesis is true and the conclusion is true –false if you can find a counterexample for which the hypothesis is true but the conclusion is false

Example 5 Show that this conditional is false by finding a counterexample: If it is February, then there are only 28 days in the month.

Example 6 Show that this conditional is false by finding a counterexample: If the name of a state contains the word New, then the state borders an ocean.

Converses A converse switches the hypothesis and the conclusion.

Example 7 Write the converse of the following conditional: If two lines intersect to form right angles, then they are perpendicular.

Example 8 Write the converse of the following conditional: If two lines are not parallel and do not intersect, then they are skew.

Summary conditionalp→qIf p, then q. converseq→pIf q, then p. p = hypothesis q = conclusion

Biconditionals when a conditional and its converse are true, you can combine them as a true biconditional connects the conditional and its converse with the word “and” written shorter by using “if and only if”

Example 9 Write the converse. If the converse is also true, combine the statements as a biconditional: If two angles have the same measure, then the angles are congruent.

Example 10 Write the converse. If the converse is also true, combine the statements as a biconditional: If three points are collinear, then they lie on the same line.

Example 11 Write two statements that form the biconditional: A number is divisible by 3 if and only if the sum of its digits is divisible by 3.

Summary biconditional statements: p↔q p if and only if q

Definitions A good definition: statement that help you identify or classify an object uses clearly understood terms precise reversible

Example 12 Show that this definition of perpendicular lines is reversible. Then, write it as a true biconditional: Perpendicular lines are two lines that intersect to form right angles.

Example 13 Show that the definition of right angle is reversible. Then, write it as a true biconditional: A right angle is an angle whose measure is 90.

Deductive Reasoning “logical reasoning” process of reasoning logically from given statements to a conclusion

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

Example 14 For the given true statements, what can you conclude? Given:If M is the midpoint of a segment, then it divides the segment into two congruent segments. M is the midpoint of AB.

Law of Syllogism If p→q and q→r are true statements, then p→r is a true statement. allows you to state a conclusion from two true conditional statements when the conclusion of one statement is the hypothesis of the other statement could be known as the transitive property

Example 15 Use the Law of Syllogism to draw a conclusion from the following true statements: If a number is prime, then it does not have repeated factors. If a number does not have repeated factors, then it is not a perfect square.

Example 16 If possible, state a conclusion using the Law of Syllogism. If it is not possible to use the Law, explain why: If a number ends in 6, then it is divisible by 2. If a number ends in 4, then it is divisible by 2.