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Geometry Stuff that should be in 2.2 and 2.3

Learn about conditional statements, their inverse, converse, and contrapositive forms, as well as biconditional statements and adjacent angles.

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Geometry Stuff that should be in 2.2 and 2.3

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  1. GeometryStuff that should be in 2.2 and 2.3 Counterexample – an example used to show something is false. A syllogism is also known as DEDUCTIVE REASONING.

  2. Inverse Statement Negate the hypothesis and conclusion of the conditional statement. Conditional: If it’s hot outside, then the sun is shining. Inverse: If it’s not hot outside, then the sun is not shining.

  3. Contrapositive Statement SWITCH the hypothesis and conclusion of the INVERSE statement. Conditional: If it’s hot outside, then the sun is shining. Inverse: If it’s not hot outside, then the sun is not shining. Contrapositive: If the sun is not shining, then it is not hot outside.

  4. Notation • Conditional statement a  b • Converse statement b  a • Inverse statement ~a  ~b • Contrapositive statement ~b  ~a

  5. Example/Practice • Using the sentence: “My dog has fleas”, write the (a) conditional, (b) converse, (c) inverse and (d) contrapositivestatements. 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.

  6. 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. ab (true) Converse: If it is a rectangle, then it is a quadrilateral with 4 right angle. 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.

  7. a b c e d Floppers Which ones are Floppers? Not Floppers 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.

  8. 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

  9. 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. Adjacent AnglesWhat 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 C X D

  10. Practice • Yes or no: are these adjacent angles? A B C

  11. Assignment • pg 95, write inverse and contrapositive for #9 and #17 • pg 103, 8 & 9, 17-27, 30-38

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