# 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.