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.
This 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.
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