

A two-column geometric proof consists of a list of statements,
and the reasons that we know those statements are true. The
statements are listed in a column on the left, and the reasons
for which the statements can be made are listed in the right
column. Every step of the proof (that is, every conclusion that
is made) is a row in the two-column proof, and the reason is
next to it.
You should always number your statements and their
corresponding reasons to prevent confusion and make your
proofs more organized.

A theorem is a mathematical statement that can
be proved. It is usually and “If-then” statement.
Theorem 1- If two angles are right angles, then they
are congruent.
Theorem 2-If two angles are straight angles, then they
are congruent.

This is an example of the format of a twocolumn proof:
Statement
Reason
______________________________
1.
2.
3.
1.
2.
3.
When asked to write a two-column proof, you
will be given a diagram and a number of statements to
assist you in your proof. These will be listed as givens. And
you should list them in your proof under statements.
Then, in the reason column, you can specify, “Given”.





Because of the definition of congruent angles, if 2 angles
have the same measure, they are congruent.
Because of the definition of congruent segments, if 2
segments have the same measure, they are congruent.
Because of the definition of a right angle, if and angle has
the measure of 90, it is a right angle.
Because of the definition of a
straight angle, if an angle has
the measure of 180, then it is a
straight angle.
All of the above definitions
can be used as reasons for
statements in your proofs.
Using the theorem 1 and the format of a proof, we
can now create a proof proving two right angles
Let’s see an example problem.

Sample Problem #1
Given:
Prove:
1.
2.
3.
A is a right
B is a right
A
B
A
B
C
D
A is a right
B is a right
A
B
1. Given
2. Given
3. If two
s are
right
s, then
they are .
Now let’s see a proof involving theorem 2.

Sample Problem #2
1. Diagram as shown 1. Given
2.
EFG is a straight 2. Assumed from diagram
Given: Diagram as shown
EFG
HFJ
3.
HFJ is a straight
3. Same as 2
4.
EFG
4. If 2
HFJ
s are straight
then they are
s,
.
As you can see, sometimes the given will be simply in the form of the diagram, since it is always
safe to assume that an angle is a straight angle when it looks like one in the diagram. Also,
when a reason is repeated twice, you can shorten the amount of writing you have to do by
stating that the reason is the same as a previous step, as shown in step 3 of the above proof.

Sample Problem #3
Given:
Prove:
A
D
1.
A = 90
1. Given
2.
ACB = 40
2. Given
A = 90
3.
ACD = 50
3. Given
ADB = 40
4.
A is a right
4. If an
ADC = 90
ADC is a right
ADC
A
then it is a right
.
5. Addition
6. Same as 4
7. If two
s are right
then they are .
BDC = 50
ADC
A
5.
B 6.
7.
C
's measure is 90
s,

Pg. 26-27 #3-5,7
#3.
Statements
1.
A is a right
2.
B is a right
3.
A
B
Reasons
1. Given
2. Given
3. If 2
s are right

s then they are
.
#4
Statements
1. CDE = 110
2. FGH = 110
3. CDE
FGH
Reasons
1. Given
2. Given
3. If 2 s have the same measure then they are .

#5
Statements
1. JK = 2.5 cm
2. NO = 2.5 cm
3. JK NO
Reasons
1. Given
2. Given
3. If 2 segments have the same measure then they
are .

#7
Statements
1. 1 = 20
2. 2 = 40
3. 3 = 30
4. XYZ = 90
5. XYZ is a right angle

Reasons
1. Given
2. Given
3. Given
4. Addition
5. If an ‘s measure is 90,
then it is a right .
"Math a Formula." Math A. 1991. Oswego City School Districts. 27
May 2008 <http://regentsprep.org/Regents/math/forsheet/Fo
rmula.htm>.
Rhoad, Richard, George Milauskas, and Robert Whipple.
Geometry for Enjoyment and Challenge. New Edition ed.
Evanston, Illinois: McDougal Littel, 1991. 23-27.