Section 2.6
A proof is a logical argument that shows a
statement is true.
The first proof format we will use is called a twocolumn proof.
A two-column proof has numbered statements and
corresponding reasons that show an argument in
logical order.
In a two-column proof, each statement in the lefthand column is either given information or the
result of applying a known property or fact to
statements already made. Each reason in the righthand column is the explanation for the
corresponding statement.
Example 1
Use the diagram to prove m1 = m4.
Given: m2 = m3, mAXD = mAXC
Prove: m1 = m4
1.
Statements
m2 = m3,
mAXD = mAXC
mAXD = m2 + m1
mAXC = m3 + m4
Reasons
1.
Given
2.
 Add. Post.
3.
m2 + m1 = m3 + m4 3.
Subst. Prop.
4.
m1 = m4
2.
4.
Subt. Prop.
Theorems
A theorem is a statement that can be proven.
Once you have proven a theorem, you can use
the theorem as a reason in other proofs.
The reason used in a proof can include
definitions, properties, postulates and theorems.
Congruence of Segments Theorem
Segment congruence is reflexive, symmetric, and
transitive.
Reflexive Property:
For any segment AB, AB  AB.
Symmetric Property:
If AB  CD, then CD  AB.
Transitive Property:
If AB  CD and CD  EF , then AB  EF .
Congruence of Angles Theorem
Angle congruence is reflexive, symmetric, and
transitive.
Reflexive Property:
For any angle A, A  A.
Symmetric Property:
If A  B, then B  A.
Transitive Property:
If A  B and B  C , then A  C.
Example 2
Name the property illustrated by the statement.
a.
If 5  3, then 3  5
Symmetric Property
b. If H  T and T  B, then H  B.
Transitive Property
In Section 2.6, most of the proofs involve showing
that congruence and equality or equivalent. You
may find that what you are asked to prove seems to
be obviously true. It is important to practice
writing these proofs so that you will be prepared to
write more complicated proofs in later chapters.
Example 3
Given: Ray BD bisects ABC
Prove: mABC = 2 ∙ m1
1.
Statements
Ray BD bisects ABC
1.
Reasons
Given
2.
m1 = m2
2.
Def. of  Bis.
3.
m2 + m1 = mABC
3.
 Add. Post.
4.
mABC = 2 ∙ m1
4.
Subst. Prop.
Writing a Two-Column Proof
1.
2.
3.
4.
Draw or copy diagrams and label given
information to help develop proofs.
Write or copy the Given and Prove
statements.
The first statement and reason pair you
write is given information
The statements are based on facts that you
know or on conclusions from deductive
reasoning.
5.
6.
7.
The reasons are definitions, postulates,
properties, or proven theorems that allow you
to state the corresponding statement.
Always remember to number the statements
and reasons.
Remember to give a reason for the last
statement which will always be the Prove
Statement.
Example 4
Write a two-column proof for the following
statement.
If you know that M is the midpoint of segment
AB, N is the midpoint of segment CD, AB = CD,
prove that AM = CN
Step 1: Draw a diagram from the previous
statement.
A
M
B
C
N
D
Step 2: Write the Given and Prove statements from
the diagram.
Given: M is the midpoint of AB
N is the midpoint of CD
AB  CD
Prove: AM  CN
Step 3: Write the two-column proof.
Statements
M is the midpoint of AB
Reasons
1. N is the midpoint of CD
AB  CD
1.
Given
2.
AM = MB; CN = ND
2.
Def. of Midpt.
3.
AM+ MB= AB
CN + ND = CD
3.
Seg. Add.
Post.
4.
AM + MB = CN + ND
4.
Subst. Prop.
5.
AM = CN
5.
Subt. Prop.