Transcript 2.4 Properties & Proofs

Geometry Warm up
•
1.
2.
Write the conditional, converse, inverse and
contrapositive of the following sentences:
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.
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
More properties
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
• Try it with numbers….
Let,
a=9+1
b=7+3
x=5+5
If, 9 + 1 = 7 + 3
and 9 + 1 = 5 + 5
Then 7 + 3 = 5 + 5
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
Mark your
AB = CD
Given
drawing
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.
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