Two-Column Proof

How to prove the given statement by writing two-column proof: definition, 1 example, and its solution.

Definition

Definition

Two-column proof is a way
to prove a statement.

To write a two-column proof:

Make a two-column form like this.
Name the left column Statement.
And name the right column Reason.

Start from the [Given] statement(s).

Then derive the [Prove] statement
using logic (known theorems, laws, etc.).

[Prove] is derived by logic.
So two-column proof is an example of
deductive reasoning.

Example

Example

Solution

Make a two-column form like this.

Name the left column Statement.
And name the right column Reason.

Start from the first Given statement.

Write [m∠1 + m∠2 = 90]
in the left column.

And write [Given]
in the right column.

Write the next Given statement.

Write [m∠2 + m∠3 = 90]
in the left column.

And write [Given]
in the right column.

The right side of [m∠1 + m∠2 = 90]
and the right side of [m∠2 + m∠3 = 90]
are both 90.

So you can substitute 90 in m∠1 + m∠2 = 90
to m∠2 + m∠3.

Then write [m∠1 + m∠2 = m∠2 + m∠3]
in the left column.

And write [Substitution]
in the right column.

You can also write [Transitive property]
in the right column.

Transitive property:
if a = c and b = c,
then a = c.

Now you might want to cancel m∠2 on both sides.

In order to do that,
first write [m∠2 = m∠2]
in the left column.

And write [Reflexive property]
in the right column.

Reflecxive property:
a = a.

Then subtract [m∠2 = m∠2]
from [m∠1 + m∠2 = m∠2 + m∠3].

Then [m∠1 = m∠3].

And write [Subtraction]
in the right column.

m∠1 = m∠3

Then, by the definition of cougruent angles,
∠1 ≅ ∠3.

From the Given statements,
you found the Prove statement [m∠1 = m∠3].

So close the two-column form
by drawing the bottom line.

So this is the two-column proof of the example.