Two-Column Proofs

Two-column proofs (also known as formal proofs) are set up in a two-value table, one being "Statement" and the other being "Reason". To prove a simple problem using this method, set up a table like the following:

Be sure to leave room for values to go in both columns. In geometry, the first row is the 'given' of the problem. This is the information that is given about a certain problem without using a picture. The last row should be the conclusion of what you are trying to prove.

Example of a Two-Column Proof

Now, suppose a problem tells you to solve ${\displaystyle x+1=2}$ for ${\displaystyle x}$, showing all steps made to get to the answer. A proof shows how this is done:

Statement	Reason
${\displaystyle x+1=2}$	Given
${\displaystyle x=1}$	Property of subtraction

We use "Given" as the first reason, because it is "given" to us in the problem.

Written Proof

Written proofs (also known as informal proofs, paragraph proofs, or 'plans for proof') are written in paragraph form. Other than this formatting difference, they are similar to two-column proofs.

Sometimes it is helpful to start with a written proof, before formalizing the proof in two-column form. If you're having trouble putting your proof into two column form, try "talking it out" in a written proof first.

Flowchart Proof

A flowchart proof or more simply a flow proof is a graphical representation of a two-column proof. Each set of statement and reasons are recorded in a box and then arrows are drawn from one step to another. This method shows how different ideas come together to formulate the proof.

Exercises

Answers to each exercise can be found separately in the Appendix.

1.

Given:

• ${\displaystyle r||s}$ r is parallel to s
• ${\displaystyle \angle 1=60^{\circ }}$ Angle 1 = 60 degrees

Prove: Find the measures of the other seven angles in the accompanying figure (above).

2.

Given:

• ${\displaystyle \angle 2\cong \angle 3}$ Angles 2 and 3 are congruent

Prove: Lines r and s parallel

3.

Given:

• ${\displaystyle \angle 1=\angle 2=90^{\circ }}$ Angles 1 and 2 are both 90 degrees

Prove: Lines a and b in the figure are parallel.

4.

Given:

• ${\displaystyle {\overline {GH}}||{\overrightarrow {DK}}}$ Line GH is parallel to ray DK
• ${\displaystyle \angle 6=75^{\circ }}$ Angle 6 = 75 degrees.
• ${\displaystyle \angle 2=30^{\circ }}$ Angle 2 = 30 degrees.

Prove: Find the measure of each numbered angle in the figure above.

A Classic Example: Proving that the Square Root of 2 is Irrational

1. Assume that ${\displaystyle {\sqrt {2}}}$ is a rational number, meaning that there exists an integer ${\displaystyle a}$ and an integer ${\displaystyle b}$ in general such that ${\displaystyle a/b={\sqrt {2}}}$.
2. Then ${\displaystyle {\sqrt {2}}}$ can be written as a simplified fraction ${\displaystyle a/b}$ such that ${\displaystyle a}$ and ${\displaystyle b}$ are coprime integers, that is, their greatest common divisor being 1.
3. It follows that ${\displaystyle a^{2}/b^{2}=2}$ and ${\displaystyle a^{2}=2b^{2}}$.   ( ${\displaystyle (a/b)^{n}=a^{n}/b^{n}}$  )
4. Therefore ${\displaystyle a^{2}}$ is even because it is equal to ${\displaystyle 2b^{2}}$. (${\displaystyle 2b^{2}}$ is necessarily even because it is 2 times another whole number and even numbers are multiples of 2.)
5. It follows that a must be even (as squares of odd integers are never even).
6. Because a is even, there exists an integer k that fulfills: ${\displaystyle a=2k}$.
7. Substituting ${\displaystyle 2k}$ from step 6 for a in the second equation of step 3: ${\displaystyle 2b^{2}=(2k)^{2}}$ is equivalent to ${\displaystyle 2b^{2}=4k^{2}}$, which is equivalent to ${\displaystyle b^{2}=2k^{2}}$.
8. Because ${\displaystyle 2k^{2}}$ is divisible by two and therefore even, and because ${\displaystyle 2k^{2}=b^{2}}$, it follows that ${\displaystyle b^{2}}$ is also even which means that b is even.
9. By steps 5 and 8 a and b are both even, which contradicts that ${\displaystyle a/b}$ is irreducible as stated in step 2.

Q.E.D.

Because there is a contradiction, the assumption (1) that ${\displaystyle {\sqrt {2}}}$ is a rational number must be false. By the law of excluded middle, that is, that a proposition can only be true or false, the opposite is proven: ${\displaystyle {\sqrt {2}}}$ is irrational.