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Geometry Warm up
<=
span
style=3D'font-size:75%'>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, t=
hen it
is not warm.
• If it is not warm, then the =
sun is
not shining.
<=
span
style=3D'font-size:75%'>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
•<=
span
style=3D'font-size:86%'>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.
•
![](3D"slide0001_image002.jpg")
Algebraic Properties of Equality
•Addition property:
If a =3D b, then a + c =3D b + c
<=
span
style=3D'font-size:75%;visibility:hidden'>•
•Subtraction property: If a =3D b, then a – c =3D =
b –
c
<=
span
style=3D'font-size:75%;visibility:hidden'>•
•Multiplication property: If a =3D b, then ac =3D bc
<=
span
style=3D'font-size:75%;visibility:hidden'>•
•Division property:
If a =3D b, then a =3D b=
, c ≠0
• c c
<=
span
style=3D'font-size:71%'>•Try it with
numbers….
Let,
a =3D 9 + 1
b =3D 7 + 3
x =3D 5 + 5
•
If, 9 + 1 =3D 7 + 3
and 9 + 1 =3D 5 + 5
Then 7 + 3 =3D 5 + 5
•
•
Substitution propert=
y: If a =3D b, then you may replace a =
with b
in=
any
true equation and the result will still be true.
For instance if a =3D =
b and a
=3D 17, then you can substitute 17 in for a and the equation =
becomes: 17 =3D b
<=
/span>
•
Generally, the substitu=
tion property looks like
this:
<=
span
style=3D'mso-special-format:nobullet;display:none'>•If a =3D
b
=
and a =3D x
=
then b =3D x
•
More properties
![](3D"slide0004_image005.gif")
Just Three more…
Transitive Property – looks like a syllogism=
:
a =3D b
b =3D c
a =3D c
•
Reflexive property: a =3D a
•
Symmetric property: a =3D b and =
b =3D a
![](3D"slide0004_image006.gif")
PROOFS… only the beginning
• A =
;
B =
;
C =
;
D
•
•Gi=
ven: AB =3D CD (this will usually be the first item in your proof table)
•Prove: =
AC =3D BD (this will be the last item in yo=
ur
proof table – don’t use the word ‘prove’ in your table –
ever!))
=
Statements =
; &n=
bsp;
Reasons
=
AB =3D CD =
;
Given
=
AB + BC =3D CD + BC Addition Property
=
AC =3D BD =
; &n=
bsp;
Segment Addition Property
•
=
You need to ask yourself – every time you ma=
ke a
statement –
=
=
; &n=
bsp; =
;
HOW DO I KNOW?
=
You need to have a mathematically sound reason =
211;
which are the
=
=
; &n=
bsp; =
;
PROPERTIES.
![](3D"slide0005_image007.gif")
![](3D"slide0005_image008.gif")
![](3D"slide0005_image009.gif")
Mark your drawing
![](3D"slide0005_image010.gif")
![](3D"slide0005_image011.gif")
![](3D"slide0005_image012.gif")
![](3D"slide0005_image013.gif")
You
just proved the
Overlapping Segme= nts Theorem!
Overlapping Segme= nts Theorem!
•
•
•
• A =
;
B =
;
C =
;
D
If AB =3D CD then AC =3D BD=
3;
•The con=
verse
is also true;
•If AC =3D BD then AB =3D CD
•How wou=
ld you
write the biconditional statement?
•AC
=3D BD if and only if AB =3D CD
![](3D"slide0006_image014.gif")
•A<=
span
style=3D'mso-spacerun:yes'> =
;
B =
;
C &nb=
sp;
D
•
•
•
•
•
• =
;
O
•If=
m >AOB =3D m > COD, then=
3;
=
m> AOC =3D m> BOD
•
=
The conve=
rse is
also true:
=
if m> AOC =3D m> BOD
=
then m >AOB =3D m > COD
•
•
![](3D"slide0007_image015.gif")
![](3D"slide0009_image016.gif")
Assignment
•2.4 A & B
•worksheet
' + ENDSHOW_MESG + '
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tes![](3D"buttons.gif")
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