Geometry Chapter 2 section 3 Pages # 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 Definitions1

The student will be able to: Write Biconditional Statements Recognize good definitions 2 Lesson 2-3: Biconditionals and Definitions

Review Notation 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) Let a = the figure is a triangle Let b = it has three sides

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.

Lesson 2-3: Biconditionals and Definitions 5 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 .

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. a  b (true) Converse: If it is a rectangle, then it is a quadrilateral with 4 right angles. b  a (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. Biconditional Statement

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. Floppers a b c d e Which ones are Floppers? Not Floppers Only b and c are floppers

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.) ABCXDABCXD  AXB &  BXC are adjacent angles. They share the vertex X and the ray XB.  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 B Yes C No these angles do not share a vertex or a side NO these angles share a side but not a vertex

Biconditional - Example Lesson 2-3: Biconditionals and Definitions 11 Symbology: “p if and only if q”

Biconditional - Example Lesson 2-3: Biconditionals and Definitions 12 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.

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 Definitions 13

Definitions Lesson 2-3: Biconditionals and Definitions 14

Definitions Lesson 2-3: Biconditionals and Definitions 15

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