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Example of two-column proof vs. paragraph proof

Here is an example comparing a proof written in two-column form or written as text. And, I will also show you MY exact thought process when I was thinking about this. I have not done these type of problems in recent years, so I do not have the proof memorized.

The idea is to show that two-column proof is NOT the only kind of proof there is, nor is it necessarily the 'best'. The idea of proving is to communicate clearly in a convincing way your argument. Sometimes that might be easier to do in just plain prose.

PROBLEM: Prove that if the two diagonals in a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Better draw a picture first of all. It's a quadrilateral with diagonals. We're supposed to prove that it is a parallelogram. I will try to draw a picture that doesn't look exactly like a parallelogram; in other words a picture that is not exact.

(Why? Because often, when looking at a picture that IS drawn exactly, we say, "Well I SEE that it's a parallelogram. No need proving it." So instead I want to draw a quadrilateral that doesn't at first sight look like a parallelogram.)

So what you have is a quadrilateral with two diagonals that bisect each other. Meaning that the intersection point is a midpoint for both of the diagonals.

Well right there it sounds like some line segments will have equal lengths. And, two lines crossing always form two pairs of vertical angles... So I will have some same angles and some same line segments. Sounds like I can easily prove that there are two congruent triangles and other two congruent triangles.

But how can one get from that to proving that the lines forming the quadrilateral are parallel?

It must be the corresponding angles stuff that will work there. I will have angles with same measure, so that makes that the lines must be parallel.

Okay, the proof is ready in my mind now. Just have to write it so others can understand.


Please look at the picture. Since the diagonals are bisecting each other, the line segments marked with one little line are equal, and similarly the line segments marked with double little lines. The two angles marked with dark blue line are equal, being vertical angles. It follows from SAS congruence theorem that the two yellow triangles are congruent.

Since they are congruent, angles A and A' have the same measure. And, angles A' and A'' are the same because they are vertical angles. So since A and A' are the same, and A' and A'' are the same, it follows that angles A and A'' are the same.

But this is equivalent to the two lines that form the top and bottom of the quadrilateral being parallel.

An identical argument using the two white triangles instead of the two yellow ones proves that the two sides of the quadrilateral are parallel.

So the quadrilateral is a parallelogram.


Argument Reason why
1. The two lines marked with one brown little line are congruent. 1. The two diagonals bisect (given).
2. The two lines marked with two brown little lines are congruent. 2. The two diagonals bisect (given).
3. The two angles marked with blue lines are congruent. 3. They are vertical angles.
4. The two yellow triangles are congruent. 4. SAS theorem and 1, 2, and 3.
5. The angles A and A' are congruent. 5. The two yellow triangles are congruent.
6. The angles A' and A'' are congruent. 6. They are vertical angles.
7. The angles A and A'' are congruent. 7. 5 and 6 together.
8. The lines that form bottom and top of the quadrilateral are parallel. 8. 7 and the theorem that says that corresponding angles being the same is equivalent to lines being parallel.
9. The lines that form the two sides of the quadrilateral are parallel. 9. Repeat steps 1-8 using the two white triangles.
10. The quadrilateral is a parallelogram. 10. 8 and 9 together.

See also

What is a mathematical proof?

Proof that square root of 2 is an irrational number

An example proof of a property of logarithms

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