I am looking about similar triangles and I always see the word corresponding but I always forget to include it in the definition. I am trying to think of an example I can use to show the importance of the word corresponding.

So here is the definition:

Ex:

“If the measures of the corresponding sides of two triangles are proportional then the triangles are similar.”

So if the word corresponding is not in the definition:

“If the measures of the sides of two triangles are proportional then the triangles are similar.”

but what is so important about the word corresponding? Any ideas?

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1 Answer

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The word “corresponding” is not such a big deal when it comes to deciding if two triangles are similar.

The concept of similarity allows reflection (in addition to scaling, translation, and rotation) as a legal operation to map one object into another. In other words, if object AAA is similar to object BB, then AA is also similar to the mirror image of BB.

In the case of two triangles, given that you have matched up one pair of sides, there are only two possible ways to match up (correspond) the remaining sides. If either of these two ways of correspondence yields proportionality, then you can declare the triangles to be similar.

The issue of correspondence is more pertinent with objects having more than three sides.

so it wouldn’t matter in isosceles triangle? If i say triangle ABC is similar to triangle DEF and AB and BC are the congruent sides and DE and EF are the congruent sides, but AB corresponds to DE and BC corresponds to EF then I can just switch the sides? so I would write: ABEF=DEBC\frac{AB}{EF}=\frac{DE}{BC}, the ratio is the same but the sides are not corresponding?

– kero

2 days ago

For similarity, it’s not so important because for other reasons having only three sides there’s no room for error. But if you are dealing quadralaterals you could have a rectangle 3x4x3x4 vs. a kite/diamond 3x3x4x4. And for proving congruenence… a side of 5 a side of 3 and a thirty degree angle between them is entirely different that a side of 5, a side of 3, and then a 30 degree angle between so side of 3 and the third unknown side.

– fleablood

2 days ago

@kero In your example it is true that (1) ABCABC is similar to DEFDEF, but also (2) ABCABC is similar to FEDFED. By matching corresponding sides, from (1) we can deduce ABDE=BCEF\frac{AB}{DE}=\frac{BC}{EF}, and from (2) we can deduce ABEF=BCDE\frac{AB}{EF}=\frac{BC}{DE}. Once you’ve establish the similarity of two triangles, the proportionality ratios that you can deduce still need to refer to corresponding sides. (What you wrote has the RHS of (2) flipped, and is not true.)

– grand_chat

2 days ago

@grand_chat I see, I’m beginning to understand. So would it be okay to define similar triangles without the word corresponding or is there an exception to the case.

– kero

2 days ago

@kero In the case of triangles, your proposed modification to the definition is acceptable, because if you’ve found that the measures of the sides are proportional, then you’re implicitly asserting a correspondence between the sides, which means you’ve satisfied the original definition of similarity. In the case of objects with more than three sides, your modified definition leads to ambiguity; since more than one correspondence is possible, you need to state which sides correspond (and also check that the corresponding angles are equal).

– grand_chat

2 days ago