Geometry Warm up Write the conditional, converse, inverse and contrapositive of the following sentences: 1.The sun is shining, it is warm. If the sun is shining, then it is warm. If it is warm, then the sun is shining. If the sun is not shining, then it is not warm. If it is not warm, then the sun is not shining. 2.All dogs shed. If it is a dog, then it sheds. If it sheds, then it is a dog. If it is not a dog, then it does not shed. If it does not shed, then it is not a dog.

2.4 Properties & Proofs I’m going to tell you about some algebraic properties. Properties that you already know and take for granted. However, in this class, we have to use the properties to PROVE stuff – so we need to be very comfortable and aware of what is around the corner.

Algebraic Properties of Equality Addition property: If a = b, then a + c = b + c Subtraction property: If a = b, then a – c = b – c Multiplication property: If a = b, then ac = bc Division property: If a = b, then a = b, c ≠0 c c

Try it with numbers…. Let, a = b = x = If, = and = Then = Substitution property: If a = b, then you may replace a with b in any true equation and the result will still be true. For instance if a = b and a = 17, then you can substitute 17 in for a and the equation becomes: 17 = b Generally, the substitution property looks like this: If a = b and a = x then b = x More properties

Just Three more… Transitive Property – looks like a syllogism: a = b b = c a = c Reflexive property: a = a Symmetric property: a = b and b = a

PROOFS… only the beginning A B C D Given: AB = CD (this will usually be the first item in your proof table) Prove: AC = BD (this will be the last item in your proof table – don’t use the word ‘prove’ in your table – ever!)) Statements Reasons AB = CD Given AB + BC = CD + BC Addition Property AC = BD Segment Addition Property You need to ask yourself – every time you make a statement – HOW DO I KNOW? You need to have a mathematically sound reason – which are the PROPERTIES. Mark your drawing

You just proved the Overlapping Segments Theorem! A B C D If AB = CD then AC = BD The converse is also true; If AC = BD then AB = CD How would you write the biconditional statement? AC = BD if and only if AB = CD

Overlapping Angles Theorem A B C D O If m >AOB = m > COD, then m> AOC = m> BOD The converse is also true: if m> AOC = m> BOD then m >AOB = m > COD

Assignment 2.4 A & B worksheet