Right, so we were discussing conditions to prove congruence in the last post and covered two. Namely the SSS (Side Side Side) and the ASA (Angle Side Angle).
I'm going to repeat what I said before- that you need to apply intuition to many of these methods. Establishing congruence can be tricky- but if you look closely and apply both the conditions you're learning and a certain amount of intuition it is guaranteed that you can figure it out!
All right, now we discuss the three remaining methods.
3) SAS (Side Angle Side):
I like to call this the sister condition of the ASA. They are switched but similar.
To arrive at this condition and to understand it, first draw a side like we did in the ASA.
Assume AB=1 cm. (Fig. 1)
Please not it is not to scale in this blog- if I neglected to mention that earlier I'm sorry. I suggest that you draw it to scale when you do it yourself, at least when you're still figuring out the conditions. Seeing is believing, they say!
Now, draw angle 30 degrees shooting out of A, and extend AC till AC=0.5 cm. You can change the numbers, but follow the steps. (Fig. 2)
Finally, draw a line to connect C and B. (Fig. 3)
The value of CB is not wholly important in this case, so I am not including the value.
If done right, it should look like the figures below.
Now do the same for DEF.
You should land up with:
Observe DEF and ABC are exactly the same. Hence ABC≅DEF
4) AAS: The Angle Angle Side
This condition is perhaps the most complex and you can understand this in a variety of ways. Let me show you my technique and I hope it helps!
Now, I want you to follow the following steps. Put away your pens and paper or keep them, but just try to imagine.
So we looked at the ASA. Let us try to use the ASA to understand the AAS.
We have a side AB= 1 cm.
Shoot an angle BAC= 30 degrees from point A.
Now we're left with line AC, but we really don't know what to do with it. How much should we extend it or retract it?
But we are given another piece of important information- the angle ACB.
I hope you're following this. This can be confusing, but it's learnt best slowly, steadily with lots of imagination. Put it this work now, and you can handle confusing sums later with a lot of confidence!
All right, getting back-
You may remember the property- sum of angles of a triangle is ALWAYS equal to 180 degrees.
When you have two angles, you can find out the third. In our case, we can find out angle CBA.
When we have angle CBA and angle BAC, along with side AB, what does that mean/
It means we're back to our good old ASA.
I'll leave the diagrams to you. Let me know in case of doubts!
5) Ah, finally. RHS. This is a special one- for right angled triangles ONLY!
This is the funnest one to prove, to me at least.
I assume you're familiar with Pythagoras's theorem:
where c= hypotenuse
a,b= sides of the right triangle
If you've forgotten, check this out.
So the RHS just states that if the hypotenuse of two right triangles are equal, and one of the other two sides are also correspondingly equal.
Look at the diagram. CB=EF (hypotenuse of triangle ABC= that of triangle DEF)
Also AC=DF
They are right triangles so they already have angle BAC= angle EDF
So, we have proved congruency. ABC≅DEF
Okay, so we're done for this session. Hope you've understood the concepts. Feel free to comment or e-mail :-)
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