Download Introduction to Two Column Proofs

Introduction to Two-Column
Proofs
Lesson 1.7
Geometry as a Language
Which is more important to a problem, the statement of
the problem or the given diagram? Explain.
Things to Assume
Straight lines and angles
Collinearity of points
Betweenness of points
Things to definitely NOT
Assume
Right angles
Congruency
Relative s size of segments or
angles
Postulate: If a statement is a postulate, then it is an
unproved assumption.
Theorem: If a statement is a theorem then it can be proved.
What are examples of postulates we have covered so far?
Why do we need them?
What do you notice about the structure of each of the
statements?
If___________, then__________.
Conditional Statement: If a statement is a conditional
statement , then it is written in the form “if…then…”
Parts of a conditional statement
• Hypothesis: the “if” part of the statement
• Conclusion: the “then” part of the statement
Hypothesis: You don’t study
Conclusion:
You won’t pass
your exam
Write the following statements as
conditional statements:
1. Expensive cars are reliable
2. Right angles are congruent.
If it’s a theorem, then…
What does a proof look like
Reason
Statement
1.
2.
3.
4.
5.
6.
This is where the
equations belong.
Where we want
to see what we
want to prove
GEOMETRIC
LANGUAGE
1.
2.
3.
4.
5.
6.
The reasoning and the
“why” for each of the
Statements
What allows you to
write each of these
statements
ENGLISH LANGUAGE
Theorem: If two angles are right angles, then they are congruent
Statement
1. Two angles are right
angles
2. 𝑚⊾ = 90
3. all right angles are
congruent
Reason
1. Given
2. Definition of Right
Angles
3. Definition of
Congruency
Things to know about proofs
1. Once a theorem is proven, it can be used to prove others
(it does not have to be proven again)
2. It’s simple for something to be proven and called a
theorem, but only things that are going to be consistently
used will classify as theorems
3. Most theorems connect different definitions like
congruency and measure (definitions are used to prove
theorems)
1. Given: A is a right angle
C is a right angle
Prove: A  C
Statement
Reason
Given: Diagram as shown
Prove: GKJ  IKH
Statement
Reason
1.Given: Diagram as shown
Prove: KHI  JHG
2. Given: mIGJ  50
mJGK  40
L is a right angle
Prove: IGK  L
3. Given: GI  2.5cm
JK  2.5cm
Prove: GI  JK