Tag: proof by contradiction

Book 1 – Proposition 7

Proposition 7

Given two straight lines constructed on a straight line [from its extremities] and meeting in a point, there cannot be constructed on the same line [from its extremities], and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it.

Desmos.com link to graphs

The strategy for this proof is to perform a proof by contradiction. We will assume that two other straight lines can be constructed in this fashion and will reach a contradiction.

Suppose two straight lines AC and CB are constructed on the straight line AB and meeting at the same point C as shown.

img0107a

Basically, we have started with a line segment AB and added a point C to complete a triangle. Proposition states that we cannot add another point D on the same side of AB such that AC is equal to AD and BC is equal to BD. Let’s add a point D on the same side of AB as C is.

img0107b

This will be a proof by contradiction. Assume that AC is equal to AD and BC is equal to BD.

Since AC is equal to AD, the angle ACD is equal to the angle ADC by Proposition 5.

Since angle DCB is smaller than angle ACD (because it is contained in that angle), we know that angle ADC is greater than angle DCB.

The following show that angle CDB is greater than angle DCB:

  • Angle CDB is greater than angle CDA (because angle CDA is contained in angle CDB).
  • Angle CDA is equal to angle ACD as stated previously.
  • Angle ACD is greater than angle DCB (because angle DCB is contained in angle ACD).

[CDB > CDA = ACD > DCB]

Now we shift our attention to triangle BCD. Since BC is equal to BD, Proposition 5 tells us that angle CDB is equal to angle DCB.

But we just showed that if our assumptions were true that angle CDB is greater than angle DCB, which is a contradiction. Therefore, there cannot be constructed on the same line [from its extremities], and on the same side of it, two other straight lines meeting in another point and equal to the former two respectively, namely each to that which has the same extremity with it.

Q.E.D.

That wraps up one week’s worth of these propositions. I hope you are finding it rewarding – George

Book 1 – Proposition 6

Proposition 6

If in a triangle two angles be equal to one another, the sides which subtend the equal angles will also be equal to one another.

(Note: If you’d like to click through the images in this proof, they are available in this graph on Desmos.com.)

In Proposition 5 we started with two equal sides and proved that the angles that were opposite of those sides were equal. In this proposition we will begin with two equal angles and prove that the two sides that are opposite those angles are equal. We will begin with the triangle ABC, and let angle ABC be equal to angle ACB.

img0106aThis proof is an example of a proof by contradiction. We will assume that sides AB and AC are not equal to one another, and as we proceed we will run into a contradiction that will tell us that our initial assumption is false and that sides AB and AC must be equal to one another.

Assume that sides AB and AC are not equal to one another. In that case one side must be longer than the other. Suppose that side AB is longer than side AC. From the longer side AB let DB be cut off equal to AC. (Proposition 3)
Join the points C and D as shown.

img0106b

We will now be comparing the triangles DCB and ACB.
We know that DB is equal to AC, and the two triangles have the side BC in common. Also, the angle DBC is equal to the angle ACB as those are the two equal angles from triangle ABC.

Since the two triangles have two pairs of equal sides with an equal angle between them, Proposition 4 tells us that triangle DCB is equal to triangle ACB. However triangle DCB is clearly less than triangle ACB, so we have reached a contradiction. Our assumption that AB is not equal to AC is absurd, and therefore the two sides AB and AC must be equal to each other.

Q.E.D.

Proof by contradiction is an often used form for proving propositions or theorems in mathematics. The U.S. court system uses this principle in its trials – we assume that the defendant in innocent and then search for evidence that contradicts this assumption. A statistical hypothesis test uses the same format – assuming that the null hypothesis is true prior to searching for evidence that it is false.

Note: I have decided to stop previewing propositions. I will cover one proposition per post – George