VIEWS: 0 PAGES: 16 POSTED ON: 10/1/2012 Public Domain
Geometry Chapter 2 section 3 Pages #98 - 100 • Biconditionals and Definitions. • Standard addressed: Construct and judge the validity of a logical argument and give counterexamples to disprove a statement. Lesson 2-3: Biconditionals and Definitions 1 The student will be able to: • Write Biconditional Statements • Recognize good definitions Lesson 2-3: Biconditionals and 2 Definitions Review Notation Let a = the figure is a triangle Let b = it has three sides 1. Conditional statement a b (If a, then b) 2. Converse statement b a (If b, then a) 3. Inverse statement ~a ~b (If not a, then not b) 4. Contrapositive statement ~b ~a (If not b, then not a) Example/Practice • Using the sentence: “My dog has fleas”, write the (a) conditional, (b) converse, (c) inverse and (d) contrapositive statements. a) If it is my dog, then it has fleas b) If it has fleas, then it is my dog. c) If it is not my dog, then it does not have fleas. d) If it does not have fleas, then it is not my dog. Biconditional • When a conditional statement and its converse are both true, the two statements may be combined into a true biconditional statement. • Use the phrase if and only if Statement: If an angle is right angle, then it has a measure of 90. Converse: If an angle has a measure of 90, then it is a right angle. Biconditional: An angle is right angle if and only if it has a measure of 90. Lesson 2-3: Biconditionals and 5 Definitions Biconditional Statement When a conditional statement AND the converse are BOTH TRUE, this creates a special case called ‘biconditional”. Conditional: If a quadrilateral has 4 right angles, then it is a rectangle. ab (true) Converse: If it is a rectangle, then it is a quadrilateral with 4 right angles. ba (true) Biconditional: A quadrilateral has 4 right angles if and only if it is a rectangle. (don’t use if and then) a b (true BOTH ways) iff means “if and only if” A biconditional is a statement that is true backwards and forwards. A biconditional is a DEFINITION. Floppers a b c d e Floppers Which ones are Floppers? Not Floppers Only b and c are floppers • Let’s write a definition: Step 1: write a conditional statement: If a figure is a Flopper, then it has one eye and two tails. (true) Step 2: write the converse: If a figure has one eye and two tails, then it is a Flopper. (true) Step 3: write the biconditional (definition) A figure is a Flopper if and only if it has one eye and two tails. Or A figure is a Flopper iff it has one eye and two tails. Quick Review If the original conditional is true AND the converse is true, then the statement is a definition. This statement is called a BICONDITIONAL Notation: p q (note the double arrow) We say: “p if and only if q” This can be abbreviated to: p iff q Adjacent Angles What does ‘adjacent’ mean? • Adjacent angles are angles that share a vertex and a side. They do not share any interior points. (They don’t overlap.) A B AXB & BXC are adjacent C angles. They share the vertex X and the ray XB. X D AXC & BXC are NOT adjacent angles. They share the vertex X but they overlap thus causing the sharing of interior points. Practice • Yes or no: are these adjacent angles? A No these angles do not share a vertex or a side B Yes C NO these angles share a side but not a vertex Biconditional - Example Symbology: pq “p if and only if q” Lesson 2-3: Biconditionals and 11 Definitions Biconditional - Example Biconditional: A ray is an angle bisector if and only if it divides an angle into two congruent angles. 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, . Lesson 2-3: Biconditionals and 12 Definitions Definitions A good definition has the following components: - Uses clearly understood terms. - Is precise (avoids words such as large, sort of, some, etc.). - Is reversible. You can write it as a true biconditional. Lesson 2-3: Biconditionals and 13 Definitions Definitions Lesson 2-3: Biconditionals and 14 Definitions Definitions Lesson 2-3: Biconditionals and 15 Definitions Objective Practices • Lesson Check: pg. 101 #1-6 – Check if off with me when finished before the end of the period • HW pgs. 101 – 104 #7 – 45 odds DUE Thursday Nov 10