The Congruence of Triangles

I have to apologize yet again for my delay in coming up with this post. There's been so many things to do, like work, work and coming up with even more excuses :-)
Anyway, let's not lose any more time and start.

So, in my last post, I had outlined the basics of congruence. Which means things being exactly the same.
Let's look at the usability of this concept. When you realize the importance of something, it's much more fun learning about it.

In many cases, we hear things like 'She looks SO MUCH like you!' or 'I have the EXACT same dress'. While this may be the truth, it may be an exaggeration as well. Looking like someone is not the same as being just like them- Anna and Melanie both might have blond hair, but Anna if Anna has green eyes and Melanie has grey eyes, the concept of congruence, of exactness, is immediately ruled out.
An exact same dress- if Anna has a brown shirt in a small size and Melanie has the same thing in a larger size, say medium, the dresses aren't exactly the same.

These may be small examples, but they are significant in learning about this concept that at many times proves frustratingly elusive.
But without this concept, we wouldn't have cola cans all being the same size, we might end up with a random size that takes all the fun out. We wouldn't have the impressive precision that mass produced products have- your favourite brand of shampoo, conditioner, chocolate, and so on.

This link talks about more examples- read it to get a further understanding.

All right, so I'm hoping you're starting to get the drift.
Congruency is exactness, fine, but how do you prove it?
A subjective way congruency can be expressed is saying something like 'I like the series 'Arrested Development' as much as I like 'Breaking Bad' '. As much as. In this case, equally.
Unfortunately, that's too arbitrary and people around you might disagree. So, to establish objectivity, we have a set of rules by which you can prove to everyone that two triangles (sorry, not series! not yet, at least!) are the same.

1)SSS: This stands for Side Side Side. Namely, if EACH side of one triangle is EQUAL to the CORRESPONDING side of the OTHER triangle, the triangles are said to be congruent.

Look at the triangles above. There's something missing. The naming of the triangle points which is VERY IMPORTANT TO ESTABLISH CONGRUENCY.

Better, I hope.
Now.
AB=DE
BC=EF
AC=DF


Yay! Thus ABC  DEF. Notice the difference from '='. It is merely the sign that represents mathematical congruence.

2)ASA: Angle Side Angle
I'm going to ask you to apply some intuition to this one. Imagine a triangle with  one of its sides AB. Imagine JUST the side. Assume it is of length 1 cm.

Now imagine two angles shooting out of its ends.
Extend the lines till they meet. Assume angles 30 degrees and 60 degrees.

OK, we have a triangle now!
Now do the same thing for line DE, again assuming DE= 1 cm etc. etc.. You land up with DEF. Try doing it yourself for practice.
You know ABC  DEF because we arrived at both the same way!

The remaining conditions will be discussed in the next post. Any questions, feel free to ask in comments or email :-)