“Indubitably.”
“The proof is in the pudding.”
Le pompt de pompt le solve de crime!"
Deductive Reasoning
Je solve le crime. Pompt de pompt pompt."
Proving Theorems
The Midpoint Theorem
If m is the midpoint of A B , then A M 
A
M
Hypothesis
Conclusion
B
1
AB
2
Written form
Simplistic form
Given: m is the m idpt . of A B
Prove: A M 
1
2
AB
Proving Theorems
The Midpoint Theorem Given:
A
M
Def of mp
B
m is the m idpt . of A B
AM = MB
AM + MB = AB
AM + AM = AB
2__
AM = __
AB
AM 
1
2
Prove: A M 
Def of mp
Statements
2
m is the m idpt . of A B
2
AB
1
AB
2
Reasons
Given
Definition of midpoint
Segment Addition Post.
Substitution
Combine like terms
Mental step
Division Prop. of Equality
As you continue to learn to prove theorems,
it is important to get accustomed to the various
forms of stating a theorems.
You need to get accustomed to using the
hypothesis and the conclusion.
This will take time. You will be given enough time.
You are learning a new way of thinking.
Understanding the importance of this
type of thinking will take even more time.
Please be patient!!! Let’s begin.
These proofs are using deductive reasoning.
Deductive reasoning is a linking of
cause and effect statements to reach a conclusion.
It is like a chain that starts and moves to other the end.
Beginning
Hypothesis
Given
End
Conclusion
To Prove
It is also like a row of dominos.
Like dominos which have two
blocks on each tile, each step of the
proof also has two parts – statements
and reasons.
Statements without reasons are
not valid or substantiated.
Each line of the proof must contain a statement
supported by a reason or justification.
When this happens, the hypothesis logical
forces you to accept the conclusion just like a row of
dominos being toppled.
How are we going to learn how to do proofs?
1] Examine the structure of completed proofs.
2] Fill in the reasons for steps of partially
completed proofs.
3] Fill in proofs that have blanks on
either the statement or the reason side.
4] Do our own proofs with simple geometry
and simple algebra.
This is a slow very process.
However, it is relatively painless.
It leads to success and understanding.
How many proofs will we do?
How many swings at a baseball
before you are a good hitter?
B
Given: m  A E C  m  B E D
A
?
1
C
2
3
?
D
E
Statements
m  AEC  m  BED
m  AEC  m  1  m  2
m  BED  m  2  m  3
m 1 m  2  m  2  m  3
m2m2
m 1 m  3
Prove: m  1  m  2
Label diagram to help visualize.
Reasons
Given
Angle Add. Postulate
Angle Add. Postulate
Substitution Prop. Of Equality
Reflexive Prop. Of Equality
Subtr. Prop. Of Equality
B
g
?
C
g
D
?
4
1
m2m3
Prove: m  A E C  m  C E F
2 3
g
Given:
m 1 m  4
g
Label diagram to help visualize.
A
E
F
Statements
Reasons
m 1 m  4
Given
m2m3
Given
m 1 m  2  m  3  m  4
m  AEC  m  1  m  2
m  CEF  m  3  m  4
m  AEC  m  CEF
Addition Prop. Of Equality
Angle Add. Postulate
Angle Add. Postulate
Substitution Prop. Of Equality
W
?
g
R
L gK
Given: WK = RN
LK = RU
g
?
U
Prove:
N
g
WL = UN
Label diagram to help visualize.
Statements
Reasons
Given
WK = RN
WK = WL + LK
____
Segment Add. Postulate
RN = RU + UN
____
Segment Add. Postulate
WL + LK = RU + UN
LK = RU
WL = UN
Substitution Prop. Of Equality
Given
Subtr. Prop. Of Equality
W
?
g
R
L gK
g
?
U
g
Notice that the given
doesn’t have to come first.
You can put it in when it is
N needed.
Statements
Reasons
Given
WK = RN
WK = WL + LK
____
Segment Add. Postulate
RN = RU + UN
____
Segment Add. Postulate
WL + LK = RU + UN
LK = RU
WL = UN
Substitution Prop. Of Equality
Given
Subtr. Prop. Of Equality
Let’s try an alternate
approach to the same
problem.
W
g
R
L gK
?
Given: WK = RN
LK = RU
g
?
U
Prove:
N
g
WL = UN
Label diagram to help visualize.
Statements
Reasons
Given
Given
WK = RN
LK = RU
WK = WL + LK
____
Segment Add. Postulate
RN = RU + UN
____
Segment Add. Postulate
WL + LK = RU + UN
Substitution Prop. Of Equality
WL = UN
Subtr. Prop. Of Equality
Did you notice that it was not as easy to
follow?
It is not wrong. But understanding is
more difficult. In geometry there are often
many correct approaches to the same
problem.
If you find it easier to put the given down
first. Do it. Eventually, you will learn
easier ways to do your proofs.
It is not uncommon to be able to complete
only 50% of yourproofs
It is not uncommon to be able to
complete only 50% of your homework.
This is the nature of geometry in the
beginning. Do not feel bad.
Many students experience frustration in
the beginning. Persevere and the
frustration will disappear.
C’est fini.
Good day and good luck.