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Chapter 2: Deductive Reasoning. 2.1 If-Then Statements and Converses. Normal Students vs. Geometry Students. “ If it rains today, then the tennis match will be cancelled. ” “ If B is between A and C, then AB + BC = AC. ”. Conditional Statement (or conditionals).
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Chapter 2: Deductive Reasoning 2.1 If-Then Statements and Converses
Normal Students vs.Geometry Students • “If it rains today, then the tennis match will be cancelled.” • “If B is between A and C, then AB + BC = AC.”
Conditional Statement (or conditionals) • Both of the previously statements are conditional statements… • IF_________________, THEN_____________. • What we put on the red line is the hypothesis. • What we put on the blue line is the conclusion.
Name the hypothesis and the conclusion for the following: • “If it rains today, then the tennis match will be cancelled.” • “If B is between A and C, then AB + BC = AC.” • Generally, we say: IF p, THEN q.
Converse • Original statement: If p, then q. • Converse: If q, then p. • Both true?
Counterexample • Conditional Statement: • If John lives in Texas, then he lives south of Canada • Converse: • Counterexample:
Biconditionals • If a conditional and its converse are both true, they can be combined into a single statement by using the words “if and only if”. The “if and only if” statement is called a biconditional. • **Every definition can be written as a biconditional.**
“iff”= “if and only if” • Definition: • Congruent segments are segments that have equal lengths. • Biconditional • Segments are congruent if and only if their lengths are equal.
