Junior Cert axioms and theorems

Squirrel

In Junior Cert I remember ini our book there were theorems which had to be learned off and axioms, which could be taken as true.

Now, one of the theorems is along the lines of if two of the angles of a triangle are the same, then prove that the two sides opposite the angles are the same. And one of the axioms stated that if two two sides are different then the angles opposite are different too, but proportionally so. In my mind this proves the theorem as if one side is longer and the corresponding angle is bigger, then surely if the two sides are the same the two angles are the same (just taken the answer from the theorem but can work in reverse).

Have I missed something? I asked this when I was in Junior Cert but was just told that it had to be learn, stating the axiom wouldn't do.

Fremen

This looks like a good place to use a mathematical concept known as the "contrapositive"
If A implies B, then you know for certain that (Not implies (Not A)
but you don't necessarily get B implies A

Setting A as "The sides are different" and B as "the angles are different", then as you said, the axiom states A implies B.

Then using the contrapositive,
the angles are not different implies the sides are not different, i.e.
angles the same implies sides the same

So yes, it looks like you can derive one result from the other using only logic. It's been a long while since I did the junior cert, so I don't really remember much of the course, and I may be missing something relevant to the course here.
As an aside, I wouldn't recommend using that in the JC: better just to do it the way they expect and not "rock the boat" too much

Squirrel

That's what I was told when I did my junior cert, it was 4 years ago and that's one thing that I remembered when helping my sister with her teorems.

emmet14

how did any of yous remember them theroms? there are 10 triangle ones with two deductions! and you have to learn circle ones aswell! is there any easy way of remembering them?

LeixlipRed

Emmet, the key to "remembering" theorems is not to remember them at all. If you understand what's actually happening (ie, what you need to prove and the method to do it) then you don't need to learn them off line by line.

Fremen

Exactly. Even for the leaving cert, I never really learned things, I just learned how the proofs worked, so I could work them out as I went along. There's usually only one or two "clever bits" in a proof, and the rest is just grinding algebra or whatever.

Sadly, it doesn't work quite so well in college...