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#1 2007-05-16 00:27:56

Identity
Member
Registered: 2007-04-18
Posts: 934

Congruency and Similarity

I reckon this is a pretty ambiguous topic... all those definitions (SSS, SAS, ASA, RHS, PPP, PAP, APA, AA) could be interchanged for one another given sufficient information on the triangle.

Well basically check out triangle 1. Definition SAS states 'two sides plus the INCLUDED angle'... But if you think about, you can conclude, can you not, that both these triangles are congruent?

Likewise, triangle 2: ASA states 'two angles plus the INCLUDED side'... yet, these triangle look similar, even if they do not follow that rule to the word?

For diagram3, the book states blatantly that the triangles are NOT CONGRUENT. However, shouldn't that answer really be: It cannot be proved whether or not these triangles are congruent?

When finding the correct ratios in similar triangles, how do you know which side to divide by which? Say in ABC, AB = 4, AC = 8, BC = 6 and in EFG, EF = 6, EG = 12, FG = 9. Should you always do biggest in ABC over biggest in EFG, smallest in ABC over smallest in EFG, and the remaining side over the remaining side?

Also, one last thing for this post... ~ means similar to, ~ (with line through it) means not similar to. Can the (with line through it) thing also be applied to congruency? What is the symbol for not congruent

thanks for help

Last edited by Identity (2007-05-16 00:41:08)

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#2 2007-05-16 02:11:18

mathsyperson
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Registered: 2005-06-22
Posts: 4,900

Re: Congruency and Similarity

I was in a picture-making mood. big_smile

For question 1, the angle does indeed need to be the included one, because if it's not then there are two possibilities for what the triangle could look like, as shown in the picture. (The one on the right with the red lines shows how to construct the other two, if you're interested)

For 2, any two angles are enough to show that triangles are similar, because if you have two identical angles then the third must be identical as well. However, to prove that they are congruent, the side you have needs to be the same on both triangles. It doesn't necessarily have to be the included side though. Your example has the side in the same places in both triangles, so they would be congruent, but just any old ASA isn't enough, as I've shown in my picture.

3's a bit tricky. The only way that those two triangles could be congruent is if they were isoceles, and if you assume that the diagram tells you about any sides and angles that match then that means that they can't be congruent. If you assume nothing about the unmarked sides then you're right though, it is possible for them to be congruent. A better statement would be "these triangles are not congruent in general."

You've got the method right for finding ratios of similar triangles. If two triangles are similar, then the biggest side on one will correspond to the biggest side on the other, and so will the smallest on each. If you already know that the triangles are similar, then you can just check one pair of sides to find the ratio. But if you don't, then you need to check all 3 pairs to verify that the ratios match.

I don't know what the symbol for not congruent would be though. Sorry.


Why did the vector cross the road?
It wanted to be normal.

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#3 2007-05-16 02:29:37

Identity
Member
Registered: 2007-04-18
Posts: 934

Re: Congruency and Similarity

Thanks heaps mathsy! Even though this is a revision unit there were just a few things i had to clarify

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