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Deductive Reasoning
“The proof is in the pudding.”“Indubitably.”
Je solve le crime. Pompt de pompt pompt."
Le pompt de pompt le solve de crime!"
Proving TheoremsThe Midpoint Theorem
If m is the midpoint of , then
A M B
AB1
2AM AB
Written form
Given:
Prove:
.m is themidpt of AB
1
2AM AB
Hypothesis
Conclusion
Simplistic form
Proving TheoremsThe Midpoint Theorem
A M B
Given:
Prove:
.m is themidpt of AB1
2AM AB
Def of mp Def of mp
.m is themidpt of ABAM = MB
AM + MB = AB
AM + AM = AB
2 AM = AB
1
2AM AB
Statements Reasons
Given
Definition of midpoint
Segment Addition Post.
Substitution
Combine like terms
Division Prop. of Equality
__ __ 2 2 Mental step
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.
BeginningHypothesisGiven
EndConclusionTo 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?
3
21
E D
C
BA Given:
Prove:
m AEC m BED
1 2m m
m AEC m BED
1 2m AEC m m
2 3m BED m m
1 2 2 3m m m m
2 2m m
1 3m m
Label diagram to help visualize.
Given
Angle Add. Postulate
Angle Add. Postulate
Substitution Prop. Of Equality
Reflexive Prop. Of Equality
Subtr. Prop. Of Equality
?
?
Statements Reasons
4
F
32
1
E
DCB
A
Given:
Prove:
2 3m m 1 4m m
m AEC m CEF
Statements Reasons
Label diagram to help visualize.
1 4m m 2 3m m
1 2m AEC m m 1 2 3 4m m m m
3 4m CEF m m
m AEC m CEF Substitution Prop. Of Equality
Angle Add. Postulate
Angle Add. Postulate
Addition Prop. Of Equality
Given
Given
g
g
gg
? ?
UR N
KLWGiven:
Prove:
Label diagram to help visualize.
WK = RNLK = RU
WL = UN
Statements Reasons
WK = RN
____ = WL + LK
____ = RU + UN
WL + LK = RU + UN
WL = UN
Given
Given
Segment Add. Postulate
Segment Add. Postulate
Substitution Prop. Of Equality
Subtr. Prop. Of Equality
LK = RU
WK
RN
g
g g
g
?
?
UR N
KLW
Statements Reasons
WK = RN
____ = WL + LK
____ = RU + UN
WL + LK = RU + UN
WL = UN
Given
Given
Segment Add. Postulate
Segment Add. Postulate
Substitution Prop. Of Equality
Subtr. Prop. Of Equality
LK = RU
WK
RN
g
g g
g
?
? Notice that the given doesn’t have to come first. You can put it in when it is needed.
Let’s try an alternate approach to the same problem.
Given:
Prove:
Label diagram to help visualize.
WK = RNLK = RU
WL = UN
Statements Reasons
WK = RNLK = RU
____ = WL + LK
____ = RU + UN
WL + LK = RU + UN
WL = UN
GivenGiven
Segment Add. Postulate
Segment Add. Postulate
Substitution Prop. Of Equality
Subtr. Prop. Of Equality
WK
RN
UR N
KLW g
g g
g
?
?
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.
