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Bellringer • Find a counterexample to show that each statement is false. – 1. You can connect any four points to form a rectangle. – 2. The square of a number is always greater than the number. • Write the converse of each statement ; determine the truth value. – 3. If today is September, then tomorrow is October 1. – 4. If AB is the perpendicular bisector of CD, then AB and CD are perpendicular. Answers • 1. 4 collinear points • 2. 0.5 • 3. If tomorrow is Oct. 1, then today is Sept; both • 4. If AB and CD are perpendicular, then AB is the perpendicular bisector of CD Geometry Chapter 2: Reasoning and Proofs 2.3 Biconditionals and Definitions Revisiting 2.2 Conditionals • Conditional– Original statement • Consists of hypothesis and conclusion • If and then statement : p→ q • Converse– Statement switches p and q – Becomes conclusion then hypothesis – If and then statement is q → p Q=then you should eat vegetables. P=If you want to be healthy 2.2 Conditional con’t • Inverse– Original statement with Negation (meaning not) both the hypothesis and conclusion • If not p then not q : ~p→~q • Contrapositive– Converse statement with Negation both the conclusion and hypothesis. • If not q then not p: ~q → ~p Statement Hypothesis Conclusion Conditional If P Then Q Converse If Q Then P Inverse If not P Then not Q Contrapostive If not Q Then not P Lesson Purpose Objective Essential Question • To write biconditionals and recognize good definitions. • How can you make a conjecture and prove that it’s true? Definitions • A bi-conditional – is a single true statement • that combines a true conditional and its converse. • Combines p→q and q→p as p ↔ q or q ↔ p. • “p if and only if q” • For instance: • A point is a midpoint if and only if it divides a segment into two congruent segments. – You can write it as two conditionals that are converses. “+” Converse =Biconditional If p then q + If q then p =p if and only if q p→q + q→p = p ↔ q or q↔p Conditional Identifying the Conditional in a Biconditional • A ray is an angle bisector if and only if it divides an angle into two congruent angles. – Let p and q represent the following: – p: A ray is an angle bisector – q: A ray divides an angle into two congruent angles • p→ q: If a ray is an angle bisector, then it divides an angle into two congruent angles. • q → p : If a ray divides an angle into two congruent angles, then it is an angle bisector. Examples • Name the two conditionals that form the biconditional. – Two numbers are reciprocals if and only if their product is 1. – If two numbers are reciprocals, then their product is 1. – If the product of two numbers is 1, then the numbers are reciprocals. Writing a Biconditional • What is the converse of the following true conditional? – If two angles have equal measure, then the angles are congruent. • Converse: If two angles are congruent, then the angles have equal measure. – Now Rewrite the statement as a biconditional. • Two angles have equal measure if and only if the angles are congruent. Examples • Indicate the converse, then write a biconditional if converse is true. – If two angles have equal measures, then the angles are congruent. – Converse: if two angles are congruent, then the angles have equal measures; true – Biconditional: Two angles have equal measures if and only if the angles are congruent Real- World Connections Recap:Summary • When a conditional and its converse are true, you can combine them as a true biconditional using the phrase if and only if. • The symbolic form of a biconditional is p can write a good definition as a true biconditional. q. You Ticket Out Ticket Out • What is the definition of a biconditional? Classwork • Sect 2.3- Pg 108-109 8, 9, 14, 16, 21,22, 28, 30, 35